Laura Vinué

Yolanda Sáenz

Beatriz Rojo-Bezares

Inés Olarte

Esther Undabeitia

Sergio Somalo

Myriam Zarazaga

Carmen Torres
email: carmen.torres@unirioja.es

Genetic environment of genes and characterisation of integrons in Escherichia coli isolates of blood origin in a Spanish hospital

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A c c e p t e d M a n u s c r i p t ABSTRACT The prevalence and characterisation of integrons and the genetic environment of sulphonamide resistance genes were studied in 135 Escherichia coli isolates recovered from blood cultures in a Spanish hospital during 2007. Class 1 and 2 integrons were identified in 54 isolates (intI1, 52 isolates; intI2, 1 isolate; and intI1 + intI2, 1 isolate). Of the 53 intI1-positive isolates, 36 (67.9%) contained the classic class 1 integron including the qacE1-sul1 region, and 11 different gene cassette arrangements were demonstrated in 33 of these isolates. Seventeen intI1-positive isolates lacked the qacE1-sul1 region, and 8 gene cassette arrangements were demonstrated in 12 of these isolates. Seventy-one isolates showed a sulphonamideresistant phenotype, 63 of which contained sul genes. The sul1 gene was associated with intI1 in 36 of 42 sul1-positive isolates, and the sul3 gene was associated with non-classic class 1 integrons in 5 of 7 sul3-positive isolates. Finally, sul2 was found associated with strA-strB genes in 32 of 35 sul2-positive isolates, identifying 11 genetic structures, 1 of them presenting the IS150 element disrupting the strB gene; this structure was included in GenBank with accession no. FJ705354. Almost onehalf of the E. coli isolates from blood cultures contained integrons and sul genes.

Moreover, sul genes were detected in different structures, one of them new, and could be important determinants in antibiotic resistance dissemination.

A c c e p t e d M a n u s c r i p t

Introduction

Integrons are genetic structures able to capture, excise and express genes, frequently included in mobile elements such as plasmids that allow their dissemination among bacteria. Class 1, 2 and 3 integrons have been implicated in the dissemination of antimicrobial resistance [START_REF] Partridge | Gene cassettes and cassette arrays in mobile resistance integrons[END_REF]. Classic class 1 integrons include in their 5'-conserved segment (5'-CS) an integrase gene (intI1) and in their 3'conserved segment (3'-CS) the qacE1 + sul1 genes encoding resistance to quaternary ammonium compounds (partially deleted) and sulphonamides, respectively [START_REF] Partridge | Gene cassettes and cassette arrays in mobile resistance integrons[END_REF]. However, non-classic class 1 integrons lacking the 3'-CS region have also been reported [START_REF] Liu | Characterization of a novel macrolide efflux gene, mef(B), found linked to sul3 in porcine Escherichia coli[END_REF][START_REF] Soufi | Prevalence and diversity of integrons and associated resistance genes in Escherichia coli isolates from poultry meat in Tunisia[END_REF][START_REF] Chuanchuen | Characterization of class 1 integrons with unusual 3' conserved region from Salmonella enterica isolates[END_REF][START_REF] Sunde | Genetic linkage between class 1 integrons with the dfrA12-orfF-aadA2 cassette array and sul3 in Escherichia coli[END_REF][START_REF] Vinué | Prevalence and diversity of integrons and associated resistance genes in faecal Escherichia coli isolates of healthy humans in Spain[END_REF][START_REF] Antunes | Dissemination of sul3-containing elements linked to class 1 integrons with unusual 3' conserved sequence region among Salmonella isolates[END_REF][START_REF] Cocchi | Distribution and characterization of integrons in Escherichia coli strains of animal and human origin[END_REF][START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF].

Sulphonamides were the first group of systemic antimicrobial agents introduced into medicine and have been extensively used since then [START_REF] Sköld | Sulfonamide resistance: mechanisms and trends[END_REF]. Three acquired genes (sul1, sul2 and sul3) implicated in sulphonamide resistance have been described in Escherichia coli. The sul1 gene is typically associated with class 1 integrons [START_REF] Partridge | Gene cassettes and cassette arrays in mobile resistance integrons[END_REF], sul2 has been found adjacent to streptomycin resistance genes [START_REF] Bean | Plasmids imparting sulfonamide resistance in Escherichia coli: implications for persistence[END_REF], and the sul3 gene has been recently described linked to 3'-CS-lacking class 1 integrons [START_REF] Liu | Characterization of a novel macrolide efflux gene, mef(B), found linked to sul3 in porcine Escherichia coli[END_REF][START_REF] Soufi | Prevalence and diversity of integrons and associated resistance genes in Escherichia coli isolates from poultry meat in Tunisia[END_REF][START_REF] Chuanchuen | Characterization of class 1 integrons with unusual 3' conserved region from Salmonella enterica isolates[END_REF][START_REF] Sunde | Genetic linkage between class 1 integrons with the dfrA12-orfF-aadA2 cassette array and sul3 in Escherichia coli[END_REF][START_REF] Antunes | Dissemination of sul3-containing elements linked to class 1 integrons with unusual 3' conserved sequence region among Salmonella isolates[END_REF]. This study presents the prevalence of sulphonamide resistance genes, characterisation of the genetic environment of these genes, and the prevalence and types of integrons in clinical E. coli isolates.

A c c e p t e d M a n u s c r i p t

Material and methods

Escherichia coli isolates and antimicrobial susceptibility testing

All E. coli isolates (n = 135) recovered from blood cultures of different patients during a 1-year period (2007) in the Hospital San Pedro (Logroño, Spain) were included in this study. Susceptibility testing to 16 antimicrobial agents was performed for all isolates by the disk diffusion method [START_REF]Performance standards for antimicrobial susceptibility testing; approved standard. Seventeenth informational supplement[END_REF]. The double-disk synergy test with cefotaxime and ceftazidime in proximity to amoxicillin/clavulanic acid was used for the detection of extended-spectrum -lactamases (ESBLs).

Study of integrons

Presence of the intI1, intI2 and intI3 genes was studied by polymerase chain reaction (PCR) in all isolates [START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF]. The 3'-CS of class 1 integrons (qacE1 + sul1 genes) and the variable region of class 1 and 2 integrons were characterised by PCR and sequencing [START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF].

Detection of sulphonamide resistance genes and their genetic environment

The presence of sul1, sul2 and sul3 genes, implicated in sulphonamide resistance, was studied by PCR in all sulphonamide-resistant (SUL R ) E. coli isolates [START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF]. The genetic environment of the sul1 gene was determined when characterisation of integrons was performed. The presence of strA-strB genes (conferring resistance to streptomycin) and its possible association with the sul2 gene was determined by a primer-walking PCR strategy with subsequent sequencing in all sul2-positive E. coli isolates [START_REF] Bean | Plasmids imparting sulfonamide resistance in Escherichia coli: implications for persistence[END_REF]. The qacH-IS440 region, associated with sul3 and with non-classical A c c e p t e d M a n u s c r i p t class 1 integrons, was analysed in all sul3-positive E. coli isolates by a primerwalking PCR strategy on the basis of the known integron sequences included in GenBank (accession nos. EF051037 and EF113389, among others). In addition, the presence of the putative macrolide efflux gene, mef(B), was investigated in the sul3positive isolates [START_REF] Liu | Characterization of a novel macrolide efflux gene, mef(B), found linked to sul3 in porcine Escherichia coli[END_REF].

Detection of other antimicrobial resistance genes and determination of phylogenetic groups

The presence of genes associated with resistance to ampicillin (bla TEM , bla SHV , bla CTX- M andbla OXA-1 ), tetracycline [tet(A)-tet(E), tet(G) and tet(M)], streptomycin (aadA),

gentamicin [aac(3)-I, aac(3)-II, aac(3)-III and aac(3)-IV], kanamycin [aph(3')-Ia and aph(3')-IIa] and chloramphenicol (cmlA and floR) was also analysed by PCR in all

SUL R E. coli isolates [START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF]. Chloramphenicol acetyltransferase activity was studied as previously described [START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF].

Identification of four major phylogenetic groups (A, B1, B2 and D) was carried out by multiplex PCR [START_REF] Vinué | Prevalence and diversity of integrons and associated resistance genes in faecal Escherichia coli isolates of healthy humans in Spain[END_REF]. Positive and negative controls from the collection of the University of La Rioja (Logroño, Spain) were included in all PCRs.

Results and discussion

Prevalence of antimicrobial resistance

The percentages of antimicrobial resistance detected among the E. coli isolates were as follows: ampicillin, 54.8%; amoxicillin/clavulanic acid, 7.4%; cefoxitin, 1.5%;

Characterisation of integrons among the isolates

Fifty-four E. coli isolates (40%) contained integrons. The intI1 gene was detected in 52 isolates (38.5%), intI2 in 1 isolate (0.7%) and both the intI1 and intI2 gene in an additional E. coli isolate (0.7%). No isolate presented the intI3 gene. It is interesting to note that integrons were only detected among SUL R isolates and not among sulphonamide-susceptible isolates. Class 1 integrons were the most frequent type of integrons detected in this study (39.3%), similar to previous reports [START_REF] Soufi | Prevalence and diversity of integrons and associated resistance genes in Escherichia coli isolates from poultry meat in Tunisia[END_REF][START_REF] Vinué | Prevalence and diversity of integrons and associated resistance genes in faecal Escherichia coli isolates of healthy humans in Spain[END_REF][START_REF] Cocchi | Distribution and characterization of integrons in Escherichia coli strains of animal and human origin[END_REF]. Table 1 shows the gene cassette arrangements of integrons detected in this study.

Of the 53 intI1-positive E. coli isolates, 36 (67.9%) harboured class 1 integrons with the classic structure based on the presence of 5'-CS and 3'-CS, and 1 of these isolates contained two class 1 integrons. Eleven different gene cassette arrangements were identified in 33 of the 36 intI1-positive isolates as follows (number of isolates): dfrA12 + orfF + aadA2 (1); aadB + aadA1 + cmlA6 (1); aadB + aadA1 + ISUnCu1 (1); dfrA17 + aadA5 [START_REF] Sköld | Sulfonamide resistance: mechanisms and trends[END_REF]; dfrA1 + aadA1 (5); dfrA15 + aadA1 (1); dfrA17 + aadA5 and aadA (1); dfrA17 (2); dfrA7 (1); dfrA5 (1); and aadA [START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF]. All PCRs behind the intI1 gene were negative in the remaining three intI1-positive isolates (Table 1). Two isolates carrying dfrA1 + aadA1 and dfrA17 + aadA5 gene cassettes, respectively, presented the intI1 gene interrupted at the 3'-end by the insertion of the IS26 element, as previously reported (i.e. GenBank accession nos. AF205943 or AJ870926).

In addition, 17 (32.1%) of the 53 intI1-positive E. coli isolates lacked the 3'-CS of classic class 1 integrons. Eight different gene cassette arrangements were detected in 12 of these 17 integrons as follows (number of isolates): dfrA5 + ereA2 (2); dfrA17 + aadA5 (1); dfrA1 + aadA1 (1); bla OXA-1 + aadA1 (1); dfrA1 (2); dfrA5 (2); dfrA17 (1); and aadA (2) (Table 1). The remaining five isolates presented long structures associated with the sul3 gene (discussed below in Section 3.3). The existence of non-classic class 1 integrons lacking the 3'-CS region has been previously reported in E. coli isolates recovered from different origins [START_REF] Soufi | Prevalence and diversity of integrons and associated resistance genes in Escherichia coli isolates from poultry meat in Tunisia[END_REF][START_REF] Sunde | Genetic linkage between class 1 integrons with the dfrA12-orfF-aadA2 cassette array and sul3 in Escherichia coli[END_REF][START_REF] Vinué | Prevalence and diversity of integrons and associated resistance genes in faecal Escherichia coli isolates of healthy humans in Spain[END_REF][START_REF] Antunes | Dissemination of sul3-containing elements linked to class 1 integrons with unusual 3' conserved sequence region among Salmonella isolates[END_REF][START_REF] Cocchi | Distribution and characterization of integrons in Escherichia coli strains of animal and human origin[END_REF][START_REF] Sáenz | Mechanisms of resistance in multiple-antibiotic-resistant Escherichia coli strains of human, animal, and food origins[END_REF].

Most of the gene cassettes found within the variable region of these class 1 integrons corresponded to different alleles of dfrA (dfrA1, dfrA5, dfrA7, dfrA12 and dfrA17), aadA (aadA1 and aadA2) and aadB genes, conferring resistance to trimethoprim, streptomycin and gentamicin/tobramycin/kanamycin, respectively. The dfrA17 + aadA5 and aadA gene cassette arrangements were the most frequently detected in our study (in 12 of 53 class 1 integrons in each case), followed by dfrA1 + aadA1 arrangement (in 6 isolates). It is interesting to note the presence of two unusual variable regions carrying both aadB and aadA1 gene cassettes. One of them presented aadB + aadA1 followed by an insertion element ISUnCu1, part of the A c c e p t e d M a n u s c r i p t IS1111 group, previously described in an E. coli isolate from sick animals in Germany [START_REF] Kadlec | Analysis and distribution of class 1 and class 2 integrons and associated gene cassettes among Escherichia coli isolates from swine, horses, cats and dogs collected in the BfT-GermVet monitoring study[END_REF], and the other showed aadB + aadA1 followed by a chloramphenicol resistance gene cassette, cmlA6, previously detected in a commensal E. coli isolate from Bolivia [START_REF] Pallecchi | Population structure and resistance genes in antibiotic-resistant bacteria from a remote community with minimal antibiotic exposure[END_REF]. ISUnCu1 is located after the first nucleotide of the 2L core site of attC of the aadA1 gene cassette [START_REF] Partridge | Gene cassettes and cassette arrays in mobile resistance integrons[END_REF], with the transposase of ISUnCu1 in opposite orientation to aadB + aadA1 cassette genes. Disruption of the aadA1 gene cassette by insertion of an IS1111-attC element will probably negatively affect its mobilisation.

The two intI2-positive E. coli isolates presented the same gene cassette arrangement in their variable region (dfrA1 + sat + aadA1), and one of them also harboured a class 1 integron including the aadA gene cassette (Table 1). It is known that dfrA1 + sat + aadA1 is the more frequently detected cassette arrangement among class 2 integrons [START_REF] Vinué | Prevalence and diversity of integrons and associated resistance genes in faecal Escherichia coli isolates of healthy humans in Spain[END_REF][START_REF] Cocchi | Distribution and characterization of integrons in Escherichia coli strains of animal and human origin[END_REF][START_REF] Kadlec | Analysis and distribution of class 1 and class 2 integrons and associated gene cassettes among Escherichia coli isolates from swine, horses, cats and dogs collected in the BfT-GermVet monitoring study[END_REF].

Genetic environment of sul genes

A total of 71 E. coli isolates showed a SUL R phenotype and it was possible to identify at least one sul gene in 63 (88.7%) of them. The following combinations of sul genes were detected (number of isolates): sul1 (22); sul1 + sul2 (17); sul1 + sul3 (3); sul2 (17); sul2 + sul3 (1); and sul3 (3). In addition, as previously indicated, 52 of the 71 SUL R isolates contained class 1 integrons, 1 isolate a class 2 integron and another isolate carried both class 1 and 2 integrons.

The genetic environment of the sul1 gene was associated with class 1 integrons in 36 (85.7%) of the 42 sul1-positive isolates, as previously indicated (Table 1).

A c c e p t e d M a n u s c r i p t

The sul3 gene was associated with a non-classic class 1 integron (lacking the 3'-CS) in five of the seven sul3-positive isolates, and the following structures were detected: dfrA12 + orfF + aadA2 + cmlA1 + aadA1 + qacH + IS440 + sul3 + orf1 + mef(B) + IS26 (2 isolates); estX + psp + aadA2 + cmlA1 + aadA1 + qacH + IS440 + sul3 + orf1 + mef(B) + IS26 (2 isolates); and dfrA12 + orfF + aadA2 + cmlA1 + aadA1 + qacH + IS440 + IS10 + sul3 + orf1 + mef(B) + IS26 (1 isolate). This last structure has been included in the GenBank database with the accession no. FJ587511 (Fig. 1), although it was previously described in a clinical human E. coli isolate [START_REF] Sunde | Genetic linkage between class 1 integrons with the dfrA12-orfF-aadA2 cassette array and sul3 in Escherichia coli[END_REF]. The sul3 gene was identified in two additional isolates, but non-related to class 1 integrons. In one of them, the sul3 gene was linked to qacH and to the macrolide efflux gene mef(B), identifying the following structure: qacH + IS440 + sul3 + orf1 + mef(B) + IS26 (Fig. 1). The genetic context of the other sul3-positive isolate could not be identified because all the performed PCRs were negative. The qacE1-sul1 fragment replaced by qacH-IS440-sul3 sequence in the sul3-positive strains has been previously reported [START_REF] Soufi | Prevalence and diversity of integrons and associated resistance genes in Escherichia coli isolates from poultry meat in Tunisia[END_REF][START_REF] Chuanchuen | Characterization of class 1 integrons with unusual 3' conserved region from Salmonella enterica isolates[END_REF][START_REF] Sunde | Genetic linkage between class 1 integrons with the dfrA12-orfF-aadA2 cassette array and sul3 in Escherichia coli[END_REF], even associated with the mef(B) gene [START_REF] Liu | Characterization of a novel macrolide efflux gene, mef(B), found linked to sul3 in porcine Escherichia coli[END_REF].

The genetic environment of the sul2 gene was identified in 32 (91.4%) of the 35 sul2positive E. coli isolates, detecting eleven different genetic structures (Fig. 2). The most frequent structure contained the repC + sul2 + strA-strB + IS26 + tnpB structure. It is interesting to note that four E. coli isolates presented the strA-strB genes but they were truncated: In this sense, the dfrA14 gene cassette is integrated into the strA gene in three isolates, and IS150 disrupts the strB gene in one isolate.

The whole genetic structure that includes IS150 truncating strB has not been previously described and has been registered in the GenBank database with the accession no. FJ705354 (Fig. 2). A frequent association of the sul2 gene with strA-A c c e p t e d M a n u s c r i p t strB genes was demonstrated in all cases, similar to previous observations [START_REF] Bean | Plasmids imparting sulfonamide resistance in Escherichia coli: implications for persistence[END_REF].

Disruption of the strA gene by insertion of a functional trimethoprim gene cassette, dfrA14, has been previously reported in human E. coli isolates in Nigeria [START_REF] Partridge | Gene cassettes and cassette arrays in mobile resistance integrons[END_REF]. In this study we found another disrupted gene, strB, but in this case by the insertion of IS150, to our knowledge the first time this has been reported. The 1443-bp E. coli transposable element IS150, a member of the widespread IS3 family, contains two consecutive out-of-phase open reading frames (ORFs), orfA and orfB, that partially overlap. These ORFs encode three proteins, InsA, InsB and InsAB, which are jointly encoded by both ORFs by means of programmed translational frame-shifting [START_REF] Haas | Escherichia coli insertion sequence IS150: transposition via circular and linear intermediates[END_REF].

Characterisation of other antimicrobial resistance genes and phylogenetic groups

The presence of other resistance genes was also studied in the 71 SUL R isolates. In total, 60 ampicillin-resistant isolates were detected, 34 of which harboured a bla TEM lactamase gene and 1 additional isolate harboured the bla OXA-1 gene. Seven of these isolates also showed resistance to cefotaxime or ceftazidime, and all of them were ESBL-producers. The gene encoding CTX-M-14a -lactamase was detected in six of these isolates and the gene encoding SHV-12 enzyme in the remaining ESBLpositive isolate, all of them containing integrons. The 71 SUL R E. coli isolates were typed to determine their phylogenetic groups.

Phylogroup B2 was detected in 26 (36.6%) of the isolates and phylogroups D, A and B1 in 29.6%, 26.7% and 7%, respectively.

In summary, 40% of the E. coli isolates from blood cultures contained integrons, mainly class 1. In addition, 52.6% showed a SUL R phenotype, most of these isolates harbouring sul genes (88.7%); sul1 and sul2 were the most frequently genes detected in SUL R isolates, one-third of them harbouring more than one sul gene. In most of the cases, sul1 and sul3 genes have been found associated with classic and non-classic class 1 integrons, respectively. On the other hand, most of sul2-positive isolates contained this gene associated with strA-strB genes, showing a variety of genetic structures, one of them not previously described. Thus, sul genes, which confer resistance to old antimicrobial agents, are very frequently detected in blood E. coli isolates, in most cases in close association with other resistance genes for nonrelated antimicrobial agents, and these determinants could be important in the selection and dissemination of antimicrobial-resistant bacteria.

A

  c c e p t e d M a n u s c r i p t performed to detect the gene cassettes (aadA, dfr, cmlA, ereA, estX, psp and tet)

  Among 40 tetracycline-resistant isolates (36 containing integrons), 15 isolates carried tet(A), 14 carried tet(B) and 2 carried tet(D) genes. The remaining nine tetracyclineresistant isolates did not contain tetracycline resistance genes. In addition, 20 gentamicin-resistant isolates were detected [7 with the aac(3)-II gene and 2 with the aac(3)-IV gene] and 18 kanamycin-resistant isolates were identified [12 with the A c c e p t e d M a n u s c r i p t aph(3')-Ia gene]. Finally 14 chloramphenicol-resistant isolates were identified, 6 of which harboured the cmlA gene, 1 presented the floR gene, and chloramphenicol acetyltransferase activity was demonstrated in other 3 isolates.

Fig. 1 .

 1 Fig. 1. Genetic structures associated with the sul3 gene (6 isolates).

Fig. 2 .

 2 Fig. 2. Genetic structures associated with the sul2 gene (32 isolates).

  , 28.9%; gentamicin, 14.9%; kanamycin, 14.1%; streptomycin, 47.4%; tetracycline, 35.5%; sulphonamides, 52.6%; trimethoprim/sulfamethoxazole, 35.5%; chloramphenicol, 11.1%; and imipenem, 0%. Resistance to one or more antimicrobial

	agents was observed in 124 isolates (91.9%). Eighty-seven isolates (64.4%)
	presented a multiresistant phenotype (resistance to three or more antimicrobial
	M families) and 25 isolates (18.5%) exhibited resistance to at least five different families of antimicrobial agents. Moreover, 10 (7.4%) of the 135 isolates exhibited a positive a n u s c r i p t ESBL test.
	A c c e p t e d
	cefotaxime, 8.1%; ceftazidime, 3%; aztreonam, 5.2%; nalidixic acid, 40.1%;

ciprofloxacin

Funding

This work was partially supported by Project SAF2006-14207-C02 from the Ministry of Education and Science of Spain and FEDER. LV has received a fellowship from

Competing interests

None declared.

Ethical approval

Not applicable.

A c c e p t e d M a n u s c r i p t

N/A, not applicable.

a intI1 was truncated by the insertion sequence IS26 in two isolates, one with dfrA17 + aadA5 and another with dfrA1 +aadA1 as gene cassettes.

b Not sequenced; determined by PCR mapping and could be aadA1 or aadA2.

c Long structure associated with sul3 and described in Fig. 1.


Eitan Altman ⋆⋆

Arnaud Legout
email: arnaud.legout@inria.fr

Yuedong Xu
email: yuedong.xu@gmail.com

Prof Altman Eitan

Network Non-neutrality Debate: An Economic Analysis ⋆

Keywords: Network Non-neutrality, Side Payment, Nash Equilibrium, Bargaining

This paper studies economic utilities and quality of service (QoS) in a two-sided non-neutral market where Internet service providers (ISPs) charge content providers (CPs) for the content delivery. We propose new models that involve a CP, an ISP, end users and advertisers. The CP may have either a subscription revenue model (charging end users) or an advertisement revenue model (charging advertisers). We formulate the interactions between the ISP and the CP as a noncooperative game for the former and an optimization problem for the latter. Our analysis shows that the revenue model of the CP plays a significant role in a non-neutral Internet. With the subscription model, both the ISP and the CP receive better (or worse) utilities as well as QoS in the presence of the side payment at the same time. With the advertisement model, the side payment impedes the CP from investing on its contents.

Introduction

Network neutrality, one of the foundations of Internet, is commonly admitted that ISPs must not discriminate traffic in order to favor specific content providers [START_REF] Hahn | The economics of net neutrality[END_REF]. However, the principle of network neutrality has been challenged recently. The main reason is that new broadband applications cause huge amount of traffic without generating direct revenues for ISPs. Hence, ISPs want to get additional revenues from CPs that are not directly connected to them. For instance, a residential ISP might want to charge Youtube in order to give a premium quality of service to Youtube traffic. This kind of monetary flows, which violates the principle of network neutrality, are called two-sided payment. We use the term side payment to name the money charged by ISPs from CPs exclusively.

On the one hand, the opponents of network neutrality argue that it does not give any incentive for ISPs to invest in the infrastructure. This incentive issue is even more severe in two cases: the one of tier-one ISPs that support a high load, but do not get any revenue from CPs; and the one of 3G wireless networks that need to invest a huge amount of money to purchase spectrum. On the other hand, advocates of network neutrality claim that violating it using side payment will lead to unbalanced revenues among ISPs and CPs, thus a market instability.

Recent work addressed the problem of network neutrality from various perspectives [START_REF] Musacchio | A two-sided market analysis of provider investment incentives with an application to the net-neutrality issue[END_REF][START_REF] Altman | A Study of Non-Neutral Networks with Usage-based Prices[END_REF][START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF][START_REF] Economides | Net neutrality on the internet: A two-sided market analysis[END_REF][START_REF] Saavedra | Bargaining power and the net neutrality debate[END_REF][START_REF] Choi | Net Neutrality and Investment Incentives[END_REF][START_REF] Ma | On Cooperative Settlement Between Content, Transit and Eyeball Internet Service Providers[END_REF][START_REF] Bangera | Impact of Prefix Hijacking on Payments of Providers[END_REF]. Among these work, [START_REF] Musacchio | A two-sided market analysis of provider investment incentives with an application to the net-neutrality issue[END_REF][START_REF] Altman | A Study of Non-Neutral Networks with Usage-based Prices[END_REF][START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF][START_REF] Hermalin | The Economics of Product-Line Restrictions With an Application to the Network Neutrality Debate[END_REF] are the closest to our work. Musacchio et al. [START_REF] Musacchio | A two-sided market analysis of provider investment incentives with an application to the net-neutrality issue[END_REF] compare one-sided and two-sided pricing of ISPs. However, they only investigate an example where the joint investments of CPs and ISPs bring revenue from advertisers to CPs. In [START_REF] Altman | A Study of Non-Neutral Networks with Usage-based Prices[END_REF], the authors show how side payment is harmful for all the parties involved such as ISP and CP. Altman et al. in [START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF] present an interesting bargaining framework to decide how much the ISP should charge the CP. However, their models might give a biased conclusion by overlooking the end users' sensitivity towards the prices of the CP and the ISP. Hermalin et.al in [START_REF] Hermalin | The Economics of Product-Line Restrictions With an Application to the Network Neutrality Debate[END_REF] study the impact of QoS differentiation (removing product-line constraint) in a two sided Internet market. The monopoly ISP is able charge a continuum of CPs based on the qualities of connections chosen by them. The CPs differ in the attractiveness of their content. The authors show that the product-line restriction has an ambiguous overall effect, but is usually harmful. In this work, they differentiate the sources of CPs' income for the first time. Different from [START_REF] Hermalin | The Economics of Product-Line Restrictions With an Application to the Network Neutrality Debate[END_REF], we investigate the QoS provision of the ISP to the CP instead of the QoS differentiation among a continuum of CPs. By using simple two-sided market models, we answer the question when the side payment is beneficial.

In this paper, we unravel the conflicts of the side payment in a more general context. We consider a simplified market composed of one ISP, one CP, some advertisers, and a large number of end users. The ISP charges end-users based on their usage and sets their QoS level according to the price paid. The CP can either have a subscription based or an advertisement based revenue model. For the subscription based revenue model, the CP gets its revenue from the subscription paid by end-users. End-users adapt their demand of contents based on the price paid to the ISP and the CP. For the advertisement based revenue model, the CP gets its revenue from advertisers. End users adapt the demand according to the price paid to the ISP and the CP's investment on its contents.

Our work differs from related work [START_REF] Musacchio | A two-sided market analysis of provider investment incentives with an application to the net-neutrality issue[END_REF][START_REF] Altman | A Study of Non-Neutral Networks with Usage-based Prices[END_REF][START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF][START_REF] Hermalin | The Economics of Product-Line Restrictions With an Application to the Network Neutrality Debate[END_REF] by: i) incorporating the QoS provided by the ISP, ii) studying different revenue models of the CP, and iii) introducing the relative price sensitivity of end users in the subscription model. Especially, in the subscription model, the relative price sensitivity decides whether the side payment is beneficial (or harmful) to the ISP and the CP. Our finding contradicts the previous work (e.g. [START_REF] Altman | A Study of Non-Neutral Networks with Usage-based Prices[END_REF]) that argues that the side payment is harmful for all parties involved. In the advertisement model, the ability of the CP's investment to attract the traffic of end users plays a key role. It determines whether the side payment is profitable for the ISP and the CP. Our main contributions are the following.

-We present new features in the mathematical modeling that include the QoS, the relative price sensitivity of end-users, and the CP's revenue models. -We analytically show that the side payment from the CP to the ISP is beneficial to the ISP and the CP in terms of profits under certain conditions. -We utilize bargaining games based on [START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF] to investigate how the side payment is determined.

The rest of this work is organized as follows. In section 2, we model the economic behaviors of ISP, CP, advertisers and end users. Section 3 and 4 study the impact of the side payment and its bargaining outcomes. Section 5 presents numerical study to validate our claims. Section 6 concludes this paper.

Basic Model

In this section, we first introduce the revenue models of the ISP and the CP. Then, we formulate a game problem and an optimization problem for the selfish ISP and the CP. Finally, we describe the bargaining games in a two-sided market.

Revenue Models

We consider a simplified networking market with four economic entities, namely the advertisers, the CP, the ISP and end users. All the end users can access the contents of the CP only through the network infrastructure provided by the ISP. The ISP collects subscription fees from end users. It sets two market parameters (p s , q) where p s is the non-negative price of per-unit of demand, and q is the QoS measure (e.g. delay, loss or block probability). End users can decide whether to connect to the ISP or not, or how much traffic they will request, depending on the bandwidth price and the QoS. The CP usually has two revenue models, the user subscription and the advertisement from clicks of users. These two models, though sometimes coexisting with each other, are studied separately in this work for clarity. The CP and the ISP interact with each other in a way that depends on the CP's revenue models. In the subscription based model, the CP competes with the ISP by charging users a price p c per-unit of content within a finite time. End users respond to p s , p c and q by setting their demands elastically. Though p c has a different unit as p s , it can be mapped from the price per content into the price per bps (i.e. dividing the price of a content by its size in a finite time). The price p c not only can stand for a financial disutility, but also can represent the combination of this disutility together with a cost per quality. Thus a higher price may be associated with some better quality (this quality would stand for parameters different from the parameter q which we will introduce later). Without loss of generality, p s and p c can be positive or 0. For the advertisement based model, instead of charging users directly, the CP attracts users' clicks on online advertisements. The more traffic demands the end users generate, the higher the CP's revenue.

To better understand network neutrality and non-neutrality, we describe the monetary flows among different components. The arrows in Figure 1 represent the recipients of money. A "neutral network" does not allow an ISP to charge a CP for which it is not a direct provider for sending information to this ISP's users. On the contrary, monetary flow from a CP to an ISP appears when "network neutrality" is violated. The ISP may charge the CP an additional amount of money that we denote by f (D) = p t D where p t is the price of per-unit of demand. We denote by δ ∈ [0, 1] the tax rate of this transferred revenue imposed by the regulator or the government. We present market demand functions for the subscription and the advertisement based revenue models.

Subscription model Let us define the average demand of all users by D that has D(p s , p c , q) = max{0, D 0 -α(p s + ρp c ) + βq},

where D 0 , α, β and ρ are all positive constants. The parameter D 0 reflects the total potential demand of users. The parameters α and β denote the responsiveness of demand to the price and the QoS level of the ISP. The physical meaning of (1) can be interpreted in this way. When the prices of the ISP and the CP increase (resp. decrease), the demand decreases (resp. increases). If the QoS of the ISP is improved, the demand from users increases correspondingly. The parameter ρ represents the relative sensitivity of p c to p s . We deliberately set different sensitivities of end users to the prices of the ISP and the CP because p c and p s refers to different type of disutilities. If ρ = 1, the prices of the ISP and the CP are regarded as having the same effect on D. When ρ > 1, users are more sensitive to the change of p c than p s . The positive prices p s and p c can not be arbitrarily high. They must guarantee a nonnegative demand D.

We denote S CP to be the pricing strategy of the CP that has S CP = {p c : p c ≥ 0}. The utility (or revenue equivalently) of the CP is expressed as

Ucp = (p c -p t )D(p s , p c , q). (2)
Note that the variable D(p s , p c , q) is interchangeable with D all the time. Next, we present the utility of the ISP with QoS consideration. We assume that the pricing strategy of the ISP is defined by S ISP = {(p s , q) : p s ≥ 0; 0 < q ≤ q max }.

To sustain a certain QoS level of users, the ISP has to pay the costs for operating the backbone, the last-mile access, and the upgrade of the network, etc. Let u(D, q) be the amount of bandwidth consumed by users that depends on the demand D and the QoS level q. We assume that u(D, q) is a positive, convex and strictly increasing function in the 2-tuple (D, q). This is reasonable because a larger demand or higher QoS usually requires a larger bandwidth of the ISP. We now present a natural QoS metric as the expected delay1 . The expected delay is computed by the Kleinrock function that corresponds to the delay of M/M/1 queue with FIFO discipline or M/G/1 queue under processor sharing [START_REF] El-Azouzi | Telecommunications Network Equilibrium with Price and Quality-of-Service Characteristics[END_REF]. Similar to [START_REF] El-Azouzi | Telecommunications Network Equilibrium with Price and Quality-of-Service Characteristics[END_REF], instead of using the actual delay, we consider the reciprocal of its square root, q = 1 √ Delay = u(D, q) -D. Thus, the cost C(D, q) can be expressed as C(D, q) = p r u(D, q) = p r D + p r q 2 , where p r is the price of per-unit of bandwidth invested by the ISP. Therefore, the cost of the ISP is denoted by C(D, q) = p r u(D, q). The utility of the ISP is defined as the difference between revenue and cost:

Uisp = (p s -p r )D(p s , p c , q) + (1 -δ)p t D(p s , p c , q) -p r q 2 .
(

) 3
Advertisement model Nowadays, a small proportion of CPs like Rapidshare and IPTV providers get their income from end users. Most of other CPs provide contents for free, but collect revenues from advertisers. The demand from users is transformed into attentions such as clicks or browsing of online advertisements.

To attract more eyeballs, a CP needs to invest money on its contents, incurring a cost c. The investment improves the potential aggregate demand D 0 in return. Let D 0 (c) be a concave and strictly increasing function of the investment cost c.

With abuse of notations we denote the strategy of the CP by S CP = {c : c > 0}.

Hence, the demand to the CP and the ISP is written as

D = D0(c) -αp s + βq. ( 4
)
The utility of the ISP is the same as that in [START_REF] Altman | A Study of Non-Neutral Networks with Usage-based Prices[END_REF]. Next, we describe the economic interaction between advertisers and the CP. There are M advertisers interested in the CP, each of which has a fixed budget B in a given time interval (e.g., daily, weekly or monthly). An advertiser also has a valuation v to declare its maximum willingness to pay for each attention. The valuation v is a random variable in the range [0, v]. Suppose that v is characterized by probability density function (PDF) x(v) and cumulative distribution function (CDF) X(v). We assume that the valuations of all advertisers are i.i.d. Let p a be the price of per attention charged by the CP. We denote by D a (p a ) the demand of attentions from advertisers to the CP. Therefore, D a can be expressed as [START_REF] Liu | Mathematical Modeling of Competition in Sponsored Search Market[END_REF]]

D a = M B • Prob(v ≥ p a )/p a = M B • (1 -X(p a ))/p a . ( 5
)
When the CP increases p a , the advertisers will reduce their purchase of attentions. It is also easy to see that the revenue of advertising, p a •D a , decreases with regard to p a either. However, the attentions that the CP can provide is upper bounded by the demand of users through the ISP. Then, we can rewrite D a as that in [START_REF] Liu | Mathematical Modeling of Competition in Sponsored Search Market[END_REF] by

D a = min{D, M B • (1 -X(p a ))/p a }. (6)
Correspondingly, subtracting investment from revenue, we obtain the utility of the CP by

Ucp = (p a -p t )D a -c. (7)
Lemma 1. The optimal demand D a is a strictly decreasing function of p a if the pdf x(v) is nonzero in (0, v).

The proofs of all lemmas and theorems can be found in [START_REF] Altman | Network Non-neutrality Debate: An Economic Analysis[END_REF]. From (6), we can observe that the optimal price p a * is obtained at

D = M B • (1 -X(p a * ))/p a * .
Here, we denote a function y(•) such that p a * = y(D). According to the demand curve of attentions, y(•) is a decreasing function of D. The utility of the CP is a function of the demand D and the cost c, i.e. U cp = y(D) • D -p t D -c.

Problem Formulation

With the subscription model, the strategy profile of the ISP is to set the 2tuple (p s , q) and that of the CP is to set p c . This is actually a game problem in which the ISP and the CP compete by setting their prices. Since the ISP's QoS is tunable, we call this game "QoS Agile Price competition". With the advertisement model, the strategy of the ISP is still the price paid by end users, while that of the CP is to determine the investment level c. The ISP and the CP maximize their own utilities selfishly, but do not compete with each other. We name this maximization as "Strategic Pricing and Investment".

Definition 1. QoS Agile Price Competition In the subscription model, the CP charges users based on their traffic demands. If the Nash equilibrium N E (1) = {p s * , p c * , q * } exists, it can be expressed as

(G1) Uisp(p s * , p c * , q * ) = max {p s ,q}∈S ISP Uisp(p s , p c * , q), (8)
Ucp(p s * , p c * , q * ) = max

p c ∈S CP Ucp(p s * , p c , q * ). ( 9
)
Definition 2. Strategic Pricing and Investment In the advertisement model, the ISP sets (p s , q) and the CP sets c to optimize their individual utilities. If there exists an equilibrium {p s * , q * , c * }, it can be solved by

(G2) Uisp(p s * , q * , c * ) = max {p s ,q}∈S ISP Uisp(p s , q, c * ), (10)
Ucp(p s * , q * , c * ) = max c∈S CP Ucp(p s * , q * , c). ( 11
)

Bargaining Game

The side payment serves as a fixed parameter in the above two problems. A subsequent and important problem is how large the side payment should be.

When the ISP decides the side payment unilaterally, it might set a very high p t in order to obtain the best utility. However, this leads to a paradox when the ISP sets p s and p t at the meantime. With the subscription based model, if the ISP plays a strategy (p t , p s , q) and the CP plays p c , the noncooperative game leads to a zero demand and hence a zero income. This can be easily verified by taking the derivative of U isp over p t (also see [START_REF] Altman | A Study of Non-Neutral Networks with Usage-based Prices[END_REF]). In other words, the ISP cannot set p t and p s simultaneously in the price competition. Similarly, with the advertisement based model, the ISP meets with the same paradox. There are two possible ways, the Stackelberg game and the bargaining game, to address this problem. Their basic principle is to let the ISP to choose p t and p s asynchronously. In this work, we consider the bargaining game in a market where the ISP and the CP usually have certain marketing powers. Our analysis in this work is close to the one presented in [START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF], but comes up with quite different observations. We here analyze the bargaining games of the side payments that are played at different time sequences. The first one, namely pre-bargaining, models the situation that the bargaining takes place before the problems (G1) or (G2). The second one, defined as post-bargaining, models the occurrence of bargaining after the problems (G1) or (G2). Let γ ∈ [0, 1] be the bargaining power of the ISP over the CP. They negotiate the transfer price p t determined by p t * = arg max p t (U isp ) γ (U cp ) 1-γ . Since the utilities can only be positive, the optimal p t maximizes a virtual utility U

p t * = arg max p t U = arg max p t (1 -γ) log Ucp + γ log Uisp. ( 12
)
We use [START_REF] Altman | Network Non-neutrality Debate: An Economic Analysis[END_REF] to find p t as a function of the strategies of the ISP and the CP.

Price Competition of the Subscription Model

In this section, we first investigate how the relative price sensitivity influences the price competition between the ISP and the CP. We then study the choice of the side payment under the framework of bargaining games.

Properties of Price Competition

This subsection investigates the impact of the side payment on Nash Equilibrium of the noncooperative game G1. Before eliciting the main result, we show some basic properties of the subscription based revenue model.

Lemma 2. The utility of the CP, U cp (p s , p c , q), in (2) is a finite, strictly concave function with regard to (w.r.t.) p c .

Similarly, we draw the following conclusion.

Lemma 3. The utility of the ISP, U isp (p s , p c , q), in (3) is a finite, strictly concave function w.r.t. the 2-tuple (p s , q) if the market parameters satisfy 4αp r > β 2 .

For the QoS Agile Price Competition, we summarize our main results as below.

Lemma 4. When the ISP and the CP set their strategies selfishly,

the Nash equilibrium (p s * , p c * , q * ) is unique; -the QoS level q * at the NE is influenced by the side payment in the ways:

• improved QoS with ρ + δ < 1;

• degraded QoS with ρ + δ > 1;

• unaffected QoS with ρ + δ = 1 if (p s * , p c * , q * ) satisfy p s * > 0, p c * > 0, 0 < q * < q max and 4αp r > β 2 .

When the NE (p s * , p c * , q * ) is not at the boundary, we can yield the following expressions by solving the best response equations.

q * = β(D0 -αp r + αp t (1 -ρ -δ)) 6αp r -β 2 , ( 13
)
p c * = 2p r (D0 -αp r + αp t (1 -ρ -δ)) ρ(6αp r -β 2 ) + p t , (14)
p s * = 2p r (D0 -αp r + αp t (1 -ρ -δ)) 6αp r -β 2 + p r -(1 -δ)p t , (15)
D * = 2p r α(D0 -αp r + αp t (1 -ρ -δ)) 6αp r -β 2 , (16)
For the case that the NE is at the boundary, interested users can find it in the technical report [START_REF] Altman | Network Non-neutrality Debate: An Economic Analysis[END_REF]. Lemma 4 means that that the QoS provision of the ISP is influenced by the side payment. We interpret the results by considering ρ and δ separately. When users are indifferent to the price set by the ISP and that by the CP (i.e., ρ = 1), a positive tax rate δ leads to the degradation of q in the presence of the side payment. Next, we let δ be 0 and investigate the impact of ρ. If users are more sensitive to the price of the ISP (i.e. ρ < 1), the side payment is an incentive of the ISP to improve its QoS. Otherwise, charging side payment leads to an even poorer QoS of the ISP. Therefore, if users are more sensitive to the CP's price, a good strategy of the ISP is to share its revenue with the CP so that the latter sets a lower subscription fee.

Bargaining of the Side Payment

To highlight the bargaining of the side payment, we make the following two simplifications: i) the tax ratio δ is 0, and ii) p t can be positive, zero or negative. We let δ = 0 because it turns out to have the similar effect as ρ. We have shown that a negative p t might benefit both the ISP and the CP in some situations. Hence, we do not require p t to be positive in the bargaining game. When q reaches q max , the QoS is reflected as a fixed parameter in the demand model. To avoid considering the boundary cases of q, we also remove the constraint q ≤ q max . Pre-bargaining: In the pre-bargaining, p t is chosen based on the NE of the ISP and the CP. The equations ( 13)∼ (15) yield the expression of U

U = 4 log D0 -αp r + αp t (1 -ρ) + constant . ( 17
)
The utility U is increasing or decreasing in p t depending on the sign of (1 -ρ).

If ρ < 1, a positive p t improves not only the QoS level of the ISP, but also the utilities of the ISP and the CP. As p t increases, p s decreases and p c increases consequently until p s hits 0. Hence, in the pre-bargaining, p s * = 0. The prices p t * and p c * are computed by

p t * = p r (4αp r + 2D0 -β 2 ) 4αp r + 2ραp r -β 2 . ( 18
)
p c * = 2p r (D0 -αp r + αp t * (1 -ρ)) ρ(6αp r -β 2 ) + p t * . ( 19
)
On the contrary, when ρ > 1, a negative p t benefits both of them. Then, p t * is a negative value such that p c * is 0. When ρ = 1, the QoS and the utilities are unaffected by any p t . Among all these cases, the selection of p t is uninfluenced by the bargaining power γ.

Post-bargaining: For the post-bargaining, the ISP and the CP compete for the subscription of users first, knowing that they will bargain over p t afterwards [START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF]. In brief, we find p t as a function of p s , p c and q first. Then, the ISP and the CP compete with each other by setting the prices. To solve the maximization in [START_REF] Altman | Network Non-neutrality Debate: An Economic Analysis[END_REF], we let dU dp t be 0 and obtain

p t = γp c -(1 -γ)(p s -p r ) + (1 -γ)p r q 2 /D. ( 20
)
Submitting (20) to U isp , we rewrite the ISP's utility by

Uisp = γ (p s + p c -p r )(D0 -α(p s + ρp c ) + βq) -p r q 2 . ( 21
)
The utility of the CP is proportional to that of the ISP, i.e.

dUisp dq = γ(β(p s + p c -p r ) -2p r q), ( 23
)
dUcp dp c = (1 -γ)(D -αρ(p s + p c -p r )). ( 24
)
The best responses of U isp and U cp will not happen at the same time unless ρ = 1 or p s + p c -p r = 0. The condition p s + p c -p r = 0 does not hold because it leads to a zero demand D and zero utilities. When ρ is not 1, only one of ( 22) and ( 24) is 0. Here, we consider the case ρ > 1. The utility U cp reaches its maximum upon D = αρ(p s + p c -p r ), while U isp is still strictly increasing w.r.t. p s . Thus, the ISP increases p s until the demand D is 0, which contradicts the condition of a nonzero D.

If D = α(p s + p c -p r ),
dUcp dp c is negative and dUisp dp s is 0. Then, the CP decreases p c until 0 and the ISP sets p s to achieve its best utility accordingly. By letting (24) be 0, we can find (p s , q) at the Nash equilibrium

q * = β(D0 -αp r ) 4αp r -β 2 and p s * = 2p r (D0 -αp r ) 4αp r -β 2 + p r .
The price of the side payment, p t * , is thus computed by

p t = -(1 -γ) D0 -αp r 2α . (25)
When ρ = 1, p s * and p c * can be arbitrary values in their feasible region that satisfy p s * + p c * = p r + 2p r (D0-αp r ) 4p r α-β 2 . Similar result has been shown in [START_REF] Altman | Nonneutral network and the role of bargaining power in side payments[END_REF]. The analysis of ρ < 1 is omitted here since it can be conducted in the same way.

Price, QoS and Investment settings of the Advertisement Model

This subsection analyzes how the side payment influences the optimal strategies of ISP and CP with the advertisement model. The bargaining games are adopted to determine the amount of the side payment. Compared with subscription based model, the advertisement based model exhibits quite different behaviors.

Properties of the Advertisement Mode

In general, the subscription model is limited to file storage CDNs, newspaper corporations, or big content owners such as movie producers. Most of CPs are not able to provide enough unique contents so that they do not charge users, but make money from online advertisements. In this subsection, we present the general properties of the advertisement model and a couple of case studies.

Lemma 5. For any feasible investment c of the CP, there exists a best strategy of the ISP, (p s , q). When c increases, the price and the QoS (i.e. p s and q) become larger.

In G2, the CP and the ISP do not compete with each other. On one hand, the ISP sets the two-tuple (p s , q) with the observation of c. On the other hand, the CP adjusts c based on (p s , q). The investment of the CP brings more demand of end users, which increases the revenues of not only the ISP, but also the CP. Hence, different from G1, the problem G2 is not a game. Instead of studying the NE, we look into the optimal strategies of the ISP and the CP in G2. Theorem 1. There exists a unique best strategy, namely (p s * , q * , c * ), with the advertisement model if the revenue of the CP, D • y(D), is a concave function w.r.t. D ≥ 0. Lemma 6. The side payment from the CP to the ISP leads to a decreased investment on the contents when the best strategy (p s * , q * , c * ) has p s * > 0, q * > 0 and c * > 0.

Case Study

In this subsection, we aim to find the best strategies of the ISP and the CP when the valuation of advertisers follows a uniform distribution or a normal distribution. Due to the page limit, the case of normal distribution is put in the technical report [START_REF] Altman | Network Non-neutrality Debate: An Economic Analysis[END_REF].

Recall that the potential aggregate demand of users, D 0 (c), is strictly increasing and concave w.r.t c. When the CP invests money on contents, D 0 becomes larger, while its growth rate shrinks. Here, we assume a log function of D 0 (c),

D0(c) = D 0 0 + K log(1 + c), (26)
where the constant K denotes the ability that the CP's investment brings the demand. The nonnegative constant D 0 0 denotes the potential aggregate demand of end users when c is zero (the CP only provides free or basic contents). The utility of the ISP remains unchanged. Uniform Distribution: Suppose v follows a uniform distribution in the range [0, v]. Then, the CDF X(p a ) is expressed as p a v . The optimal price p a is obtained when D = MB p a • (1 -p a v ) in the range [0, v] (see subsection 2.1). Alternatively, there has

p a = M Bv M B + Dv . ( 27
)
The above expressions yield the utility of the CP by

Ucp = M BvD M B + Dv -c -p t D. (28)
Deriving U cp over c, we obtain

dUcp dc = ( (M B) 2 v (M B + Dv) 2 -p t ) • K 1 + c -1. ( 29
)
We let (29) be 0 and get

c = K( (M B) 2 v (M B + Dv) 2 -p t ) -1. ( 30
)
The rule of the ISP to decide (p s , q) is the same as that in the subscription model, except that the aggregate demand is not a constant, but a function of c,

c = exp( D 2p r α (4p r α -β 2 ) -D 0 0 + αp r -(1 -δ)p t α K ) -1. (31)
Note that (30) is strictly decreasing and (31) is strictly increasing w.r.t. D. They constitute a fixed-point equation for the 2-tuple (D * , c * ). In the beginning, we assume that the optimal strategies are not on the boundary. When D approaches infinity, (30) is negative while (31) is positive. When D is zero, if (30) is larger than (31), there exists a unique fixed-point solution. In this fixed point, the ISP and the CP cannot benefit from changing their strategy unilaterally. We can solve c * and D * numerically using a binary search. If (30) is smaller than (31) when D is 0, the best strategy of the CP is exactly c = 0. The physical interpretation is that the increased revenue from advertisers cannot compensate the investment on the contents. Once D * and c * are derived, we can solve p s * and q * subsequently. In this fixed-point equation, p t greatly influences the optimal investment c * . When p t grows, the right sides of (30) and (31) decrease at the mean time. The crossing point of two curves, (30) and (31), shifts toward the direction of smaller c as shown in Lemma 6. Intuitively, the contents of the CP become less when the ISP charges a positive p t . The boundary case of the optimal strategies as well as the bargaining games over p t are analyzed in [START_REF] Altman | Network Non-neutrality Debate: An Economic Analysis[END_REF].

Evaluation

We present some numerical results to reveal how the QoS, prices of the ISP and the CP, as well as their utilities evolve when the price of the side payment changes. The impact of bargaining power on the side payment is also illustrated.

More numerical examples are demonstrated in the technical report [START_REF] Altman | Network Non-neutrality Debate: An Economic Analysis[END_REF]. Subscription Model: We consider a networking market where the demand function is given by D = 200 -10(p s + ρp c ) + 0.5q. The operational cost of per-unit of bandwidth is set to p r = 1. Two situations, ρ = 0.5 and ρ = 1.5, are evaluated. The tax rate δ is set to 0 for simplicity. According to the above analysis, the side payment is beneficial (resp. harmful) to the ISP and the CP if ρ is less (resp. greater) than 1. In figure 2, p t has different impacts on utilities of the ISP and the CP. When ρ > 1, end users are more sensitive to the change of p c than p s . A positive p t leads to the increase of p c , causing a tremendous decrease of demand. Hence, both the ISP and the CP lose revenues w.r.t. a positive p t . Figure 3 further shows that a positive p t yields a better QoS if ρ < 1 and a worse QoS if ρ > 1.

Next, the ISP and the CP bargain with each other to determine p t . We relax the choice of p t so that it can be negative. In the pre-bargaining game, p t is independent of the bargaining power γ. The optimal p t is obtained when p s * decreases to 0 for ρ > 1 and p c * decreases to 0 for ρ < 1. We evaluate p t by changing ρ and α in figure 4. When ρ increases from 0.2 to 2, p t decreases until it becomes negative. A negative p t means that the ISP needs to transfer revenue to the CP instead. When ρ = 1, p t can be an arbitrary value as long as the sum of p s * and p c * is the fixed value computed in section 3.2. Figure 4 also shows that a larger α results in a smaller absolute value of p t . . The total budget of advertisers is set to 1000. We conduct two sets of experiments. The first one is to evaluate the impact of the side payment on the best strategies of the ISP and the CP. The second one is to find the optimal p t in the pre-bargaining game. In figure 5, the CP's investment is a decreasing function of p t . When p t is large enough, c reduces to 0. Figure 6 illustrates the utility of the CP when p t and K change. The CP's utility increases first and then decreases with K = 10 when p t increases. For the cases K = 20 and 30, the increase of p t usually leads to the decrease of revenues. In figure 7, the utility of the ISP with K = 10 and 20 increases first and then decreases when p t grows. These curves present important insights on the interaction between the CP and the ISP. If the contents invested by the CP can bring a large demand, the side payment is not good for both the ISP and the CP. On the contrary, when the efficiency K is small, a smaller investment of the CP might not significantly reduce the demand from users. The ISP is possible to be better by extracting revenues from the CP with the low efficiency K.

Conclusion and Future Work

In this paper, we first under what situations the side payment charged by the ISP is beneficial for the ISP (or the CP). Then, we study how the price of the side payment is determined. Our models take account of three important features, the relative price sensitivity, the CP's revenue models, and the QoS provided by the ISP. With the subscription model, the relative price sensitivity determines whether the ISP should charge the side payment from the CP or not. With the advertisement model, the charge of the side payment depends on the ability of the CP's investment to attract the demand.

Fig. 1 .

 1 Fig. 1. Money flow of a non-neutral network.

  dUisp dp s = γ(Dα(p s + p cp r )),(22)

Fig. 2 . 5 Fig. 3 .

 253 Fig. 2. Subscription Model: Utilities of the ISP and the CP

Fig. 4 .

 4 Fig. 4. Subscription Model: Prebargaining of p t

Fig. 5 .

 5 Fig. 5. Advertisement Model: CP's investment at the equilibrium Advertisement Model: In the advertisement model, we consider the demand function D = K log(1 + c) -10p s + 0.5q. The coefficient K reflects the efficiency of the CP's investment to attract end users. The valuation of each click/browsing follows uniform distribution in the range [0, 10]. The total budget of advertisers is set to 1000. We conduct two sets of experiments. The first one is to evaluate the impact of the side payment on the best strategies of the ISP and the CP. The second one is to find the optimal p t in the pre-bargaining game. In figure5, the CP's investment is a decreasing function of p t . When p t is large enough, c reduces to 0. Figure6illustrates the utility of the CP when p t

Fig. 6 .Fig. 7 .

 67 Fig. 6. Advertisement Model: The ISP's utility at the equilibrium

The QoS metric can be the functions of packet loss rate or expected delay etc.

The work was supported by the INRIA ARC Meneur on Network Neutrality.


David Burguet

Symbolic extensions and continuity properties of the entropy

In this short note we give a new presentation of the entropy theory of symbolic extensions. Then we deduce from the main results of this theory some continuity properties of the entropy regarding the smoothness of the dynamical system. We also prove that generic continuous interval maps have nowhere continuous entropy function.

1 Background

Monotone operator on a complete lattice and some basic dynamical applications

We first recall some elements of set theory. A (partially) ordered set (L, ≤) is a complete lattice if every subset of L has both a supremum and an infimum. A map T of L into itself is an increasing operator if T f ≤ T g for all f ≤ g. We recall now an elementary fixed point theorem due to Tarski and Knaster.

Theorem 1 [START_REF] Tarski | A lattice-theoretical fixpoint theorem and its applications[END_REF] Let L be a complete lattice and let T : L → L be an increasing operator. Then the set of fixed points of T in L is also a complete lattice. In particular it is not empty and there is a least fixed point.

The least fixed point of T can be obtained in a inductive way as the stationary limit of the transfinite sequence (T α 0) α where 0 is the least element of L and T α 0 is T (T α-1 0) for successor ordinals α and T α 0 is the least upper bound of T β 0 over β < α for limit ordinals α. By the increasing property of T any fixed point of T is larger than T α 0 for all ordinals α. Then the transfinite sequence (T α 0) α is obviously stationary after any ordinal β with cardinal larger than the cardinality of L. Let us denote by α(T ) or just α * the least ordinal α such that T α 0 = T α+1 0. This ordinal is called the order of accumulation of T .

We illustrate this formalism with two basic examples. One naive example of increasing operator is given by a nondecreasing map f of the unit interval endowed with the usual order. Then one obtains the least fixed point of f as the stationary limit of the transfinite sequence (f α (0)) α . Clearly the order of accumulation is countable (indeed for any α < α(f ) the interval ]f α (0), f α+1 (0)] is nonempty and thus contains a rational number). Moreover it is not difficult to construct examples of nondecreasing maps with a given countable ordinal as order of accumulation.

We present now another context where this formalism applies. By a dynamical system we mean a continuous map T : X → X on a compact metrizable space X to itself. Recall a point x of X is nonwandering if for every neighborhood U of x there is an integer n ≥ 1 such that T n (U ) ∩ U is nonempty. The set of all nonwandering points is called the nonwandering set of T , and is denoted by Ω(T ). The nonwandering set Ω(T ) of a dynamical system (X, T ) differs in general from the nonwandering set of (Ω(T ), T | Ω(T ) ). The set of forward invariant compact subsets of X ordered with the reverse inclusion ⊃ is clearly a complete lattice. Then the operator T Ω which associates to any forward invariant compact subset K the nonwandering set of T | K is increasing. It is well known that the least fixed point of T Ω is the closure of the set of the recurrent points of (X, T ) (Theorem 5.08 of [START_REF] Nemyckii | Qualitative Theory of Differential Equations[END_REF]). Moreover by an argument of compacity the order of accumulation of T Ω is countable and any countable ordinal can be achieved as the order or accumulation of the operator T Ω associated to some dynamical system [START_REF] Kato | The depth of centers of maps on dendrites[END_REF].

Symbolic extensions

We address the question whether a dynamical system can be encoded with a finite alphabet. More precisely, does a given dynamical system (X, T ) admit a symbolic extension, that is a subshift (Y, S) of a full shift over a finite alphabet, along with a continuous surjection π : Y → X such that π • S = T • π? We are also interested in minimizing the entropy introduced by the code. For example can we choose these extensions to be principal, i.e. such that they preserve the entropy of measures? We focus particularly on smooth dynamical systems. In this case, the answer highly depends on the smoothness of the dynamical system :

• there exist C 1 examples with no symbolic extensions [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF], [START_REF] Asaoka | Hyperbolic set exhibing C 1 -persistent homoclinic tangency for higher dimensions[END_REF], [START_REF] Burguet | Examples of C r interval map with large symbolic extension entropy[END_REF];

• there exists C r examples with r > 1 with no principal symbolic extensions [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF], [START_REF] Burguet | Examples of C r interval map with large symbolic extension entropy[END_REF]. However C r interval maps [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF] with r > 1 and C 2 surface diffeomorphisms [START_REF] Burguet | C 2 surface diffeomorphism have symbolic extensions[END_REF] always admit symbolic extensions (T.Downarowicz and S.Newhouse have conjectured the existence of symbolic extensions for any C r map with r > 1 on a compact manifold (Conjecture 1.1 of [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF]));

• C ∞ dynamical systems always admit principal symbolic extensions [START_REF] Boyle | Residual entropy, conditional entropy and subshift covers[END_REF].

The main tool which allows to validate or invalidate the existence of (principal) symbolic extensions is the powerful Symbolic Extension Entropy Theorem [START_REF] Boyle | The entropy theory of symbolic extension[END_REF], due to M.Boyle and T.Downarowicz, which relates the entropy of symbolic extensions with the emergence of entropy at different small scales. This emergence is captured by the convergence of entropy structures which we present in the next subsection. First, we recall some notations and definitions.

We denote by M(X, T ) the set of Borel invariant probability measures of the dynamical system (X, T ) endowed with the weak star topology and we denote by dist a distance on M(X, T ). Given a symbolic extension π : (Y, S) → (X, T ) we will consider the function

h π ext : M(X, T ) → R + : ∀µ ∈ M(X, T ), h π ext (µ) = sup π * ν=µ h(µ)
where h denotes the usual Kolmogorov-Sinai entropy of (X, T ) and π * : M(Y, S) → M(X, T ) is the map induced by π on measures. Finally the symbolic extension entropy function is the function h sex = inf π h π ext , which is related to the topological entropy of the symbolic extensions (Y, S) of (X, T ) by the following variational principle (Theorem 8.1 of [START_REF] Boyle | The entropy theory of symbolic extension[END_REF]) :

sup µ h sex (µ) = inf S h top (S)

Entropy structure

The entropy structures of a dynamical system (X, T ) are special nondecreasing sequences (h k ) k∈Z + of nonnegative functions which converge pointwise to the Kolmogorov-Sinai entropy function. They recover the usual entropy invariants but also the new ones arising from the entropy theory of symbolic extensions. Two entropy structures (h k ) k and (g k ) k satisfy the following property :

∀γ > 0 ∀k ∈ Z + ∃l ∈ Z + , g l > h k -γ and h l > g k -γ (1)
The above relation ensures that the main invariant of the entropy theory of symbolic extensions whose definition involves an entropy structure, namely the operator T sex introduced in the next section, does not depend on the choice of the entropy structure. Moreover most of the usual appraoches to entropy lead to an entropy structure [START_REF] Downarowicz | Entropy structure[END_REF], so that entropy structures may be considered as a master entropy invariant unifying the previous theories of entropy. For example if (P k ) k is a decreasing sequence of partitions whose diameter goes to zero and with small boundaries, i.e. µ(∂P k ) = 0 for all µ ∈ M(X, T ), then the functions h k = h(., P k ) define an entropy structure.

We refer to [START_REF] Downarowicz | Entropy structure[END_REF] for a complete definition as well as further properties and examples.

Symbolic Extension Entropy Theorem revisited

A real function f on a compact metric space E is said to be upper semicontinous if lim sup y→x f (y) ≤ f (x) for all x ∈ E. When f is a real function defined on E we denote by ⌈f ⌉ the smallest upper semicontinuous function larger than or equal to f if f is bounded from above and ⌈f ⌉ is the constant function equal to +∞ if not.

Using the framework of operators on lattices presented in the previous section we restate the Symbolic Extension Entropy Theorem. Let (X, T ) be a dynamical system. We denote by S(X, T ) the set of nonnegative upper semicontinuous functions defined on M(X, T ) to which we add the function constant equal to +∞. The set S(X, T ) endowed with the usual order is a complete lattice : if F is a family of upper semicontinuous functions then inf f f ∈ F is itself upper semicontinous and then it is the infimum of F in S(X, T ), and sup f ∈F f is clearly the supremum of F in S(X, T ).

Let H = (h k ) k be an entropy strucuture, we define the operator T sex on S(X, T ) as follows :

T sex : S(X, T ) → S(X, T ) f → lim k ⌈f + h -h k ⌉
One easily checks from (1) that T sex does not depend on the choice of the entropy structure (h k ) k . Clearly T sex is an increasing operator. Moreover, the functions hh k being nonnegative, T sex f ≥ f for all f ∈ S(X, T ). By using the affine structure of the set of invariant probability measures M.Boyle and T.Downarowicz proved that the least fixed point of T sex coincides with the infimum of the affine fixed points of T sex . The Symbolic Extension Entropy Theorem can then be restated as follows :

Theorem 2 (Theorem 5.5 of [START_REF] Boyle | The entropy theory of symbolic extension[END_REF]) The affine fixed points of T sex are exactly the functions h π exth, i.e. f is a nonnegative affine upper semicontinuous function on M(X, T ) fixed by T sex if and only if there exists a symbolic extension π such that f = h π exth. Moreover h sexh is the least fixed point of T sex . In [START_REF] Boyle | The entropy theory of symbolic extension[END_REF] the authors considered entropy structures (h k ) k such that h 0 = 0 and h k+1h k is upper semicontinuous for all k ∈ Z + . Then the Symbolic Extension Entropy Theorem is stated in [START_REF] Boyle | The entropy theory of symbolic extension[END_REF] as follows : a nonnegative affine function f is equal to h π exth for some symbolic extension π : (Y, S) → (X, T ) if and only if fh k is upper semicontinuous for all k ∈ Z + . According to Lemma 2.1.6 of [START_REF] Downarowicz | Entropy structure[END_REF] this last condition is equivalent to T sex f = f . Since T sex does not depend on the choice of the entropy structure (h k ) k the above Theorem 2 follows at once.

In this setting the transfinite sequence (u α ) α and the order of accumulation of entropy introduced in [START_REF] Boyle | The entropy theory of symbolic extension[END_REF] are respectively the sequence (T α sex 0) α and the ordinal α(T sex ). By an easy argument of compacity the order of accumulation is a countable ordinal. It was proved in [START_REF] Burguet | Orders of accumulation of entropy[END_REF] that any countable ordinal is realized as the order of accumulation of entropy of a dynamical system.

As we already mentioned in Section 1.2 it is not known whether dynamical systems of intermediate smoothness (C r with 1 < r < +∞) always admit symbolic extensions. When T is a C 1 dynamical system on a compact manifold M of dimension d let us denote by (χ i (x)) i=1,...,d the Lyapunov exponents at a regular point x. The sum of the positive Lyapunov exponents of an invariant measure µ is then given by i χ + i (µ) := i χ + i (x)dµ(x) where we use the notation a + := max(a, 0) for all real numbers a. It is easily checked that i χ + i defines an affine upper semicontinuous function on M(X, T ) [START_REF] Burguet | C 2 surface diffeomorphism have symbolic extensions[END_REF]. We conjecture that Conjecture 1 Let T : M → M be a C r map, with 1 < r, defined on a compact manifold M of dimension d. Then

d i χ + i r-1 is a fixed point of T sex . In particular it admits a symbolic extension π such that h π ext -h = d i χ + i r-1 .
It was proved for interval maps [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF] and C 2 surface diffeomorphisms [START_REF] Burguet | C 2 surface diffeomorphism have symbolic extensions[END_REF]. We have already noted that C ∞ dynamical systems always admit principal symbolic extensions, i.e. symbolic extensions π : (Y, S) → (X, T ) with h π ext = h. Therefore the above conjecture also holds true in this case.

Continuity of the entropy

This last section deals with continuity properties of the entropy function. As for symbolic extensions the picture highly depends on the smoothness of the dynamical system :

• there exist (continuous) dynamical systems such that the entropy function h is nowhere continuous;

• there exist C 1 examples such that the restriction of h to some nonempty compact subset of the set of Borel invariant probability measures is nowhere continuous;

• there exist C r examples with r > 1 where h is not upper semicontinuous [21][26]. However for C r interval maps and C 2 surface diffeomorphisms h is a difference of upper semicontinuous functions : in particular the set of continuity points of h is a dense G δ set;

• h is upper semicontinuous for C ∞ dynamical systems [START_REF] Newhouse | Continuity properties of entropy[END_REF].

As the last item, concerning the special case of C ∞ maps, is a famous result of S.Newhouse [START_REF] Newhouse | Continuity properties of entropy[END_REF] we will not revisit it. We explain in the following how the three first items follow from previous works on entropy structure and symbolic extensions.

Continuous Examples

The first item follows from the complete characterization of possible entropy functions of (continuous) dynamical systems due to T.Downarowicz and J.Serafin. Recall a Choquet simplex K is a compact convex set satisfying the following property. For any x ∈ K there exists a unique Borel probability measure µ x on K supported by the extremal set such that f (x) = f (y)dµ x (y) for all affine continuous functions f : K → R. Then a function g : K → R is said to be harmonic if g(x) = g(y)dµ x (y). The harmonic extension of a Borel map f : K → R supported on the extremal set of K is the harmonic function x → K f (y)dµ x (y).

Theorem 3 [START_REF] Downarowicz | Possible entropy functions[END_REF] Let K be a Choquet simplex and f : K → R + be a nonnegative function. Then there exists a dynamical system (X, T ) and a affine homeomorphism a : M(X, T ) → K such that f • a is the entropy function of (X, T ) if and only if f is a pointwise limit of a nondecreasing sequence of nonnegative harmonic upper semicontinuous functions.

We only need to construct a Choquet simplex K and an increasing sequence of nonnegative harmonic upper semicontinuous functions f k : K → R whose limit f is nowhere continuous. Let K be the Choquet simplex M([0, 1]) of the Borel probability measures on [0, 1] and let f k be the harmonic extension of the characteristic function of the set of Dirac measures {δ a1 , ..., δ a k }, where (a n ) n∈Z + is a countable enumeration of the rational numbers of [0, 1]. The functions f k are just given by f k (µ) = k l=1 µ({a k }) for all µ ∈ M([0, 1]). In particular they are upper semicontinuous. Moreover by the theorem of monotone convergence the limit f is just the harmonic extension of the characteristic function of the Dirac measures at rational numbers. Clearly f is nowhere continuous because any probability measure on [0, 1] is a weak limit of atomic measures supported by either rational or irrational numbers.

In fact this phenomenon is generic among continuous interval maps endowed with the topology of uniform convergence : Theorem 4 The entropy function of a generic continuous interval map is nowhere continuous.

We conjecture the above theorem also holds in higher dimensions. We first begin with the following adaptation of Theorem 3.8 of [START_REF] Abdenur | Nonuniform hyperbolicity for C 1generic diffeomorphisms[END_REF].

Proposition 1 There exists a residual set S of continuous interval maps such that the periodic measures of any f ∈ S are dense in the set of ergodic measures.

The proof is based on the following elementary perturbative result : Lemma 1 Let µ be an ergodic measure of an interval map f . Then for every C 0 -neighborhood U of f and every neighborhood V of µ in the set of probability measures with the weak star topology, there exists g ∈ U admitting a periodic point p such that the associated periodic measure γ p belongs to V.

Proof : Clearly one can assume µ is not periodic for f . By the ergodic theorem

1 n n-1 k=0 δ f k x n
converges to µ for µ-almost all x. Fix such a point x and let n be an integer such that1 n n-1 k=0 δ f k x ∈ V. One can also assume x to be recurrent and then choose f n x close enough to x so that x is a periodic point of period n of g ∈ U with 1 n n-1 k=0 δ g k x ∈ V after a small C 0 -perturbation of f . Given a positive integer n, let K n denote the set of continuous interval maps f satisfying the following property :

Each periodic orbit O of period n is transversal, i.e. on any neighborhood U of p ∈ O there exist x 1 < p < x 2 in U such that (f n (x 1 ) -x 1 ) (f n (x 2 ) -x 2 ) < 0. Lemma 2 K = n≥1 K n is a residual set in the set of continuous interval maps.
Proof : It is enough to prove that K n is itself residual for each n. For any pair of distinct rational numbers (a, b) let

F n (a, b) := {f ∈ C([0, 1]), ∀x ∈ [a, b], f n (x) ≥ x and ∃x 0 ∈ [a, b], f n (x 0 ) = x 0 }∪{f ∈ C([0, 1]), ∀x ∈ [a, b], f n (x) ≤ x and ∃x 0 ∈ [a, b], f n (x 0 ) = x 0 }. The sets F n (a, b
) are closed and have empty interior because the set of Kupka-Smale maps

1 of C 1 interval maps is C 0 -dense in K [25]. Therefore K n ⊃ C([0, 1]) \ a<b F n (a, b) is a residual set.
We notice that transversal periodic orbits are persistent in the following sense. Assume O is a transversal periodic orbit of period n, then for any neighborhood U of p ∈ O there exists a neighborhood U of f such that any g ∈ U admits a periodic orbit O ′ of period n with O ′ ∩ U = ∅ (this last periodic orbit O ′ may be not isolated among g-periodic points of period n, thus O ′ is not transversal in general).

Let X be a subset of C([0, 1]) endowed with the induced topology and let K(Y ) be the set of compact subsets of a compact metric space Y endowed with the Hausdorff metric. A map φ : X → K(Y ) is said to be lower semicontinuous (resp. upper semicontinuous) if for any f in X and for any open set V in Y with V ∩ φ(f ) = ∅ (resp. V ⊃ φ(f )) there exists a neighborhood U of f in X such that V ∩ φ(g) = ∅ (resp. V ⊃ φ(g)) for every g ∈ U . The set of continuity points of such (upper or lower semicontinuous) functions is a residual set of X [START_REF] Kuratowski | Topology II[END_REF].

Proof of Proposition 1 : Let X g be the closure of the periodic measures of g ∈ K in the set M([0, 1]) of probability measures of [0, 1] endowed with the weak star topology. Transversal periodic points are persistent so that the map g → X g from K to K(M([0, 1])) is lower semicontinuous. Since K is a residual set it follows that the set S of continuity points of g → X g is also a residual set of C([0, 1]). Let f ∈ S be such a point of continuity and let µ be an ergodic f -invariant measure. By the previous lemma µ is approximated by a periodic measure of a continuous map arbitrarily close to f . After another small C 0 perturbation one can assume this last map in K. By continuity of g → X g at f the measure µ belongs then to the closure of the periodic measures of f . Let f : [0, 1] → [0, 1] be a continuous interval map. We recall that a family J = (J 1 , ..., J p ) of closed disjoint intervals is called a p-horseshoe of f if J k ⊂ f (J i ) for all i, k. Let us denote H J := n∈Z f n J and ({1, ..., p} Z + , σ) the one sided shift with p symbols. The map π : (H J , f ) → ({1, ..., p} Z + , σ) defined by (π(x)) k = q iff f k (x) ∈ J q is a semi-conjugacy. In particular h top (f ) ≥ log p. In fact horseshoes characterize entropy of continuous interval maps [START_REF] Misiurewicz | Horseshoes for mappings of the interval[END_REF] : if f is a continuous interval map with entropy h top (f ) > 0 then for all h < h top (f ) there exists a p-horseshoe for f N with entropy log p/N > h.

We show now as in [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF] that horseshoes accumulate on periodic orbits for generic maps. For a given continuous interval map f : [0, 1] → [0, 1] and a given positive integer n we say that a set S of periodic points of f satisfies the property (P n ) when :

For each periodic point p in S there exists a horseshoe with entropy log 3 such that any invariant measure supported on this horseshoe lies in the open

1 n -neighborhood of γ p .
We observe that if S satisfies the property (P n ) then the closure of S satisfies the property (P n-1 ). Let R n denote the set of continuous interval maps f such that the set of all periodic points satisfies the above property (P n ).

Proposition 2 R = n≥1 R n is residual in the set of continuous interval maps.

Proof : For any interval map f ∈ K and for any integer m we denote by P er m (f ) the set of periodic points of period less than m. This set is a closed subset of the interval with empty interior because all the periodic points of f are transversal. Therefore, if for all positive integers l and m we let P er l,m (f ) = {p ∈ P er m (f ), s.t. [p, p + 1 l [∩P er m (f ) = {p} or ]p -1 l , p] ∩ P er m (f ) = {p}}, the union l∈Z + P er l,m (f ) is a dense subset of P er m (f ). Let R n (l, m) be the subset of K such that the property (P n ) holds for P er l,m (f ). Then we have the following inclusions

R n ⊂ l,m R n (l, m) ⊂ R n-1 so that R = l,m,n R n (l, m). Therefore we only need to prove R n (l, m) is an open dense set in K because K is residual by Lemma 2.
Let us first prove that this set is open in K. Since horseshoes are persistent under small C 0 perturbations it is enough to show that f → P er l,m (f ) is upper semicontinuous from K to K([0, 1]). For any f ∈ K there exists a finite subset {p 1 < p 2 < ... < p N } of P er m (f ) containing P er l,m (f ) such that either |p ip i+1 | < 1 l or |p ip i+1 | ≥ 1 l and ]p i , p i+1 [∩P er m (f ) = ∅ for i = 0, ..., N with p 0 = -1 and p N +1 = 2. In the second case we remark that p i and p i+1 belong to P er l,m (f ) ∪ {p 0 , p N +1 }. Let I = i, |p ip i+1 | < 1 l and let r > 0 be such that 2r + |p ip i+1 | < 1 l for all i ∈ I. The transversal periodic points p 1 , ..., p N are persistent so that any g ∈ K close enough to f admits periodic points q 1 , ..., q N of period less than m with q i ∈ B(p i , r) for i = 1, ..., N . We can also assume that [p i + r, p i+1r] ∩ P er m (g) = ∅ for all i / ∈ I. Then it is easily seen that P er l,m (g) is contained in the open r-neighborhood of P er l,m (f ).

To prove the density it is enough to approximate any C 1 Kupka-Smale maps by a map in R n (l, m). To do it one just blows up the finitely many periodic points of period m in small intervals where we "put" an interval map of entropy log 3 (a 3-tent map for example). We refer to [START_REF] Mcgoff | Orders of accumulation of entropy on manifolds[END_REF] for the technical details of the construction. In doing so we may introduce other periodic points of period n but they are all 1 n -close to a horseshoe with entropy log 3 when the inserted intervals are small enough. Theorem 4 follows immediately from Proposition 1 and Proposition 2. Indeed ergodic measures of any f ∈ S ∩ R are approximated by periodic measures (with zero entropy) and by ergodic measures with log 3 entropy (which correspond to the measures of maximal entropy of the horseshoes closed to the periodic points). Therefore any invariant measure is approximated by a convex combination of ergodic measures with either zero entropy or with log 3 entropy.

Remark 1 By using the robustness of horseshoes it is easily seen that generic continuous maps on a compact manifold have infinite topological entropy (see Section 5 of [1]

). In particular they do not admit symbolic extensions.

C 1 examples

In [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF], examples of C 1 surface volume preserving diffeomorphisms (M, T ) without symbolic extensions are built by accumulating horseshoes on periodic points. These examples 2 satisfy more precisely the following property (see in [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF] the proof of Theorem 1.3 after Lemma 5.1) :

There are a nonempty compact subset E of M(M, T ) and a positive real number ρ 0 > 0 such that :

• the periodic measures are dense in E;

• ∀µ ∈ E, lim sup ν∈E,ν→µ h(ν) > ρ 0
In particular the entropy function restricted to E is nowhere continuous. Indeed any invariant measure in E with positive entropy is a limit of periodic measures in E according to the first item. Then, if µ ∈ E is an invariant measure 2 They are generic in the complementary set of the Anosov diffeomorphisms in the set of C 1 volume preserving diffeomorphisms on a surface M .

with zero entropy, the second item implies that µ is a limit of invariant measures in E with entropy larger than ρ 0 > 0.

Entropy as a difference of upper semicontinuous functions

A way to estimate how a nonnegative function h defined on a compact metric space fails to be upper semicontinuous consists in wondering whether it is a difference of nonnegative upper semicontinuous functions and in determining the "smallest" writing as a difference of nonnegative upper semicontinuous functions : what is the smallest upper semicontinuous function f ≥ 0 such that f + h is upper semicontinuous, which can be written as f = inf{g ≥ 0 : g and g + h is upper semicontinuous}? To this end we consider the operator T usc :

T usc : S(X, T ) → S(X, T ) f → µ → lim sup ν→µ (h(ν) -h(µ)) + + f (ν)
As T sex the operator T usc is increasing and satisfies T usc f ≥ f for all f ∈ S(X, T ).

Lemma 3 h = gf with f, g ∈ S(X, T ) if and only if f is a fixed point of T usc .

Proof : Assume h + f is upper semicontinuous. We have lim sup

ν→µ (h(ν) -h(µ)) + + f (ν) ≤ max lim sup ν→µ f (ν), lim sup ν→µ (h(ν) -h(µ) + f (ν))
and then by upper semicontinuity of f and h + f :

lim sup ν→µ (h(ν) -h(µ)) + + f (ν) ≤ f (µ)
Conversely assume T usc (f ) = f . This implies that lim sup

ν→µ (h(ν) -h(µ) + f (ν)) ≤ lim sup ν→µ (h(ν) -h(µ)) + + f (ν) ≤ f (µ)
and thus f + h is upper semicontinuous.

We denote by h usc the smallest upper semicontinuous function larger than h such that h usch is itself upper semicontinuous. In the previous setting h usch is the smallest fixed point, T α(Tusc) usc 0, of the monotone operator T usc .

Lemma 4 For all f ∈ S(X, T )

T sex f ≥ T usc f
In particular the fixed points of T sex are fixed points of T usc and h sex ≥ h usc .

Proof : Any dynamical system admits an entropy structure (h k ) k with upper semicontinuous functions (h k ) k (see for instance the remark following Theorem 7.0.1 of [START_REF] Downarowicz | Smooth interval maps have symbolic extensions[END_REF]). Consider such an entropy structure (h k ) k of (X, T ). Let f ∈ S(X, T ) and µ ∈ M(X, T ). For all k ∈ Z + we have by upper semicontinuity of

h k lim sup ν→µ (h(ν) -h(µ)) + + f (ν) ≤ lim sup ν→µ (h -h k )(ν) + (h k (ν) -h(µ)) + + f (ν) ≤ lim sup ν→µ ((h -h k )(ν) + f (ν))
and finally by taking the limit in k we get for all µ ∈ M(X, T )

lim sup ν→µ (h(ν) -h(µ)) + + f (ν) ≤ T sex f (µ)
The inclusion of the sets of fixed points follows then immediately from the inequality T sex f ≥ f which holds for all f ∈ S(X, T ).

From this last lemma we deduce easily the following corollary :

Corollary 1 Let (X, T ) be a dynamical system admitting a symbolic extension. Then the entropy function h is a difference of nonnegative upper semicontinuous functions. In particular it is the case for C r interval maps with r > 1 and C 2 surface diffeomorphisms; more precisely we have then that h +

d i χ + i r-1
is upper semicontinuous.

On the other hand there exist dynamical systems with continuous entropy function h but without symbolic extension (Example 2.20 of [START_REF] Boyle | The entropy theory of symbolic extension[END_REF]). In particular h sex = h usc and there exist fixed points of T usc which are not fixed points of T sex . However recall that when (X, T ) admits symbolic extensions then h sex and thus h usc coincide with h on a dense G δ set of M(X, T ) (Proposition 3.1 of [START_REF] Boyle | The entropy theory of symbolic extension[END_REF]). When the entropy is a difference of nonnegative upper semicontinuous function then h usc coincides also with h on a residual set for similar reasons.

To compare with the Realization Theorem of Downarowicz and Serafin (Theorem 3) observe that any difference of nonnegative upper semicontinuous functions is a nondecreasing limit of upper semicontinuous functions. Finally let us mention that a real function φ defined on M(X, T ) can be written as a difference of nonnegative upper semicontinuous functions if and only if there exists a sequence of continuous functions (φ n ) n with φ(µ) = n φ n (µ) and n φ n (µ) ≤ C for all µ ∈ M(X, T ) and for some constant C [START_REF] Haydon | On certain classes of Baire-1 functions with applications to Banach space theory[END_REF].

As a function of the first Baire class the set of continuity points of any difference of upper semicontinuous functions is a dense G δ set. In particular the examples of T.Downarowicz and S.Newhouse discussed in Section 3.2 have no symbolic extensions by Corollary 1 (consider the restriction of h to E). This gives an alternative proof of Proposition 4.4 of [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF]. We recall that for a large class of dynamical systems the continuity points of the entropy function are measures of zero entropy. Indeed periodic measures are often dense in the set of invariant probability measures (which implies also that generic invariant measures have zero entropy, see Proposition 5.4 of [START_REF] Abdenur | Nonuniform hyperbolicity for C 1generic diffeomorphisms[END_REF]). For example it is the case of dynamical systems satisfying the so-called specification property (uniformly hyperbolic dynamical systems [START_REF] Sigmund | Generic properties of invariant measures for Axiom Adiffeomorphisms[END_REF], continuous piecewise monotone maps [START_REF] Hofbauer | Generic properties of invariant measures for continuous piecewise monotonic transformations[END_REF],... ) and it is also the case of C 1 generic diffeomorphisms [START_REF] Abdenur | Nonuniform hyperbolicity for C 1generic diffeomorphisms[END_REF]. Continuity points are also zero entropy measures for C 1+α surface diffeomorphisms because in this case the invariant measures with positive entropy are approximated by periodic measures [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF].

C r examples and related questions

In all the known smooth examples with large symbolic extension entropy [START_REF] Asaoka | Hyperbolic set exhibing C 1 -persistent homoclinic tangency for higher dimensions[END_REF], [START_REF] Downarowicz | Symbolic extension entropy in smooth dynamics[END_REF], [START_REF] Burguet | Examples of C r interval map with large symbolic extension entropy[END_REF], we accumulate horseshoes on periodic measures at infinitely many scales. The same process gives large lower bounds on h usc according to the following lemma. It follows that these examples are relevant from the point of view of the continuity properties of the entropy independently of the question of symbolic extensions.

Lemma 5 Let (X, T ) be a dynamical system and let µ be a T -invariant probability measure. We assume that for all k ∈ Z + , there exist periodic points (p (i1,...,i 2k+1 ) ) (i1,...,i 2k+1 )∈Z +2k+1 and invariant probability measures (µ (i1,...,i 2k ) ) (i1,...,i 2k )∈Z +2k (we put Z + 0 = {∅} and µ ∅ = µ) such that :

• for all (i 1 , ..., i 2k ) ∈ Z + 2k , the periodic measures γ p (i 1 ,...,i 2k+1 ) are converging to µ (i1,...,i 2k ) when i 2k+1 goes to +∞;

• for all (i 1 , ..., i 2k+1 ) ∈ Z + 2k+1 , the measures µ (i1,...,i 2k+2 ) are converging to γ p (i 1 ,...,i 2k+1 ) when i 2k+2 goes to +∞;

• the limits lim

TQuestion 1 Question 2 Question 3

 123 h(µ (i1,...,i 2k+2 ) ) ... exist; Then for all n ∈ Z + : h(µ (i1,...,i 2l ) ) ...Proof :The proof is done by induction on n. Assume the lemma is true for n -1 and let us prove it for n. By the induction hypothesis we have for every µ i1,i2 h(µ (i1,...,i 2l ) ) ...ThenT n usc 0(γ pi 1 ) ≥ lim sup i2 h(µ i1,i2 ) + T n-1 usc 0(µ i1,i2 ) (i1,...,i 2l ) ) ...Then by upper semicontinuity we conclude thatT n usc 0(µ) ≥ lim sup i1 (i1,...,i 2l ) ) ...We finally ask three questions :We mentioned examples of (topological) dynamical systems with h sex = +∞ and h usc = h. Do there exist examples of C r dynamical systems with 1 ≤ r < +∞ with h sex = h usc ? with h sex = +∞ and h usc = h? The entropy function of a (topological) dynamical system may be continuous and nonzero. What about C r dynamical systems with 1 ≤ r ≤ +∞? If T is a C r dynamical system with 1 ≤ r ≤ +∞, must every continuity point of the entropy function have zero entropy? We have seen that the existence of symbolic extensions is stronger than the property of the entropy function to be a difference of nonnegative upper semicontinuous functions. Regarding Conjecture 1 one can ask directly : Is the entropy function of any C r dynamical system with r > 1 a difference of nonnegative upper semicontinuous functions? Does it characterize the entropy function of C r systems with r > 1, i.e. given a Choquet simplex K and a nonnegative function f : K → R + which is a difference of nonnegative upper semicontinuous functions, does there exist a C r dynamical system (M, T ) with r > 1 on a compact manifold M and an affine homeomorphism a : M(M, T ) → K such that f • a is the entropy function of (M, T )?

A C 1 interval map f is said Kupka-Smale when every periodic point p of f is hyperbolic, i.e. f P ′ (p) = 1 where P is the period of p.

Acknowledgements : I would like to thank T.Bousch for a useful discussion about Knaster-Tarski theorem.