after many years , the fringe topic of _ weak values _ has entered the spotlight of mainstream physics .weak values are said to be a `` new '' kind of physical quantity which `` can be interpreted as a form of conditioned average associated with an observable '' .here we give an operational framework for this conditioned average and show that the anomalous values are a ubiquitous feature of generic statistical models not limited to quantum theory . this places serious constraints on the explanatory power of weak values .why is such an operational framework important ? in 1964 , bell described a statistical thought experiment which involved two stations which accepted binary inputs and produced binary events conditional on these inputs .the setup contained _ absolutely no reference to physical theories _ although was devised to teach us a lesson about physics .this was necessary and crucial since the framework was designed to test theories themselves and thus was required to be model independent .bell showed two things with this remarkably simple set up : ( 1 ) an inequality on the value of a certain statistic calculated from the data was satisfied by _ every _ local hidden variable model ; and , ( 2 ) entangled quantum systems could violate this inequality .the consequences of bell work are innumerable . by showing a separation between quantum and classical theory for an _ operational _ task, bell paved the way for considering quantum mechanical effects not as a nuisance , but as a resource . herewe stand on the shoulders of bell and propose an analogous operational framework for what is known as pre- and post - selected ( pps ) measurement .that is , we propose a model independent statistical experiment and give an inequality on a certain statistic derived from the data called the _ post - selected shift _ , which can be thought of as an operational generalization of the _ weak value _ from quantum theory .the inequality is satisfied for any statistical model lacking a certain kind of correlation .since we brought up bell , one might expect that the only a theory with correlations as rich as quantum mechanics might violate this inequality .however , we show that classically correlated systems , even without disturbance , can violate the inequality .we call the violation of the inequality an _ anomalous post - selected shift_. such an effect would , for example , be a necessary consequence of observing anomalous weak values in quantum mechanical systems .if weak values are truly `` quantum '' , this work proves it is not the anomalous shift of measurement pointers which signals it .the statistical model is that of three dichotomous events , , and .we need not make any assumptions about the temporal ordering of these events but it is helpful to consider the sequence . in doing so , we refer to the events as follows : as pre - selection , as the intermediate measurement , and as post - selection .this situation is depicted in fig . [ all ] ( a ) .the most important thing to remember is that there is a single object which encodes all the information we have about the situation : the joint probability this point can not be emphasized enough and could potentially go a long way to resolving other statistically natured paradoxes in quantum theory so we will restate it : _ complete information about any statistical problem is contained in the joint probability distribution of the random variables_. every other probability distribution of interest can be derived from the rules of probability from this object .a couple examples are :   without loss of generality , we take . considering the power series of any function of these variables , it is not difficult to see that any statistical model of these variables can be written .\end{aligned}\ ] ] clearly , there must be constraints on the coefficients . the normalization of the distribution , that is implies for all models . by considering all possibilities of the values of the variables ,it is easy to see that the positivity condition amounts to 8 inequalities which feature sums and differences of the coefficients , which we will not write out explicitly . because every model can be written in this form , a model is equivalently specified by its coefficient vector :   the experimentally accessible quantities we consider here are the pre - selected shift  ] .here we have used the notation that =\sum_x \pr(x|y ) f(x) ] , and for our example  = ( 1 + \lambda s\psi \cos\theta)/2 $ ] .using the conditional state above and , the joint distribution of and given is by considering all possibilities , it can be shown that the joint distribution is given by the coefficient vector here qw will refer to this weak quantum measurement model . notice the difference between the strong and weak model ; the terms ( and ) which contain are multiplied by and changes in a subtle way .now consider the pre- and post - selected averages in the weak measurement model : & = \lambda\cos\theta\psi,\label{qw_pre}\\ \mathbb e^{\rm qw}_{s|\psi,\phi}[s ] & = \frac{\lambda\cos\theta(\psi-\phi)}{1-\cos 2\theta \psi\phi}.\end{aligned}\ ] ] if and are different , we have = \frac{\lambda\psi}{\cos\theta},\ ] ] and thus , \right|\geq\left| \mathbb e^{\rm qw}_{s|\psi}[s ] \right|.\ ] ] this model possesses apss .in the quantum mechanical literature , this is the _ weak value _ phenomenon . indeed , =\lambda z_w \psi,\ ] ] where , when , is called the _ weak value _ .but , does eq .signify something uniquely quantum mechanical has happened ?does it even have physical significance ? is the weak value a physical quantity ?the proponents would have us believe the answer is , emphatically , yes .the reason is as follows .one can view eq .  as the expectation value of the course - grained shift of a meter performing the measurement .notice that the value of the observable is proportional to this shift : = \lambda \langle \psi|z|\psi\rangle.\ ] ] but when we perform the pre- and post - selection , we have eq . .this appears as if the meter has shifted by an amount proportional to and , moreover , this shift can be larger than the expected shift without post - selection .this had led some to suggest that is a new kind of physical property of quantum systems .be wary , however , that such ideas are guided by falsely applying physical intuition to logical inference which leads to the far worse problem of imposing physical meaning on logical deductions .we hope by understanding the conditions under which apss occurs , this deep misunderstanding of the purpose and role statistics plays in physical theories can be overcome .  as a first example , we need not look farther that the strong measurement model above . in the strong measurement model , the expectation value of in the distributions of and are & = \cos\theta\psi,\label{qs_shift}\\ \mathbb e^{\rm qs}_{s|\psi,\phi}[s ] & = \frac{\cos\theta(\psi-\phi)}{1-\cos^2\theta \psi\phi}.\end{aligned}\ ] ] notice that for and any , we have \right|\geq\left| \mathbb e^{\rm qs}_{s|\psi}[s ] \right|.\ ] ] the strong measurement model possesses apss as well .thus , the distinction between weak " and strong " measurement is a red herring apss can occur with or without the notion of measurement strength .hence , in the remainder , we work with the fully general model .the defining feature of apss is the reversal of the inequality in moving from eq .  to eq ., as occurs in the quantum models above see eq .and eq . . herewe show that any model in which and are conditionally independent given can not possess apss .  in a general 3 variable dichotomous model , if , then and \right|\leq\left| \mathbb e_{s|\psi}[s ] \right|.\ ] ]  the proof is as follows .consider : & = \sum_s s\pr(s|\psi,\phi),\\ & =  \sum_s s\frac{\pr(s,\phi |\psi)}{\pr(\phi|\psi)},\\ & =  \sum_s s\pr(s|\psi),\\ & = \mathbb e_{s|\psi } \left[s\right].\end{aligned}\ ] ] thus , implies eq . .now , eqs .  andare true for any model , but the equations become too unwieldy with all 8 coefficients . seeing the proof with ( which halves the number of terms we have to consider in the expansions below )should make the general proof obvious .we will proceed by comparing to .\end{aligned}\ ] ] matching coefficients reveals that the only two equations , which are not trivially satisfied , are eqs .  and .note also that conditional independence is necessary but not sufficient . in a correlated model ,the post - selected shift is necessarily _ different _ than the pre - selected shift it just need not be _ larger_.in ref .  we devised a classical model specified by where was interpreted in a physical model of coin toss as `` disturbance '' .the model gives = \frac{\lambda}{1-\delta } > \lambda =  \mathbb e_{s|\psi=+1}[s].\ ] ] a quick check also reveals eq .is not satisfied .either of these facts imply the model is correlated , as required .however , let us be quick to note that physically interpreting the correlation as disturbance is not necessary .the simplest way to violate the inequality is to have all terms but one of or not equal to zero ( necessarily equally one ) . for example , take the model with only : or this model implies and .using these relations , notice that \right|>\left| \mathbb e_{s|\psi}[s ] \right| = 0,\ ] ] a maximally anomalous shift ! a physical realization of this model is shown in fig . [ all ] ( c ) .a ball is placed in one of six boxes which are arranged in a tetrahedral pattern .the measuring devices can ask the binary questions shown in the figure ( left - vs - right , front - vs - back and top - vs - bottom ) .so , for example , when we first find `` right '' , we still do not know whether the ball is in the front or back .but if the final measurement finds `` bottom '' , necessarily the intermediate measurement gave `` back '' .indeed , that is , the pre - selected average is zero while the post - selected average is .this is not surprising more data ought to change ones knowledge of unknown quantities .moreover , this ( rather large ) difference does not signify a physical change in the system itself , only our knowledge of it .the last example shows beyond any doubt that the apss effect is revealing the correlations present in the model .the example is unequivocally classical by any measure .observing apss therefore does not imply anything more exotic than classical statistics .in particular , apss does not bear any relation to physical notions , such as amplitudes , invasiveness , disturbance , contextuality or interference .these properties are all consequences interesting ones , no doubt of the structure of quantum theory , some of which are present in various classical theories .but we have shown here that if that task is to observe an anomalous post - selection shift , the structure of quantum theory is not required .what is missing for most of the proposed physical meanings for weak values is an understanding of how they arise as a resource for some operational task .though , another example of an operational inequality ( derived in the context of so - called leggett - garg inequalities ) has been given that holds for any hidden variable model without disturbance .a similar result would hold here on a hidden variable view . in this context , the role of the hidden variables is to provide a `` common cause '' for the correlations between and , given .such a cause would render and conditionally independent given both and the posited hidden variable .such a demand would require that the consequence of our theorem [ eq .] always holds . in other words , within a hidden variable model , a change in the hidden variable in moving from the intermediate to post - selected measurement disturbance would be necessary to observe apss .measurement is both a fundamental conceptual and practical tool in all areas of physics , and science more broadly .one of the core consequences of quantum mechanics was in forcing us to rethink the meaning of measurement .but some of the cherished niceties of classical physics that we had to give up were never even present in other branches of science .many branches of science do not even have the kinds of fixed and rigid models we take for granted in physics , and hence the notions of observables or physical quantities are irrelevant . as a consequence, the word classical " , when used in fundamental studies of quantum theory , must be taken with a grain of salt is it really the most general classical argument to compare to ? or , a more revealing question , can such claims be verified by an experiment provably separating theories ?  herewe have given a simple probabilistic model which reveals anomalous shifts upon post - selection not requiring disturbance .a theme in quantum foundations and information research is to develop classical models for aspects of quantum theory ( e.g , ) . in this way, we hope to whittle down all the features of quantum theory until we are left with the kernel that is `` truly quantum '' . can the correlations which cause anomalous weak values be considered a uniquely quantum feature ?the above argument casts doubt on this .* acknowledgements*the authors thank justin dressel , asger ipsen , and ruth kastner for helpful comments and discussions .this work was supported in part by national science foundation grant nos .phy-1212445 and by the canadian government through the nserc pdf program .jc was also supported in part by cerc , nserc , and fxqi .justin dressel , mehul malik , filippo m. miatto , andrew n. jordan and robert w. boyd , _ colloquium : understanding quantum weak values : basics and applications _ , http://dx.doi.org/10.1103/revmodphys.86.307[reviews of modern physics * 86 * , 307 ( 2014 ) ] .yakir aharonov , david z. albert and lev vaidman , _ how the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100 _ ,http://dx.doi.org/10.1103/physrevlett.60.1351[physical review letters * 60 * , 1351 ( 1988 ) ] .christopher ferrie and joshua combes , _ how the result of a single coin toss can turn out to be 100 heads _ , http://dx.doi.org/10.1103/physrevlett.113.120404[physical review letter * 113 * , 120404 ( 2014 ) ] .kastner , _ weak values and consistent histories in quantum theory _ , http://dx.doi.org/10.1016/j.shpsb.2003.02.001[studies in history and philosophy of science part b : studies in history and philosophy of modern physics * 35 * , 57 ( 2004 ) ] .       j. dressel , c. j. broadbent , j. c. howell and a. n. jordan , _ experimental violation of two - party leggett - garg inequalities with semiweak measurements _ ,http://dx.doi.org/10.1103/physrevlett.106.040402[physical review letters * 106 * , 040402 ( 2011 ) ] .

Abstract: the question of what is genuinely quantum about weak values is only ever going to elicit strongly subjective opinions  it is not a scientific question . good questions , when comparing theories , are operational they deal with the unquestionable outcomes of experiment . we give the anomalous shift of weak values an objective meaning through a generalization to an operational definition of anomalous post - selected averages . we show the presence of these averages necessitate correlations in every model giving rise to them quantum or classical . characterizing such correlations shows that they are ubiquitous . we present the simplest classical example without the need of disturbance realizing these generalized anomalous weak values .


a human crowd is a specific system , which is of interest for various specialists for different reasons .a physicist is willing to treat a crowd as a gas or a fluid of interacting particles , a psychologist can concentrate on the process of self - categorization in crowds , and a sociologist asks for emergence of norms in a crowd . in an interdisciplinary approach , these perspectives overlap . in the social force model ( sfm ) , designed by dirk helbing and coworkers in the 90 s , physical interactions are combined with action of the social norm of keeping distance to unknown persons . although the differential equations used there can be considered as computationally complex , the sfm seems to describe properly the collective effects in crowd , which appear in particular in emergency situations , as an evacuation . in simpler techniques , such as cellular automata or lattice gas models ,a prescribed area is reserved for each pedestrian , and the influence of other pedestrians is reduced to short range interactions . even if some specific effects such as clogging and arching are reproduced ( as , for example , in ) , dynamics of a pedestrian in these techniques is fully determined by her / his local environment . on the contrary , in most of crowd disastersthe crowd size was essential . for reviews on the sfm and other techniques and a list of literaturewe refer to .  herewe are interested in forces exerted by masses of pedestrians , when physical interactions accumulate and the crowd size does matter .namely , we intend to investigate how the number of pedestrians in a room influences the flow through an exit .therefore we designed a numerical experiment as follows .a number of pedestrians is waiting , crowded , at the exit . at ,the exit is opened and the crowd is pushing towards it .let us denote the number of pedestrians who remain in the room at time as . in this setup ,an experiment with pedestrians provides data on experiments for all , because there is no stage when people gather at the exit .provided that the flow at the exit depends on the number of pedestrians in the room , we should observe this dependence by just measuring the curve .alternatively , we can measure the time gaps between successive crossings of exit .one can write where is the time instant when the number of pedestrians in the room changes from to .we are going to concentrate on the series of the time lags .is it stationary ?are there long - term correlations and/or regimes with characteristically different behavior ?  up to our knowledge , this question was not posed directly in the literature , but the shape of the function was obtained several times by different authors . in the next section we gather the results obtained by other authors which are directly close to our specific interest . for reasons explained above , we do not refer to simulations done with the cellular automata and lattice gas model . in the third section the sfm is briefly described . in sectioniv we describe the technique of the data analysis . finally , we show our numerical results ( section v ) and discuss them ( section vi ) .in , the model equation of motions of pedestrians were formulated . among other results ,an effect of pressure of crowd was demonstrated ; out of two groups attempting to cross the door in opposite directions , the larger group was prevailing until the larger group became smaller . in this paper , a noise term is included to the equation of motion , hence the term langevin equations. in , the same sfm equations were used without the noise term .there , the desired velocity was associated with the level of panic .also , the evacuation time dependence on the desired velocity was found to display a minimum . also , when the desired velocity was increased , a change of the process from a laminar to a clogging mode of the crowd behavior was observed . as the authors remarked , the effect was less pronounced for wider exits .note that , as noted in , the experimental data collected in planes do not show abrupt changes of the effectivity of evacuation when the exit width is changed from 0.6 m to 1.8 m. as shown in with more experimental data , the relation between the bottleneck width and the flow of pedestrians does not show any threshold .  in ,three curves are shown , obtained by simulations with using the sfm , on the number of pedestrian who left the room against time .the curves were obtained for 200 pedestrians and three values of the desired velocity : 0.8 , 2.0 and 6.0 m / s .first curve ( 0.8 ) shows that at the last stage of evacuation the flow decreases .this effect exists also , but is weaker , for the second curve ( 2.0 ) , but not for the third one ( 6.0 ) . instead, the latter curve was found to be particularly noisy . in fig .3 of , the distribution of clogging delays is shown for up to 160 people and the three above given values of the desired velocity .each curve shows a clear maximum between 0.2 and 0.4 s. in this and subsequent paper , the cluster size distribution is also investigated , where a cluster means a group of people in physical contact between them . for the laminar and the turbulent flow , this distributionis found to be qualitatively different .  in ,the formalism of optimization , developed by authors for other purposes , has been applied to the evacuation problem .both clogging and arching have been observed in the simulation .the evacuation time dependence on the number of people was found to decrease in a non - linear way , but no minimum of this curve was found . in ,the influence is investigated of the desired velocity on the evacuation time , the latter being a measure of panic .the mode of motion when the evacuation time increases with the desired velocity has been classified as turbulent .the effect of wider exit was investigated directly : the desired velocity where the evacuation time displays a minimum was shifted towards larger value with the exit width . in andreferences cited therein , an experiment performed in a wardroom with a group of 70 soldiers is described .it was found among other results that the clogging is more likely if the number of persons is larger than 45 .  for completenesswe remark also our two recent papers , where the sfm was used to investigate the chances that persons in the crowd can decide about themselves .the stationarity of the process of evacuation was not investigated there .the simulation is based on the model of crowd dynamics , described by helbing et al . . in this model ,the equation of motion of a person of mass is written as where the first term on the right hand side is the tentative acceleration of a person who intends to have the velocity , ( its length commonly termed as the desired velocity ) dependent on the coordinates . as a rule , the vector points to the exit center ( large distance from the person to the exit ) or to the closest point of the exit ( small distance ) .further , is the characteristic time of this acceleration , is the actual velocity of -th person , is the force exerted on -th person by -th person , and is the force exerted on -th person by a wall .the force contains three components ; social interaction which describes the tendency of and to keep distance between each other , and two physical interactions between their bodies : radial force and slide friction .the social part of interaction is also adapted from .it is given by where and are constants , is the mean radius of the vertical projection of the human body , and is the position of -th agent .the parameters of the simulation are adapted from : the amplitude of the social force , the constant which is responsible for the spatial dependence of the social force is , the radii of agents , their masses , the characteristic time of acceleration is and the absolute value of the desired velocity is . as remarked in ,these values allow to reproduce experimental interpersonal distances and flows through bottlenecks .also , the same values are assumed for all persons , to minimize the number of parameters .the instant values of the velocities allow to update the positions as well .the equations of motion are solved with the runge - kutta method of 4-th order .the simulation was performed as follows . pedestrians were gathered at a the closed exit of width of 1 m , which was opened at .the time lags were measured between crossing of the exit by subsequent pedestrians .the simulation was repeated 100 times .in our analysis procedure , we split the data of each run into ten non - overlapping parts corresponding to 100 persons leaving the room .each part is analyzed independently , but averages over all 100 simulation runs are calculated to improve statistics .quantitatively , correlations between time lags separated by people are defined by the ( auto- ) correlation function , where is the length of the considered data part and is the average time lag in this part .if the time lags are uncorrelated , is zero for positive .if correlations exist up to a certain number of people , the correlation function will be positive up to and vanish above .for the relevant case of long - range correlations , decays as a power law , a direct calculation of is hindered by the non - stationarities and trends in the data , since is not constant .we thus apply return interval statistics and detrended fluctuation analysis to study short - term and long - term correlations in the data , respectively .usually , return interval statistics ( ris ) study the time intervals between extreme events that exceed a given threshold . in a sequence of uncorrelated values ( white noise ), these return intervals are also uncorrelated and distributed according to a poisson distribution , where is the mean return interval for the given threshold . for long - correlated data , on the other hand , a stretched exponential distribution \label{stretched}\ ] ]has been observed , where the exponent is the correlation exponent from eq .( [ gamma ] ) , and the parameters and are independent of  .if , on the other hand , the data is nearly deterministic ( and not random ) , all return intervals will fluctuate weakly around the typical value , giving rise to , e. g. , a gaussian distribution , ,  \label{gauss}\ ] ] for with the small standard deviation .  herewe consider each event of a person leaving the room as an extreme event , so that the return intervals are identical with the time lags , and is identical with .there is thus no threshold for extreme events , but we get much more statistics .our ris focus on short - term correlations ( between successive persons ) .detrended fluctuation analysis ( dfa ) has become a widely - used technique for the detection of long - range correlations in noisy , nonstationary time series .the dfa procedure consists of four steps .first we determine the profiles , .secondly , we divide into non - overlapping segments of equal length .thirdly , we calculate the local trend for each segment by a least - square fit of the data .linear , quadratic , cubic , or higher order polynomials can be used in the fitting procedure ( conventionally called dfa1 , dfa2 , dfa3 , ) .then we determine the variance of the differences between profile and fit in each segment .fourthly , we average over all segments and take the square root to obtain the fluctuation function . since we are interested in how depends on the time scale , we have to repeat steps 2 to 4 for several .apparently , increases with increasing .if data are long - range power - law correlated according to eq .( [ gamma ] ) , increases , for large values of , as a power - law , to determine the asymptotic scaling behavior of this fluctuation function we plot as a function of on double logarithmic scales and calculate the slope by a linear fit in the regime . this way , short - term correlations affecting less than 10 persons subsequently exiting the room are ignored in the analysis .against the number of pedestrians remaining in the room . data from 100 simulations with people are included .the strong fluctuations at the beginning of the simulations ( close to 1000 ) are transient effects due to the opening of the exit .one can distinguish two regimes ( approximately for and ) when comparing averages and medians.,title="fig:",width=453 ]  in fig . 1 ,results are shown for the average values and medians of against the number of pedestrians remaining in the room .as we see , two parts of the process can be distinguished . in the first regime from to ,averages and medians differ significantly and the fluctuations of are rather strong .however , the fluctuations seem to decrease gradually , and so do the measured values . in the second regime from to 0 , where fluctuations are small , averages and medians are nearly identical , and the measured values increase slightly .this change of behavior can be interpreted as a cross - over from a stage with temporal cloggings to a laminar stage . in the latter case ,the crowd behind pedestrians at the exit is large enough to push them out but not large enough to cause clogging .the appearance of the second stage of evacuation agrees with the character of two out of three discharge curves , presented in fig . 2 of , for small and moderate desired velocity .and ( b ) scaled distributions of the time lags with mean for to 800 ( black squares ) , to 600 ( red circles ) , to 400 ( blue triangles up ) , to 200 ( green triangles down ) , and to 0 ( violet diamonds ) persons remaining in the evacuated room .the unscaled distributions in ( a ) show that the nearly gaussian peak for short time lags is hardly changing , while the exponential tail is decaying with decreasing .the inset in ( a ) shows the fitted slopes of the exponentials decays ( black squares together with the pearson correlation coefficients of the fits ( blue open circles ) .the scaled distributions in ( b ) show that characterizes the exponential peak fairly well.,title="fig:",width=566 ]  in fig .2(a ) , we show the distributions of return intervals , i. e. gathered in shorter parts of the data , where the departure from stationary flow can be approximately neglected . however , comparing the statistics , we see the differences . initially , for large numbers of people in the room , there is a large exponential tail of the distribution , formed by the clogging events and corresponding to the poisson distribution eq .( [ simple ] ) with a modified prefactor .in addition , there is a distinguished maximum near the time lag s , which is approximately described by a gaussian distribution eq .( [ gauss ] ) , also with a modified prefactor .the center of this peak agrees approximately with the results shown in fig . 3 of .we see that there are apparently two distinct components in the time lag distributions : typical short time lags around 0.2 s ( probably due to persons successively exiting without delays ) and exponentially distributed longer time lags ( probably due to interruptions in the flow of exiting people because of clogging or arching effects ) .  for smaller numbers of people in the room the exponential tail decreases , to nearly vanish during the last stage , i. e. for to 0 .simultaneously , the nearly gaussian peak for short time lags is hardly changing . while the behavior of the short time lags is well characterized by the nearly constant median of ( see fig . 1 ) , the changing averages of , i. e. , characterize the exponential behavior for long time lags .this is confirmed by the plot of scaled distributions shown in fig .the application of dfa to the data from each of the ten parts yields fluctuation exponents very close to 0.5 , which proves the absence of relevant long - term correlations during all stages of the evacuation procedure . specifically , we obtain and 0.54 for the first two parts ( between 1000 and 800 people in the room ) and values between 0.50 and 0.53 for all other parts . since the systematic error of such fluctuation exponents is around 0.05 for time series of just 100 values ,all of these numerical results are fully consistent with the null hypothesis of only short - term correlations in the data .is shown versus the time that measures the time interval till only persons remain in the room .curves are shown for ( red ) , 50 ( black ) , 100 ( blue ) , and 200 ( violet ) .each curve is fitted with in the scaling regime .,title="fig:",width=453 ]  finally , we are interested in the scaling behavior of the so - called discharge curve , i. e. the number of people remaining in the room against time . however , as we observe in fig . 1 , for the last approximately 50 persons the character of the curve changes . then , for each numerical experiment we determine the time when persons are left in the room in experiment , and we investigate the dependence of on for .again , is the number of pedestrians in the room .this dependence is averaged over all 100 experiments .the result is shown in fig . 3 in the log - log scale .fitting the result to a straight line we get an exponent , which indicates if and how the evacuation speed depends on the crowd size in the first stage of evacuation .a result means lack of this dependence ; if , the evacuation is stationary at least in its first stage .however , our result is that clogging events make the evacuation slower , and these events are more likely if the number of pedestrians in the room is larger .the latter effect agrees with the experimental result in .  effectively ,our exponent is smaller than one ; we get for the fitting range and parameter as cutoff point of minimal number of persons in the room . fig .3 also shows that the numerical value of is depending on , reaching a bit smaller values for smaller and larger values for larger .we admit that perhaps the scaling regime is not fully reached yet because the crowd may be too small .still , the conclusion that is close to 0.85 seems to be well grounded .this form of the scaling relation allows to extrapolate the results for larger crowds .the results indicate that the probability distribution of the time lags changes in time . in other words ,the evacuation process simulated here is not stationary .one of the consequences is that the total evacuation time depends on the number of pedestrians in the room , and therefore it is not a good measure of the efficiency of the process . if the size of the crowd is large , effects of clogging appear which are absent for small numbers of pedestrians .although the case of larger desired velocity is not investigated here , we can reasonably expect that our findings will be particularly important if large desired velocity happens to be combined with large crowd .effects of victims , who become obstacles , can only enhance the nonstationary character of the process .a straightforward interpretation of our result is that the sfm successfully describes the effect of cumulation of the physical forces between agents at the exit .keeping the hydrodynamic analogy , the pressure at the exit increases with the crowd size . if this pressure exceeds some critical value , pedestrians at the exit are not able to move , even if they are close to the exit .this is the origin of the observed large values of the time lags , and the dependence of the size of the exponential tail of the lag distribution on the size of the crowd .  on the other hand, the data analysis with the return interval statistics and detrended fluctuation analysis shows that there is no phase transition between the first stage , when the mean time lag decreases , and the second stage , when the mean time lag increases .also , correlations between pedestrians leaving the room in subsequent times are broken by the clogging events . as a result, the observed tail of the probability distribution of is poissonian .d. helbing , _ a fluid dynamic model for the movement of pedestrians _ , complex systems * 6 * ( 1992 ) 391. s. reicher , _ the psychology of crowd dynamics _hogg and r.s .tindale ( eds . ) , blackwell handbook of social psychology : group processes , blackwell , oxford 2001 , pp .182 - 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Abstract: an evacuation process is simulated within the social force model . thousand pedestrians are leaving a room by one exit . we investigate the stationarity of the distribution of time lags between instants when two successive pedestrians cross the exit . the exponential tail of the distribution is shown to gradually vanish . taking fluctuations apart , the time lags decrease in time till there are only about 50 pedestrians in the room , then they start to increase . this suggests that at the last stage the flow is laminar . in the first stage , clogging events slow the evacuation down . as they are more likely for larger crowds , the flow is not stationary . the data are investigated with detrended fluctuation analysis and return interval statistics , and no phase transition is found between the stages of the process .


the advent of automated computation led to a series of great successes and achievements in the study of dynamical problems .the use of computers and simulations allowed to compute and forecast the behavior of many important natural phenomena and , on the other hand , led to the discovery of important general aspects of dynamics .this motivates the huge work that was made by hundred of scientists to improve practical simulation and computation techniques .it also motivates the study of the theoretical limits of simulation and computation techniques , and the theoretical understanding of related problems .  in this paperwe want to focus on some of these aspects related to _ rigorous _ computation and simulation of ( discrete time ) dynamical systems .the simulation and investigation of dynamics started with what we call the `` naive '' approach , in which the user just implements the dynamics without taking rigorous account of numerical errors and roundings .then he `` looks '' to the screen to see what happens .  of course , the sensitivity to initial conditions , and the typical instability of many interesting systems ( to perturbations on the map generating the dynamics ) implies that what it is seen on the screen could be completely unrelated to what was meant to be simulated .  in spite of this, the naive approach turns out to work unreasonably well in many situations and it is still the most used one in simulations . the theoretical reasons why this method works and its limits , in our opinion are still to be understood ( some aspects have been investigated in ,, e.g. ) .  on the opposite side from the naive approach, there is the absolutely rigorous approach , which will be the main theme of this paper : the user looks for an algorithm which can give a description of the object which is meant to be computed , up to any desired precision .  in this point of view , for example the constant is a computable number because there is an algorithm that is able to produce a rational approximation of at any given precision ( for example finding the right and calculating such that the error is smaller than requested ) .  in an absolutely rigorous simulation ( computation ) the initial point or some initial distribution is supposed to be known up to any approximation and the transition map is also supposed to be known up to any accuracy ( suitable precise definitions will be given below ) .this allows the evolution of the system to be simulated with any given accuracy , and the question arise , if interesting and important objects related to the dynamics ( invariant sets or invariant measures e.g. ) can be computed from the description of the system ( or maybe adding some additional information ) .  in this paperwe would like to give a survey of a group of results related to these computational aspects , restating and updating them with some new information .we will see that in many cases the interesting objects can be computed , but there are some subtleties , and cases where the interesting object can not be computed from the description of the system or can not be computed at all ( again , up to any given precision ) . in particular , this happen case for the computation of invariant measures and julia sets .  hence , these results set theoretical limits to such computations .  * computing invariant measures .*  an important fact motivating the study of the statistical properties of dynamical systems is that the pointwise long time prediction of a chaotic system is not possible , whereas , in many cases , the estimation or forecasting of averages and other long time statistical properties is .this often corresponds in mathematical terms to computing invariant measures , or estimating some of their properties , as measures contain information on the statistical behavior of the system and on the potential behavior of averages of observables along typical trajectories of the system ( see section [ ds ] ) .an invariant measure is a borel probability measure on such that for each measurable set it holds .they represent _ equilibrium _ states , in the sense that probabilities of events do not change in time .rigorously , compute an invariant measure means to find an algorithm which is able to output a description of the measure ( for example an approximation of the measure made by a combination of delta measures placed on rational points ) up to any prescribed precision .we remark that once an interesting invariant measure is computed , it is possible to deduce from it several other important information about the dynamics : lyapunov exponents , entropy , escape rates , etc ... for example , in dimension one , once an ergodic invariant measure has been approximated , the lyapunov exponent can be estimated using the formula , where denotes the derivative of the map generating the dynamics . in higher dimensions ,similar techniques can be applied ( see e.g. for more examples of derivation of dynamical quantities from the computation of the invariant measure ) .  before giving more details about the computation of invariant measures , we remark that , since there are only countably many algorithms ( computer programs ) , whatever we mean by approximating a measure by an algorithmwould imply that only countable many measures can be computed whereas , in general , a dynamical system may have uncountably many invariant measures ( usually an infinite dimensional set ) .so , a priori most of them will not be algorithmically describable .this is not a problem because we should put our attention on the most meaningful ones .an important part of the theory of dynamical systems is indeed devoted to the understanding of physicallyrelevant invariant measures . informally speaking , these are measures which represent the asymptotic statistical behavior of many ( positive lebesgue measure ) initial conditions ( see section [ ds ] for more details ) .the existence and uniqueness of physical measures is a widely studied problem ( see ) , which has been solved for some important classes of dynamical systems .these measures are some of the good candidates to be computed .the more or less _ rigorous _ computation of such measures is the main goal of a part of the literature related to computation and dynamics .a main role here is played by the transfer operator induced by the dynamics .indeed , the map  defining the dynamics , induces a dynamics on the space of probability measures on  . is called the transfer operator associated to ( definition and basic results about this are recalled in section [ ds ] ) .invariant measures are fixed points of this operator .the main part of the methods which are used to compute invariant measures deals with suitable finite dimensional approximation of this operator . in section [ ds ] we will review briefly some of these methods , and some references.we then consider the problem from an abstract point of view , and give some general result on rigorous computability of the physical invariant measure . in particular we will see that the transfer operator is computable up to any approximation in a general context ( see thm [ l_comp ] ) .the invariant measure is computable , provided we are able to give a description of a space of regular measures where the physical invariant measure is the only invariant one ( see thm [ comp_inv ] and following corollaries ) .we will also show how the description of the space can be obtained from the one of the system in a class of examples ( piecewise expanding maps ) by using the lasota - yorke inequality .after such general statements one could conjecture that all computable dynamical systems should always have a computable invariant measure .we will see that , perhaps surprisingly , this is not true .not all systems that can be described explicitly ( the dynamics can be computed up to any prescribed approximation ) have computable invariant measures ( see section [ lst ] ) .the existence of such examples reveals some subtleties in the computation of invariant measures .  to further motivate these results, we finally remark that from a technical point of view , computability of the considered invariant measure is a requirement in several results about relations between computation , probability , randomness and pseudo - randomness ( see section [ pseudorandom ] and e.g. )  * computability in complex dynamics *  polynomial julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system .one of the reasons for their popularity is the beauty of the computer - generated images of such sets .the algorithms used to draw these pictures vary ; the nave approach in this case works by iterating the center of a pixel to determine if it lies in the julia set .there exists also more sophisticated algorithms ( using classical complex analysis , see ) which work quite well for many examples , but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution . moreover , there are examples , even in the family of quadratic polynomials , where no satisfactory pictures of the julia set exist .  in the rigorous approach ,a set is computable if , roughly speaking , its image can be generated by a computer with an arbitrary precision . under this notion of computability ,the question arise if julia sets are always computable . in a series of papers ( )it was shown that even though in many cases ( hyperbolic , parabolic ) the julia set is indeed computable , there exists quadratic polynomials which are computable ( again , in the sense that all the trajectories can be approximated by an algorithm at any desired accuracy ) , and yet the julia set is not .so we can not simulate the set of limits points on which the chaotic dynamics takes place , but , what about the statistical distribution ? .in fact , it was shown by brolin and luybich that there exists a unique invariant measure which maximizes entropy , and that this measure is supported on the julia set .the question of whether this measure can be computed has been recently solved in , where it is proved that the brolin - lyubich measure is always computable . so that even if we can not visualize the julia set as a spatial object , we can approximate its distribution at any finite precision .  * computing the speed of convergence and pseudorandom points . *  in several questions in ergodic theory the knowledge of the speed of convergence to ergodic behavior is important to deduce other practical consequences . in the computational framework ,the question turn out to be the effective estimation of the speed of convergence in the ergodic theorems such that  from less than a given error for each . ] . from the numerical - practical point of viewthis has been done in some classes of systems , having a spectral gap for example . in this casea suitable approximation of the transfer operator allows to compute the rate of decay of correlations and from this , other rates of convergence can be easily deduced .other classes of systems could be treated joining the above spectral approach , with combinatorial constructions ( towers , see e.g. ) , but the general case need a different approach .  in it was shown that much more in general , _ if the system can be described effectively , then the rate of convergence in the pointwise ergodic theorem can be effectively estimated_. we give in section [ speed ] a very short proof of a statement of this kind ( see theorem corollaryx ) for ergodic systems , and show some consequences . among these , a _ constructive _ version of pointwise ergodic theorem . if the system is computable ( in some wide sense that will be described ) then , it is possible to compute points having typical statistical behavior .such points could be hence called _ pseudorandom _ points in the system ( see section pseudorandom ) .since the computer can only handle computable initial conditions , any simulation can start only from these points .pseudorandom initial conditions are hence in principle good points where to start a simulation .we remark that it is widely believed that naive computer simulations very often produce correct statistical behavior .the evidence is mostly heuristic .most arguments are based on the various shadowing results ( see e.g. chapter 18 ) . in this kind of approach ( different from ours ) , it is possible to prove that in a suitable system every pseudo - trajectory , as the ones which are obtained in simulations with some computation error , is close to a real trajectory of the system .however , even if we know that what we see in a simulation is near to some real trajectory , we do not know if this real trajectory is typical in some sense .a limit of this approach is that shadowing results hold only in particular systems , having some strong hyperbolicity , while many physically interesting systems are not like this . in our approach we consider real trajectories instead of `` pseudo '' ones and we ask if there is some computable point which behaves as a typical point for the dynamics .we would like to thank the abdus salam international centre for theoretical physics ( trieste , it ) for support and hospitality during this research .to have formal results and precise assumptions on the computability ( up to any given error ) of continuous objects , we have to introduce some concepts .we have to introduce some recursive version of open and compact sets , and characterize the functions which are well suited to operate with those sets ( computable functions ) .we explain this theory in this section trying the explanation to be to be as much as possible simple and self contained .the starting point of recursion theory was to give a mathematical definition making precise the intuitive notions of _ algorithmic _ or _ effective procedure _ on symbolic objects .several different formalizations have been independently proposed ( by church , kleene , turing , post , markov ... ) in the 30 s , and have proved to be equivalent : they compute the same functions from to .this class of functions is now called the class of _ recursive functions_. as an algorithm is allowed to run forever on an input , these functions may be _ partial _ , i.e. not defined everywhere .the _ domain _ of a recursive function is the set of inputs on which the algorithm eventually halts .a recursive function whose domain is is said to be _total_.  we now recall an important concept from recursion theory .a set is said to be * _ recursively enumerable ( r.e . )_ * if there is a ( partial or total ) recursive function enumerating , that is .if , can be effectively converted into a total recursive function which enumerates the same set .   strictly speaking , recursive functions only work on natural numbers , but this can be extended to the objects ( thought as `` finite '' objects ) of any countable set , once a numbering of its elements has been chosen .we will use the word _ algorithm _ instead of _ recursive function _ when the inputs or outputs are interpreted as finite objects .the operative power of algorithms on the objects of such a numbered set obviously depends on what can be effectively recovered from their numbers .more precisely , let and be numbered sets such that the numbering of is injective ( it is then a bijection between and ). then any _ recursive function _ induces an _ algorithm _ .the particular case will be much used .for instance , the set of rational numbers can be injectively numbered in an _ effective _ way : the number of a rational can be computed from and , and vice versa .we fix such a numbering : from now and beyond the rational number with number will be denoted by .now , let us consider computability notions on the set of real numbers , introduced by turing in .let be a real number .we say that :  * is * _ lower semi - computable _ * if the set is r.e .* is * _ upper semi - computable _ * if the set is r.e .* is * _ computable _ * if it is lower and upper semi - computable .equivalently , a real number is computable if and only if there exists an algorithmic enumeration of a sequence of rational numbers converging exponentially fast to .that is :  a real number is * _ computable _ * if there is an algorithm such that for all .* uniformity .* algorithms can be used to define computability notions on many classes of mathematical objects .the precise definitions will be particular to each class of objects , but they will always follow the following scheme :  an object is * _ computable _ * if there is an algorithm   which computes in some way .each computability notion comes with a uniform version .let be a sequence of computable objects :   is computable * _ uniformly in _ * if there is an algorithm   such that for all , computes .for instance , the elements of a sequence of real numbers are uniformly computable if there is a algorithm such that for all .  in other wordsa set of objects is computable uniformly with respect to some index if they can be computed with a `` general '' algorithm _ starting _ from the value of the index .  in each particular case, the computability notion may take a particular name : computable , recursive , effective , r.e .so the term `` computable '' used above shall be replaced .a computable metric space is a metric space with an additional structure allowing to interpret input and output of algorithms as points of the metric space .this is done in the following way : there is a dense subset ( called ideal points ) such that each point of the set is identified with a natural number .the choice of this set is compatible with the metric , in the sense that the distance between two such points is computable up to any precision by an algorithm getting the names of the points as input .using these simple assumptions many constructions on metric spaces can be implemented by algorithms .a * _ computable metric space _ * ( cms ) is a triple , where  1 . is a separable metric space . is a dense , numbered , subset of called the set of * _ ideal points_*. 3 .the distances between ideal points are all computable , uniformly in ( there is an algorithm such that ) . is a numbered set , and the information that can be recovered from the numbers of ideal points is their mutual distances . without loss of generality, we will suppose the numbering of to be injective : it can always be made injective in an effective way .[ recconv]we say that in a metric space , a sequence of points converges _ _ recursively _ _ to a point if there is an algorithm such that for all .[ comp_points]a point is said to be * _ computable _ * if there is an algorithm such that converges recursively to .we define the set of * _ ideal balls _ * to be where is an open ball .we fix a numbering which makes the number of a ball effectively computable from its center and radius and vice versa . is a countable basis of the topology .[ reop ] we say that an open set is _ effective _ if there is an algorithm such that .observe that an algorithm which diverges on each input enumerates the empty set , which is then an effective open set .sequences of uniformly effective open sets are naturally defined . moreover ,if is a sequence of uniformly effective open sets , then is an effective open set .an * _ effective -set _ * is an intersection of a sequence of uniformly effective open sets .  obviously , an uniform intersection of effective -sets is also an effective -set .let and be computable metric spaces .let also and be enumerations of the ideal balls in and .a computable function is a function whose behavior can be computed by an algorithm up to any precision . for thisit is sufficient that the pre - image of each ideal ball can be effectively enumerated by an algorithm .[ comp_func ] a function is * _ computable _ * if is an effective open set , uniformly in .that is , there is an algorithm such that for all .a function is * _ computable on _ * if there are uniformly effective open sets such that   intuitively , a function is computable ( on some domain ) if there is a computer program which computes ( for ) in the following sense : on input , the program , along its run , asks the user for approximations of , and eventually halts and outputs an ideal point satisfying .this idea can be formalized , using for example the notion of _ oracle computation_. the resulting notion coincides with the one given in the previous definitions .recursive compactness is an assumption which will be needed in the following . roughly , a compact set is recursively compact if the fact that it is covered by a finite collection of ideal balls can be tested algorithmically ( for equivalence with the -net approach and other properties of recursively compact set see ) .  [ rec.compact ]a set is _ * recursively compact * _ if it is compact and there is a recursive function such that halts if and only if is a covering of .let be a metric space , a borel measurable map and a -invariant borel probability measure. a set is called -invariant if .the system is said to be ergodic if each -invariant set has total or null measure . in such systems the famous birkhoff ergodic theorem says that time averages computed along -typical orbits coincides with space average with respect to more precisely , for any it holds  almost each , where   this shows that in an ergodic system , the statistical behavior of observables , under typical realizations of the system is given by the average of the observable made with the invariant measure .we say that a point belongs to the basin of an invariant measure if ( [ birkhoff ] ) holds at for each bounded continuous . in case a manifold ( possibly with boundary ) , a physical measure is an invariant measure whose basin has positive lebesgue measure ( for more details and a general survey see ) .computation of such measures will be the main subject of this section .a function between metric spaces naturally induces a function between probability measure spaces .this function is linear and is called transfer operator ( associated to ) .measures which are invariant for are fixed points of .let us consider a computable metric space and a measurable function .let us also consider the space of borel probability measures on   let us define the linear function by duality in the following way : if then is such that    the computation of invariant measures ( and many other dynamical quantities ) very often is done by computing the fixed points ( or the spectrum ) of this operator in a suitable function space .the most applied and studied strategy is to find a suitable finite dimensional approximation of ( restricted to a suitable function space ) so reducing the problem to the computation of the corresponding relevant eigenvectors of a finite matrix .an example of this is done by discretizing the space by a partition and replacing the system by a ( finite state ) markov chain with transition probabilities is the lebesgue measure on  ( see e.g. ) , then , taking finer and finer partitions it is possible to obtain in some cases that the finite dimensional model will converge to the real one ( and its natural invariant measure to the physical measure of the original system ) . in some casethere is an estimation for this speed of convergence ( see eg . for a discussion ) , but a rigorous bound on the error ( and then a real rigorous computation ) is known only in a few cases ( piecewise expanding or expanding maps , see ) .similar approaches consists in applying a kind of faedo - galerkin approximation to the transfer operator by considering a complete hilbert base of the function space and truncating the operator to the action on the first elements ( see ) .another approach is to consider a perturbation of the system by a small noise .the resulting transfer operator has a kernel .this operator then can be approximated by a finite dimensional one , again by faedo - galerkin method and relevant eigenvectors are calculated ( see e.g. ) then , if we prove that the physical measure of the original system can be obtained as a limit when the size of the noise tends to zero ( this happen on uniformly hyperbolic system for example ) we have a method which in principle can rigorously compute this measure .variations on the method of partitions are given in , while in a different method , fastly converging , based on periodic points is exploited for piecewise analytic markov maps .another strategy to face the problem of computation of invariant measures consist in following the way the measure can be constructed and check that each step can be realized in an effective way . in some interesting examples we can obtain the physical measure as limit of iterates of the lebesgue measure ( recall that is the lebesgue measure ) .to prove computability of the main point is to explicitly estimate the speed of convergence to the limit .this sometimes can be done using the spectral properties of the system ( ) .  concluding , if the goal is to rigorously compute an invariant measure , most of the results which are in the today literature are partial . indeed , beside being applied to a quite restricted class of systems , to compute the measure those methods need additional information .for example , the way to compute rigorously the finite dimensional approximation is often not done effectively , or the rate of convergence of the approximation is computed up to some constants depending on the system , which are not estimated .  in the remaining part of the section we present some results , mainly from explaining some result about rigorous computations of invariant measures .these results have the advantage to give in principle an effective method for the rigorous computation of an invariant measure and give a quite general result , where all the needed assumptions are explicited .they have the disadvantage to not optimize computation time .so it is not clear if they can be implemented and used in practice .the rigorous framework into they are proved , however allows to see them as an investigation about the theoretical limits of rigorous computation of invariant measures .moreover by this , also negative results can be proved . in particular , we give examples of _ computable _ systems having no computable invariant measures .   in this sectionwe explain precisely what we mean by computing a measure .this means , having an algorithm who is able to approximate the measure by `` simple measures '' up to any given approximation .let us consider the space of borel probability measures over .let be the set of real - valued bounded continuous functions on .we recall the notion of weak convergence of measures :   is said to be _ * weakly convergent * _ to if for each .let us introduce the wasserstein - kantorovich distance between measures .let and be two probability measures on and consider :    where is the space of functions on having lipschitz constant less than one .the distance has the following useful properties  [ ambros ]  1 . is a distance and if is bounded , separable and complete , then with this distance is a separable and complete metric space .2 .  if is bounded , a sequence is convergent for the metrics if and only if it is convergent for the weak topology .if is compact is compact with this metrics .  item ( 1 ) has an effective version : inherits the computable metric structure of .indeed , given the set of ideal points of we can naturally define a set of ideal points in by considering finite rational convex combinations of the dirac measures supported on ideal points .this is a dense subset of .the proof of the following proposition can be found in ( hr07 ) .if bounded then is a computable metric space .a measure is then computable if there is a sequence converging recursively fast to in the metric ( and hence for the weak convergence ) .invariant measures can be found as fixed points of the transfer operator , i.e. as solutions of the equation there is an abstract theorem allowing the computation of isolated fixed points ( see galhoyroj3 corollary 3 ) of computable maps . to apply it is necessary to consider a space where the transfer transfer operator is computable( this space hopefully will contain the physical invariant measure ) .we remark that if is not continuous then is not necessarily continuous ( this can be realized by applying to some delta measure placed near a discontinuity point ) hence not computable .still , we have that is continuous ( and its modulus of continuity is computable ) at all measures which are far enoughfrom the discontinuity set .this is technically expressed by the condition .[ l_comp ] let be a computable metric space and be a function which is computable on .then is computable on the set of measures   from a practical point of view this proposition provides sufficient conditions to rigorously approximate the transfer operator by an algorithm .the above tools allow us to ensure the computability of on a large class of measures .this will enable us to apply a general result on computation of fixed points and obtain  [ comp_inv ] let be a computable metric space and a function which is computable on .suppose there is a recursively compact set of probability measures such that for every , holds .then every invariant measure isolated in is computable .moreover the theorem is uniform : there is an algorithm which takes as inputs finite descriptions of and an ideal ball in which isolates the isolating ball is not necessary . ]an invariant measure , and outputs a finite description of .a trivial and general consequence of theorem [ comp_inv ] is the following :  if a system as above is uniquely ergodic and its invariant measure satisfies , then it is a computable measure .the main problem in the application of theorem [ comp_inv ] is the requirement that the invariant measure we are trying to compute be isolated in . in general the space of invariant measures in a given dynamical systemcould be very large ( an infinite dimensional convex subset of ) and physical measures have often some kind of particular regularity . to isolate a particular measurewe can restrict and consider a subclass of regular measures .let us consider the following _ seminorm _ :  if and are computable and is recursively compact then recursively compact ( ) .this implies  [ 2]let be recursively compact and be computable on , with .then any invariant measure isolated in with is computable . once again , this is uniform in .the above general propositions allow us to obtain as a corollary the computability of many absolutely continuous invariant measures .for the sake of simplicity , let us consider maps on the interval .[ prop_dim ] if  ] is said piecewise expanding if  1 .there is a finite set of points such that is and can be extended to a on  ] defined by  (x)=\sum_{y\in t^{-1}x}\frac{f(y)}{t^{\prime } ( y)}.\ ] ]  we now show how to find a bound for such density , starting from the description of the system , and then compute the associated invariant measure .the following proposition was proved in the celebrated paper  ( thm . 1 and its proof ) andit is now called lasota - yorke inequality .we give a precise statement where the constants are explicited .[ proe]let a piecewise hyperbolic map .if is of bounded variation in  ] .then -\{d_{1}, ... ,d_{n}\}}{\sup } ( |\frac{t^{\prime \prime } ( x)}{(t^{\prime } ( x))^{2}}|)}{\underset{x\in \lbrack 0,1]-\{d_{1}, ... ,d_{n}\}}{\inf } ( \frac{1}{t^{\prime } ( x)})}+\frac{2}{\min ( d_{i}-d_{i+1})}.\ ] ]  the following is an elementar fact about the behavior of real sequences  [ sequ]if a real sequence is such that for some , then  [ densibound]if is the density of the physical measure of a piecewise expanding map as above and .then is defined as above .( sketch ) the topological mixing assumption implies that the map has only one invariant physical measures ( see ) .let us use the above results iterating the constant density corresponding to the lebesgue measure .proposition [ proe ] and lemma [ sequ ] give that the variation of the iterates is bounded by .suppose that the limit measure has density . by compactness of in ,the above properties give a bound on the variation of ( see proof of thm .1 ) .  the following is a trivial consequence of the fact that .  in the above situation .the bound on the density of the invariant measure , together with corollary [ prop_dim ] gives the following _ uniform _ result on the computation of invariant measures of such maps .[ pwexpmu]suppose a piecewise expanding map and also its derivative is computable on -\{d_{1}, ...,d_{n}\} ] are computable .then , the physical measure can be computed starting from a description of the system ( the program computing the map and its derivative ) .( sketch ) since  and are computable on each interval  ] is hence a computable dynamical system . is non - decreasing , and if and only if .this system has a physical ergodic invariant measure which is , the dirac measure placed on .the measure is physical because attracts all the interval at its left .since is not computable then is not computable .we remark that coherently with the previous theorems is not isolated .it is easy to prove , by a simple dichotomy argument , that a computable function from $ ] to itself must have a computable fixed point .hence it is not possible to construct a system over the interval having no computable invariant measure ( we always have the over the fixed point ) . with some more workwe will see that such an example can be constructed on the circle .indeed the following can be established  [ bomb ] there is a computable , continuous map on the circle having no computable invariant probability measure .  for the description of the system and for applications to reverse mathematics see ( proposition 12 ) .let be a compact subset of the plane . informally speaking , in order to draw the set on a computer screen we need a program which is able to decide , given some precision , if pixel has to be colored or not . by representing pixel by a ball where , the question one would like to answer is : does intersects ?the following definition captures exactly this idea :  a compact set is said to be * _ computable _ * if there is an algorithm such that , upon input :  * halts and outputs `` _ _ yes __ '' if , * halts and outputs `` _ _ no _ _ '' if , * run forever otherwise .we remark that if a compact set is computable under the definition above , then it is recursively compact in the sense of definition [ rec.compact ] .the converse is however false .the question of whether julia sets are computable under this definition has been completely solved in a series of papers by binder , braverman and yampolsky .see . herewe review some of their results . for simplicity ,we give the definition of the julia set only for quadratic polynomials .consider a quadratic polynomial   obviously , there exists a number such that if , then the iterates of under will uniformly scape to .the * _ filled julia set _ * is then defined by :    that is , the set of points whose orbit remains bounded . for the filled julia set one has the following result :  the filled julia set is always computable .  the * _ julia set _ * can be now defined by :    where denotes the boundary of .the julia set is the _ repeller _ of the dynamical system generated by . for all but finitely many points ,the limit of the -th preimages coincides with the julia set .the dynamics of on the set is chaotic , again rendering numerical simulation of individual orbits impractical . yetjulia sets are among the most drawn mathematical objects , and countless programs have been written for visualizing them .  in spite of this ,the following result was shown in .there exist _ computable _ quadratic polynomials such that the julia set is _ not computable . _this phenomenon of _ non - computability _ is rather subtle and rare . for a detailed exposition ,the reader is referred to the monograph .thus , we can not accurately simulate the set of limit points of the preimages , but what about their statistical distribution ?the question makes sense , as for all and every continuous test function , the averages where is the _ brolin - lyubich probability measure _ b65 , l82 supported on the julia set .we can thus ask whether the the brolin - lyubich measure is computable .even if is not a computable set , the answer does not _ a priori _ have to be negative .in fact , the following result holds :  the brolin - lyubich measure is always computable .the proof of the previous result does not involve ( perhaps surprisingly ) much analytic machinery , but it follows from some general principles in the same spirit of the ideas introduced in section [ section.measures ] .another important object in complex dynamics is the mandelbrot set , which is defined as follows :    that is , the set of parameters of quadratic polynomials for which the orbit of remains bounded . as for julia sets ,there exists many computer programs to visualize it , and the question arise whether this set is actually computable in a rigorous sense .this is , up to date , an open question .however , some partial results have been obtained .for instance , in it is proved that the mandelbrot set is recursively compact in the sense of definition [ rec.compact ] .there exists also analytical questions about the mandelbrot set that remain unsolved .for instance , it is unknown whether it is _ locally connected_. it has been conjectured that this is indeed the case .computability of the mandelbrot set is closely related to this conjecture , as it has been observed in :  if the mandelbrot set is locally connected , then it is computable .it is known that local connectivity implies another well known conjecture , namely the _ hyperbolicity conjecture _ , which in turn also implies computability .for a detailed exposition in this subject the reader is referred to .as recalled before , the birkhoff ergodic theorem tells that , if the system is ergodic , there is a full measure set of points for which the averages of the values of the observable along its trajectory ( time averages ) coincides with the spatial average of the observable .similar results can be obtained for the convergence in the norm , and others . * * * * many , more refined results are linked to the speed of convergence of this limit . andthe question naturally arise , if there is a possibility to compute this speed of convergence in a sense similar to definition recconv .  in the paper abstract results imply that in a computable ergodic dynamical system , the speed of convergence of such averages can be algorithmically estimated .on the other hand it is also shown that there are non ergodic systems where this kind of estimations are not possible . in a very short proof of this result for ergodic systems is shown .we expose the precise result ( theorem [ corollaryx ] ) and the short proof in this section .we first precise what is meant by `` compute the speed '' in some pointwise a.e . convergence .  a * _ random variable _ * on is a measurable function .  random variables * _ effectively converge in probability _ * to if for each , converges effectively to , uniformly in . that is , there is a computable function such that for all , .[ def_rec_conv]random variables * _ effectively converge almost surely _ * to if effectively converge in probability to .[ a.e.compufunct]a * _ computable probability space _ * is a pair where is a computable metric space and a computable borel probability measure on .let be a computable metric space .a function is * _ almost everywhere computable _ * ( a.e .computable for short ) if it is computable on an effective -set of measure one , denoted by and called the _ domain of computability of .  a * _ morphism _ * of computable probability spaces is a morphism of probability spaces which is a.e . computable .[ com_sum ] a sequence of functions is uniformly a.e .computable if the functions are uniformly computable on their respective domains , which are uniformly effective -sets .observe that in this case , intersecting all the domains provides an effective -set on which all are computable . in the followingwe will apply this principle to the iterates of an a.e .computable function , which are uniformly a.e .computable .[ l1comp ] the space ( resp . ) can be made a computable metric space , choosing some dense set of bounded computable functions as ideal elements .we say that an integrable function is -computable if its equivalence class is a computable element of the computable metric space . of course , if -a.e . , then is -computable if and only if is .basic operations on , such as addition , multiplication by a scalar , , etc .are computable . moreover ,if preserves and is a.e .computable , then ( from to ) is computable ( see ) .  let us call a * _ computable ergodic system _ * if is a computable probability space where is a measure preserving morphism and is ergodic .let denote the norm or the norm .let be a computable ergodic system .let be a computable element of ( resp . ) .  the convergence ( resp . convergence ) of the birkhoff averages is effective .that is : there is an algorithm such that for each  .moreover the algorithm depends effectively on .  * proof . * replacing with , we can assume that .the sequence is computable ( see remark [ l1comp ] ) and converges to by the ergodic theorems .  given , we write as with . then    let .we can compute some such that .then we can compute some .the function is computable and for all , .   now we use the above result to find a computable estimation for the a.s .speed of convergence .[ corollaryx]let be a computable ergodic system .if is -computable , then the a.s .convergence is effective .moreover , the rate of convergence can be computed as above starting from .  this will be proved by the following  [ theorem_ergodic_effective_as_convergence ] if is -computable as above , and is bounded , then the almost - sure convergence is effective ( uniformly in and a bound on and on , ) .  to prove thiswe will use the maximal ergodic theorem :  [ lemma_maximal ] for and ,   the idea is simple : compute some such that is small , apply the maximal ergodic theorem to , and then there is , that can be computed , such that is close to for .* proof * ( of proposition [ theorem_ergodic_effective_as_convergence ] ) let .compute such that . applying the maximal ergodic theorem to :   now , is not far from : expanding , one can check that  . so if , then . as a result , if for some , then . from ( [ bound_g ] ) , we then derive   as can be computed from and , we get the result .  [ remarkboundunif]this result applies uniformly to a uniform sequence of computable observables .we now extend this to -computable functions , using the density of in .* proof . *( of theorem [ corollaryx ] ) let . for ,let us consider defined as   compute such that . applying proposition [ theorem_ergodic_effective_as_convergence ] to gives some such that . applying lemma lemma_maximal to gives .as a result , .  [ remarkcompul1 ] we remark that a bounded a.e .computable function , as defined in definition [ a.e.compufunct ] is a computable element of ( see ) .conversely , if is a computable element of then there is a sequence of uniformly computable functions that effectively converge -a.e . to  again be a computable metric space and a computable probability measure on it .suppose is complete .points satisfying the above recalled pointwise ergodic theorem for an observable , will be called _ _ typical for_ _ _ _ _ _ .points which are typical for each which is continuous with compact support are called _ _ typical for the measure __ ( and for the dynamics ) .the set of computable points in ( see definition [ comp_points ] ) being countable , is a very small ( invariant ) set , compared to the whole space .for this reason , a computable point could be rarely be expected to be typical for the dynamics , as defined before .more precisely , the birkhoff ergodic theorem and other theorems which hold for a full measure set , can not help to decide if there exist a computable point which is typical for the dynamics .nevertheless computable points are the only points we can use when we perform a simulation or some explicit computation on a computer .a number of theoretical questions arise naturally from all these facts . due to the importance of forecasting and simulation of a dynamical system s behavior, these questions also have some practical motivation .  since simulations can only start with computable initial conditions , given some typical statistical behavior of a dynamical system ,is there some computable initial condition realizing this behavior ? how to choose such points ?such points could be called _ pseudorandom_ points , and a result showing its existence and their computability from the description of the system could be seen as a _ constructive _ version of the pointwise ergodic theorem .meaningful simulations , showing typical behaviors of the dynamics can be performed if computable , pseudorandom initial conditions exist ( and can be computed from the description of the system ) .thanks to a kind of effective borel - cantelli lemma , in the above problem was solved affirmatively for a class of systems which satisfies certain technical assumptions which includes systems whose decay of correlation is faster that .after the results on the estimation of the rate of convergence given in the previous section we can remove the technical assumption on the speed of decay of correlations. we will moreover show how the result is uniform also in and ( the pseudorandom points will be calculated from a description of the system ) .the following result ( , theorem 2 or ) shows that given a sequence which converges effectively a.s . to and given its speed of convergence then there it is possible to compute points for which .[ theorem_convergence_borel_cantelli ] let be a complete metric space .let be uniformly a.e .computable random variables .if effectively converges almost surely to then the set contains a sequence of uniformly computable points which is dense in .moreover , the effective sequence found above depends algorithmically on and on the function giving the rate of convergence  since by the results of the previous section , can be calculated starting from , and .this result hence can be directly used to find typical points for the dynamics .indeed the following holds ( see for the details )  [ thm1]if is a computable ergodic system , is and a.e .computable then there is a uniform sequence of computable points which is dense on the support of such that for each  this sequence can be computed starting from a description of , and .  since the above result is uniform in and it is possible to construct a r.e .set of computable observables which is dense in the space of compactly supported continuous functions we can also obtain the following ( see again for the details )  if is a computable ergodic system then there is a uniform sequence of computable points which is dense on the support of such that for each , is .moreover this sequence can be computed starting from a description of   now , as an application we get together the uniform results about computation of invariant measures for the class of piecewise expanding functions shown in subsection [ pwsec ] to show that in that class the pseudorandom points can be calculated starting from a description of the map . indeed since in that class the physical invariant measure can be calculated starting from ( see theorem [ pwexpmu ] ) .we hence obtain  each piecevise expanding map , with computable derivative , as in the assumptions of theorem [ pwexpmu ] has a sequence of pseudorandom points which is dense in the support of the measure .moreover this sequence can be computed starting from the description of the system .  hence establishing some kind of constructive versions of the ergodic theorem .in this article we have reviewed some results about the rigorous computation of invariant measures , invariant sets and typical points .here , the sentence _ rigorous computation _ means `` computable by a turing machine '' .thus , this can be seen as a theoretical study of which infinite objects in dynamics can be arbitrarily well approximated by a modern computer ( in an absolute sense ) , and which can not . in this line, we presented some general principles and techniques that allow the computation of the relevant objects in several different situations . on the other hand, we also presented some examples in which the computation of the relevant objects is not possible at all , stating some theoretical limits to the abilities of computers when used to simulate dynamical systems .the examples of the second kind , however , seem to be rather rare .an important question is therefore whether this phenomenon of non - computability is robust or prevalent in any sense , or if it is rather exceptional .for example , one could ask whether the non - computability is destroyed by small changes in the dynamics or whether the non - computability occurs with small or null probability .  besides , in this article we have not considered the efficiency ( and therefore the feasibility ) of any of the algorithms we have developed .an important ( and more difficult ) remaining task is therefore the development of a _ resource bounded _version of the study presented in this paper . in the case of julia sets , for instance, it has been shown in that hyperbolic julia sets , as well as some julia sets with parabolics , are computable in polynomial time .on the other hand , in bby06 it was shown that there exists computable siegel quadratic julia sets whose time complexity is arbitrarily high .  for the purpose to compute the invariant measure form the description of the system , in section [ pwsec ] we had to give explicit estimations on the constants in the lasota yorke inequality .this step is important also when techiques different from ours are used ( see e.g. ) .similar estimations could be done in other classes of systems , following the way the lasota yorke inequality is proved in each class ( although , sometimes this is not a completely trivial task and requires the effectivization of some step in the proof ) . a more general method to have an estimation for the constants or other ways to getinformation on the regularity of the invariant measure would be useful .g froyland _ extracting dynamical behaviour via markov models _n alistair mees , editor , nonlinear dynamics and statistics : proceedings , newton institute , cambridge , 1998 , pages 283 - 324 , birkhauser , 2001 .p. gcs , m. hoyrup and c. rojas , _ randomness on computable probability spaces a dynamical point of view _ ,( susanne albers and jean - yves marion , editors ) , in 26 international symposium on theoretical aspects of computer science , ( stacs 2009 ) , 469480 .s. galatolo , m. hoyrup and c. rojas _ computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems _ proc .conf . on computability and complexity in analysis http://arxiv.org/abs/1006.0392v1 ( 2010 ) .

Abstract: we survey an area of recent development , relating dynamics to theoretical computer science . we discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems . we will focus on central objects of the theory of dynamics , as invariant measures and invariant sets , showing that even if they can be computed with arbitrary precision in many interesting cases , there exists some cases in which they can not . we also explain how it is possible to compute the speed of convergence of ergodic averages ( when the system is known exactly ) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour ( a sort of constructive version of the pointwise ergodic theorem ) .