A Denjoy Theorem for commuting circle diffeomorphisms with mixed Holder
  derivatives
arXiv:0704.1006v1  [math.DS]  8 Apr 2007
A Denjoy Theorem for commuting circle diffeomorphisms with
mixed Hölder derivatives
Victor Kleptsyn & Andrés Navas
Abstract. We prove that if d ≥ 2 is an integer number and fk, k ∈ {1, . . . , d}, are C
1+τk commuting
circle diffeomorphisms with τk ∈]0, 1[ and τ1 + · · · + τd > 1, then the fk’s are simultaneously
(topologically) conjugate to rotations provided that their rotation numbers are independent over
the rationals.
Keywords: Denjoy Theorem, centralizers, Hölder derivative.
Introduction
Starting from the seminal works by Poincaré [13] and Denjoy [3], a deep theory for the dy-
namics of circle diffeomorphisms has been developed by many authors [1, 7, 8, 17], and most of
the fundamental related problems have been already solved. Quite surprisingly, the case of several
commuting diffeomorphisms is rater special, as it was pointed out for the first time by Moser [9]
in relation to the problem of the smoothness for the simultaneous conjugacy to rotations. Roughly
speaking, in this case it should be enough to assume a joint Diophantine condition on the rotation
numbers which does not imply a Diophantine condition for any of them (see the recent work [5] for
the solution of the C∞ case of Moser’s problem).
A similar phenomenon concerns the classical Denjoy Theorem. Indeed, in [4] it was proved
that if d ≥ 2 is an integer number and τ > 1/d, then the elements f1, . . . , fd of any family of
C1+τ commuting circle diffeomorphisms are simultaneously (topologically) conjugate to rotations
provided that their rotation numbers are independent over the rationals (that is, no non trivial
linear combination of them with rational coefficients equals a rational number). In other words,
the classical (and nearly optimal) C2 hypothesis for Denjoy Theorem can be weakened in the case
of several commuting diffeomorphisms. The first and main result of this work is a generalization of
this fact to the case of different regularities.
Theorem A. Let d ≥ 2 be an integer number and τ1, . . . , τd be real numbers in ]0, 1[ such that
τ1+ · · ·+ τd > 1. If fk, k ∈ {1, . . . , d}, are respectively C
1+τk circle diffeomorphisms which have ro-
tation numbers independent over the rationals and which do commute, then they are simultaneously
(topologically) conjugate to rotations.
Since the probabilistic arguments of [4] cannot be applied to the case of different regularities, the
preceding result is much more than a straightforward generalization of Theorem A of [4]. Indeed,
for the proof here we use a key new argument which is somehow more deterministic.
Theorem A is (almost) optimal (in the Hölder scale), in the sense that if one decreases slightly the
regularity assumptions then it is no longer true. The following result relies on classical constructions
by Bohl [2], Denjoy [3], Herman [7], and Pixton [12], and its proof consists on an easy extension of
the construction given by Tsuboi in [16].
http://arxiv.org/abs/0704.1006v1
Denjoy Theorem for commuting diffeomorphisms 2
Theorem B. Let d ≥ 2 be an integer number and τ1, . . . , τd be real numbers in ]0, 1[ such that
τ1 + · · ·+ τd < 1. If ρ1, . . . , ρd are elements in R/Z which are independent over the rationals, then
there exist C1+τk circle diffeomorphisms fk, k ∈ {1, . . . , d}, having rotation numbers ρk, which do
commute, and such that none of them is topologically conjugate to a rotation.
It is well known that the techniques developed for Denjoy Theory can be applied to the study of
group actions on the interval. In this direction we should point out that the methods of this paper
also allow to extend (in a straightforward way) the so called “Generalized Kopell Lemma” and
the “Denjoy-Szekeres Type Theorem” (Theorems B and C of [4] respectively) for Abelian groups
of interval diffeomorphisms under analogous hypothesis of different regularities. Furthermore, the
construction of counter-examples for both of them when these hypothesis do not hold can be also
extended to this context. We leave the verification of all of this to the reader.
Acknowledgments. It is a pleasure to thank Bassam Fayad and Sergey Voronin for their encour-
agements, as well as the Independent University of Moscow for the hospitality during the conference
“Laminations and Group Actions in Dynamics” held in February 2007. The first author was sup-
ported by the Swiss National Science Foundation. This work was also funded by the RFBR grants
7-01-00017-a and CNRS-L−a 05-01-02801, and by the CONICYT grant 7060237.
1 A general principle revisited
As it is well known since the classical works by Denjoy, Schwartz and Sacksteder [3, 14, 15], if I is
a wandering interval1 for the dynamics of a finitely generated semigroup Γ of C1+lip diffeomorphisms
of the closed interval or the circle (on which we will always consider the normalized length), one
can control the distortion of the elements of Γ over (a slightly larger interval than) I in terms of
the sum of the lengths of the images of I along the corresponding sequence of compositions and a
uniform Lipschitz constant for the derivatives of the (finitely many) generators of Γ. If τ belongs
to ]0, 1[ and Γ consists of C1+τ diffeomorphisms, the same is true provided that the sum of the
τ -powers of the lengths of the corresponding images of I is finite (this last condition does not follow
from the disjointness of these intervals !): see for instance [4], Lemma 2.2. It is not difficult to
prove a similar statement for the case of different regularities, and this is precisely the content of
the following lemma. However, to the difference of [4], here we will deal with finite sequences of
compositions by a technical reason which will be clear at the end of the next section.
Lemma 1.1. Let Γ be a semigroup of (orientation preserving) diffeomorphisms of the circle or the
closed interval which is generated by finitely many elements gk, k ∈ {1, . . . , l}, which are respectively
of class C1+τk , where τk∈]0, 1]. Let Ck denote the τk-Hölder constant of the function log(g
k), and
let C = max{C1, . . . , Cl} and τ = max{τ1, . . . , τl}. Given n0 ∈ N, for each n ≤ n0 let us chose
kn ∈ {1, . . . , l}, and for a fixed interval I let S > 0 be a constant such that
∣gkn · · · gk1(I)
τkn+1 . (1)
If n ≤ n0 is such that gkn · · · gk1(I) does not intersect I but is contained in the L-neighborhood of
I, where L := |I|/2 exp(2τCS), then gkn · · · gk1 has a hyperbolic fixed point.
1We say that an interval is wandering if its images by different elements of the underlying semigroup are disjoint.
Denjoy Theorem for commuting diffeomorphisms 3
Proof. Let J = [a, b] be the (closed) 2L-neighborhood of I, and let I ′ (resp. I ′′) the connected
component of J \I to the right (resp. to the left) of I. We will prove by induction on j∈{0, . . . , n0}
that the following two conditions are satisfied:
(i)j |gkj · · · gk1(I
′)| ≤ |gkj · · · gk1(I)|,
(ii)j sup{x,y}⊂I∪I′
(gkj ···gk1)
(gkj ···gk1)
≤ exp(2τ CS).
Condition (ii)0 is trivially satisfied, whereas condition (i)0 is satisfied since |I
′| = 2L ≤ |I|.
Assume that (i)i and (ii)i hold for each i ∈ {0, . . . , j − 1}. Then for every x, y in I ∪ I
′ we have
(gkj · · · gk1)
(gkj · · · gk1)
∣ log(g′ki+1(gki · · · gk1(x))) − log(g
(gki · · · gk1(y)))
Cki+1
∣gki · · · gk1(x)− gki · · · gk1(y)
τki+1
|gki · · · gk1(I)|+ |gki · · · gk1(I
)τki+1
≤ C 2τ
|gki · · · gk1(I)|
τki+1
≤ C 2τS.
This shows (ii)j. To verify (i)j first note that there must exist x ∈ I and y ∈ I
′ such that
|gkj · · · gk1(I)| = |I| · (gkj · · · gk1)
′(x) and |gkj · · · gk1(I
′)| = |I ′| · (gkj · · · gk1)
′(y).
Therefore, by (ii)j ,
|gkj · · · gk1(I
|gkj · · · gk1(I)|
(gkj · · · gk1)
(gkj · · · gk1)
|I ′|
≤ exp(2τCS)
|I ′|
which proves (i)j . Obviously, similar arguments show that (i)j and (ii)j also hold for every
j ∈ {0, . . . , n0} when we replace I
′ by I ′′.
Now for simplicity let us denote hj = gkj · · · gk1 . Assume that hn(I) is contained in the L-
neighborhood of the interval I (see Figure 1). Then property (i)n gives hn(J) ⊂ J , and this already
implies that hn has a fixed point x in J . (The reader will see that the existence of this fixed point
together with the fact that hn 6= id is the only information that we will retain for the proof of
Theorem A.)
To conclude we would like to show that the fixed point x is hyperbolic. To do this just note
that, if hn(I) does not intersect I, then there exists y ∈ I such that
h′n(y) =
|hn(I)|
Therefore, by (ii)n,
h′n(x) ≤ h
n(y) exp(2
τCS) ≤
L exp(2τCS)
and this finishes the proof. �
Denjoy Theorem for commuting diffeomorphisms 4
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hn hn
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Figure 1
hn(I)I
hyperbolic
fixed point
2 Proof of Theorem A
Recall the following well known argument (see for instance [6], Proposition 6.17, or [11], Lemma
4.1.4). If f1, . . . , fd are commuting circle homeomorphisms, then there is a common invariant
probability measure µ on S1. Moreover, if the rotation number of at least one of them is irrational,
then there is no finite orbit for the group action, and the measure µ has no atom. Therefore, the
distribution function
Fµ : S
1 → R/Z, Fµ(x) := µ([0, x[),
gives a (simultaneous) semiconjugacy between the maps f1, . . . , fd and the rotations corresponding
to their rotation numbers. Thus, for the proof of Theorem A we have to show that this semiconju-
gacy is in fact a conjugacy, and our strategy for proving this (under the hypothesis of the Theorem)
is the classical one and goes back to Schwartz [15]. Indeed, in the contrary case the support of
µ would be a (minimal) invariant Cantor set, and the connected components of its complement
would correspond to the maximal wandering open intervals. Fixing one of these intervals, say I,
we will search for a sequence of compositions hn = fkn · · · fk1 satisfying the hypothesis of Lemma
1.1. This will allow us to conclude that some hn has a (hyperbolic) fixed point, thus implying that
its rotation number is equal to zero. However, this is in contradiction to the fact that the rotation
numbers of the fk’s are independent over the rationals (it is easy to verify that the rotation number
restricted to any group of circle homeomorphisms which preserves a probability measure on S1 is a
group homomorphism: see again [6] or [11]).
In order to ensure the existence of the sequence (hn) the main idea of [4] was to endow the
space of all (infinite) sequences of compositions with a natural probability measure, and then to
prove that the “generic ones” satisfy many nice properties as for instance the convergence of the
sum (1) as n0 goes to infinity. It seems that such a probabilistic argument cannot be applied to
the case of different regularities, and we will need to introduce a new argument which is somehow
more deterministic, since it gives partial information on the sequence that we find. For simplicity
we will first deal with the case d=2.
2.1 The case d = 2
Although not explicitly stated in [4], the main probabilistic argument for the proof of the
Generalized Denjoy Theorem therein is not a dynamical issue, but it is just a statement concerning
the finiteness of the sum of the τ -powers of some positive real numbers. To be more concrete (at
least in the case d = 2 and when τ > 1/2), if (ℓi,j) is a double-indexed sequence of positive numbers
with finite total sum (where i and j are non negative integers), then with respect to some natural
probability distribution on the space of infinite paths (i(n), j(n))n≥0 satisfying i(0) = j(0) = 0,
i(n+1) ≥ i(n), j(n+1) ≥ j(n) and i(n+1)+ j(n+1) = 1+ i(n)+ j(n), one has almost everywhere
Denjoy Theorem for commuting diffeomorphisms 5
the convergence of the sum
ℓτi(n),j(n).
The first goal of this section is to prove the existence of paths sharing a similar property in the
case of different exponents τ1, τ2 in ]0, 1[ (with τ1 + τ2 > 1). A substantial difference here is that
we will construct our sequence by concatenating infinitely many finite paths, and each one of these
paths will be chosen among finitely many ones. To do this we begin with the following elementary
lemma.
Lemma 2.1. Let ℓi,j be positive real numbers, where i ∈ {1, . . . ,m} and j ∈ {1, . . . , n}. Assume
that the total sum of the ℓi.j’s is less than or equal to 1. If τ belongs to ]0, 1[, then there exists
k ∈ {1, . . . , n} such that
ℓτi,k ≤
Proof. We will show that the mean value of the function k 7→
i=1 ℓ
i,k is less than or equal to
m1−τ/nτ , from where the claim of the lemma follows immediately. To do this first note that, by
Hölder’s inequality, for each fixed k ∈ {1, . . . , n} one has
ℓτi,k =
(ℓτi,k)
i=1, (1)
∥(ℓτi,k)
· ‖(1)mi=1‖1/(1−τ) =
m1−τ .
Thus, by using Hölder’s inequality again one obtains
ℓτi,k
, (1)
· ‖(1)
k=1‖1/(1−τ)
which finishes the proof. �
Now we explain the main idea of our construction. Let us assume that the total sum of the
double-indexed sequence of positive numbers ℓi,j is ≤ 1, and suppose that the numbers τ1∈]0, 1[ and
τ2∈]0, 1[ such that τ1+τ2 > 1 are fixed. Denoting by [[a, b]] the set of integers between a and b (with
a and b included when they are in Z), let us consider any sequence of rectangles Rm ⊂ N0×N0 such
that R0 = {(0, 0)}, R2m+1 = [[im, im+1]]× [[jm, jm+2]] and R2m+2 = [[im, im+2]]× [[jm+1, jm+2]],
where (im)m≥1 and (jm)m≥1 are strictly increasing sequences of non negative integers numbers
satisfying i0 = i1 = 0 and j0 = j1 = 0 (see Figure 2). Denoting by Xm and Ym respectively the
number of points on the horizontal and vertical sides of each Rm, a direct application of Lemma
2.1 gives us, for ε := 1− τ1 − τ2 > 0 and each m ≥ 0:
Denjoy Theorem for commuting diffeomorphisms 6
– an integer r(2m+ 1) ∈ [[im, im+1]] such that
r(2m+1),j
Y 1−τ22m+1
Y τ12m+1
· Y −ε2m+1,
– an integer r(2m+ 2) ∈ [[jm+1, jm+2]] such that
i,r(2m+2)
Y τ12m+2
Y τ12m+2
·X−ε2m+2.
Starting from the origin and following the corresponding horizontal and vertical lines, we find
an infinite path (i(n), j(n))n≥0 satisfying
i(0) = j(0) = 0, i(n + 1) ≥ i(n), j(n+ 1) ≥ j(n), i(n+ 1) + j(n + 1) = 1 + i(n) + j(n),
and such that the sum
τα(n)
i(n),j(n)
is bounded by
· Y −ε2m+1 +
·X−ε2m+2
, (3)
where α(n) := 1 if |i(n + 1)− i(n)| = 1 and α(n) := 2 if |j(n + 1)− j(n)| = 1.
Figure 2
i0 = i1 i2 i3 i4 i5
j0 =j1
•••••
•••••••••••••••••••••••
•••••••••••••••••••••••••••••
R5 R6
R7 R8
Denjoy Theorem for commuting diffeomorphisms 7
Now let us consider any choice such that im = [4
mτ1 ] and jm = [4
mτ2 ] for m large enough.
Writing am ≃ bm when (am) and (bm) are sequences of positive numbers such that (am/bm) remains
bounded and away from zero, for such a choice we have Xm ≃ 2
mτ1 and Ym ≃ 2
mτ2 . Thus,
(2mτ1)τ2
(2mτ2)τ1
and therefore there exists C > 0 such that, for each m ≥ 0,
Y τ1m
This implies that the sum in (3) is bounded by
S := C
 = C
4τ2ε − 1
4τ1ε − 1
, (4)
and so the value of the sum (2) is finite (and also bounded by S).
We can now proceed to the proof of Theorem A in the case d=2. Assume by contradiction that
fk, k∈{1, 2}, are respectively C
1+τk commuting circle diffeomorphisms which are not simultaneously
conjugate to rotations and which have rotation numbers independent over the rationals. Let I be a
connected component of the complement of the invariant minimal Cantor set for the group action,
and let ℓi,j = |f
2 (I)|. We obviously have
i,j ℓi,j ≤ 1, and so we can apply all our previous
discussion to this sequence. In particular, there exists an infinite path (i(n), j(n)) starting at the
origin and such that the sum
τα(n)
i(n),j(n)
is bounded by the number S > 0 defined by (4). If for n ≥ 1 we let kn = α(n − 1) ∈ {1, 2}, then
we obtain a sequence of compositions hn = fkn · · · fk1 such that the preceding sum coincides term
by term with
|fkn · · · fk1(I)|
τkn+1 .
Thus, in order to apply Lemma 1.1 to get a contradiction, we just need to verify that, for some
n ≥ 1, the hypothesis that hn(I) = fkn · · · fk1(I) is contained in the L-neighborhood of I is satisfied
(where L := |I|/2 exp(2τCS), τ := max{τ1, τ2}, and C := max{C1, . . . , Cd}, with Ck being the τk-
Hölder constant for the function log(f ′k)).
To to this first note that, if we collapse all the connected components of the complement of the
minimal invariant Cantor set, then we obtain a topological circle Ŝ1 on which the original diffeomor-
phisms induce naturally minimal homeomorphisms f̂1 and f̂2 which are simultaneously conjugate
to rotations. Moreover, the L-neighborhood of I becomes a non degenerate interval Û ; thus, there
exists N ∈ N such that the intervals f̂−11 (Û), . . . , f̂
1 (Û), as well as f̂
2 (Û), . . . , f̂
2 (Û), cover the
circle Ŝ1. This easily implies that for any image I0 of I by some element of the semigroup generated
by f1 and f2 there exists k and k
′ in {1, . . . , N} such that fk1 (I0) and f
2 (I0) are contained in the
L-neighborhood of I. Now it is easy to see that, for the sequence of compositions that we found,
for every N̄ ∈ N there exists some integer r ∈ N such that kr = kr+1 = . . . = kr+N̄ . For N̄ = N
this obviously implies that at least one of the intervals hr+1(I), . . . , hr+N (I) is contained in the
L-neighborhood of I, thus finishing the proof.
Denjoy Theorem for commuting diffeomorphisms 8
Figure 3
•••••••••
•••••••••••••
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
R′2 R
R′4 R
R′6 R
x′0 x
y′0=y
y′2=y
y′4=y
y′6=y
We would like to close this section by giving a different type of choice for the sequence of
rectangles which is simpler to describe and for which the preceding arguments are also valable. (For
simplicity, we will use a similar construction to deal with the case d > 2, altough the preceding
one still applies). This sequence (R′m)m≥0 is of the form [[0, x
m]] × [[0, y
m]], where (x
m) and (y
are non decreasing sequences of positive integer numbers such that x′0 = y
0 = 0, x
m > x
m−1 and
y′m = y
m−1 if m is odd, and x
m = x
m−1 and y
m > y
m−1 if m is even. If (ℓi,j) is a double-indexed
sequence of positive real numbers with total sum ≤ 1, we chose these integer numbers in such a
way that x′2m+1 = x
2m+2 = [4
mτ1 ] and y′2m = y
2m+1 = [4
mτ2 ] for m large enough. As before, inside
the rectangle Rm there is a “good” vertical (resp. horizontal) segment of line Lm for m even (resp.
odd). Therefore, for each M0 ∈ N we can concatenate these segments between Lm−1 ∩ Lm and
Lm ∩ Lm+1 at the m
th step for m < M0, and between LM0−1 ∩ LM0 and the point of LM0 on the
boundary of RM0 at the last step (see Figure 3). In this way we obtain a path (starting at the
origin) of finite length n(M0)− 1 for which the sum
n(M0)−1
τα(n)
i(n),j(n)
is bounded by some number S > 0 which is independent of M0.
Now let fk, k ∈{1, 2}, be two commuting circle diffeomorphisms of class C
1+τk which are not
simultaneously conjugate to rotations. Fix again one of the maximal wandering open intervals for
the dynamics, say I, and let ℓi,j = |f
2 (I)|. (Note that
i,j ℓi,j ≤ 1.) The method above gives us
a family of finite paths, and each of these paths determines uniquely a sequence of compositions.
Remark however that there is a little difference here, since we allow the use of the inverses of f1
and f2. Therefore, in order to apply Lemma 1.1, we will need to consider now {f1, f
1 , f2, f
2 } as
being our system of generators, and therefore we put τ = max{τ1, τ2} and C = max{C1, C2, C
where Ci (resp. C
i) is a τi-Hölder constant for the function log(f
i) (resp. log((f
′)). As in the
previous proof, we need to verify that, for some M0 ∈ N, there exists a non trivial element in the
sequence of compositions (hn) associated to its corresponding finite path which sends I inside the
L-neighborhood of itself, where L := |I|/2 exp(2τCS). As before, for proving this it suffices to show
that for every N there exists r ∈ N such that one has hr+i+1 = f1hr+i for each i ∈ {0, . . . , N − 1},
or hr+i+1 = f2hr+i for each i ∈ {0, . . . , N − 1}. However, this last property is always satisfied if
Denjoy Theorem for commuting diffeomorphisms 9
M0 is big enough so that the number of points with integer coordinates in the line segment LM0
contained in RM0 \ RM0−1 is greater than N . Note that it is in this last argument where we use
the fact that we keep only finite sequences of compositions, altough our method combined with a
diagonal type argument easily shows the existence of an infinite sequence for which the sum (2)
converges.
2.2 The general case
In the case d = 2, the “good” paths leading to the sequence of compositions which allows to apply
Lemma 1.1 were obtained by concatenating horizontal and vertical lines. When d > 2 we will need
to concatenate lines in several (namely d) directions, and the geometrical difficulty for doing this
is evident: in dimension bigger than 2, two lines in different directions do not necessarily intersect.
To overcome this difficulty we will use the fact that, at each step (i.e. inside each rectangle), there
is not only one finite path which is good, but this is the case for a “large proportion” of finite paths.
We first reformulate Lemma 2.1 in this direction.
Lemma 2.2. Let ℓi,j be positive real numbers, where i ∈ {1, . . . ,m} and j ∈ {1, . . . , n}. Assume
that the total sum of the ℓi.j’s is less than or equal to 1. If τ belongs to ]0, 1[ and A > 1, then for
a proportion of indexes k ∈ {1, . . . , n} greater than or equal to (1− 1/A) we have
ℓτi,k ≤ A
Proof. As in the proof of Lemma 2.1, the mean value of the function
ℓτi,k (5)
is less than or equal to m1−τ/nτ . The claim of the lemma then follows as a direct application
of Chebychev’s inequality: the proportion of points for which the value of (5) is greater than this
mean value times A cannot exceed 1/A. �
Now let (ℓi1,...,id) be a multi-indexed sequence of positive real numbers having total sum ≤ 1,
and let τ1, . . . , τd be real numbers in ]0, 1[. Starting with R0 = [[0, 0]]
d, let us consider a sequence
(Rm)m≥0 of rectangles of the form Rm = [[0, x1,m]]× · · · × [[0, xd,m]] satisfying xk,m ≥ xk,m−1 for
each k ∈ {1, . . . , d}, with strict inequality if and only if k ≡ m (mod d). For each m ≥ 1 denote by
s(m) ∈ {1, . . . , d} the residue class (mod d) of m, and denote by Fm the face
[[0, x1,m]]× · · · × [[0, xs(m)−1,m]]× {0} × [[0, xs(m)+1,m]]× · · · × [[0, xd,m]]
of Rm. For each (i1, . . . , is(m)−1, 0, is(m)+1, . . . , id) belonging to this face Fm we consider the sum
xs(m),m
τs(m)
i1,...,is(m)−1,j,is(m)+1,...,id
By Lemma 2.2, if Am > 1 then the proportion of points in Fm for which this sum is bounded by
(1 + xs(m),m)
1−τs(m)
j 6=s(m)
(1 + xj,m)
τs(m)
= Am ·
1−τs(m)
s(m),m
j 6=s(m)
τs(m)
is at least equal to (1− 1/Am), where Xj,m := 1+xj,m. In order to concatenate the corresponding
lines we will use the following elementary lemma.
Denjoy Theorem for commuting diffeomorphisms 10
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Figure 4
(i1, . . . , is(m+1)−2, 0, 0, is(m+1)+1, . . . , id)
admissible in Cm
(i1, . . . , is(m+1)−1, 0, 0, is(m+1)+2, . . . , id)
admissible in Cm+1
Lemma 2.3. Let us chose inside each rectangle (Rm)m≥1 a set L(m) of (complete) lines in the
corresponding s(m)-direction whose proportion (with respect to all the lines in that direction inside
(Rm)) is at least (1 − 1/Am). If M0∈N is such that
m=1 1/Am<1, then there exists a sequence
of lines Lm ∈ L(m), m ∈ {0, . . . ,M0}, such that Lm+1 intersects Lm for every m < M0.
Proof. Let us denote by Cm the (d− 2)-dimensional face of Rm given by
[[0, x1,m]]× · · · × [[0, xs(m)−1,m]]× {0} × {0} × [[0, xs(m)+2,m]]× · · · × [[0, xd,m]].
Call a point (i1, . . . , is(m)−1, 0, 0, is(m)+2 , . . . , id) ∈ Cm admissible if there exists a sequence of lines
Li∈L(i), i∈{0, . . . ,m}, such that Li intersects Li+1 for every i∈{0, . . . ,m− 1}, and such that Lm
projects in the s(m)-direction into a point (i1, . . . , is(m)−1, 0, is(m)+1, is(m)+2, . . . , id) ∈ Fm for some
is(m)+1∈ [[0, xs(m)+1,m+1]]. We will show that the proportion of admissible points in CM0 is greater
than or equal to
P := 1−
Am > 0.
To prove this, for each m ≥ 0 let us denote by Pm the proportion of admissible points in Cm.
Since R0 reduces to the origin, it suffices to show that, for all m ≥ 0,
Pm+1 ≥ Pm −
To prove this inequality first note that each line Lm+1 ∈ L(m + 1) determines uniquely a point
(i1, . . . , is(m+1)−1, 0, is(m+1)+1, . . . , id)∈Fm+1. The projection into Cm of this line then corresponds
to the point
(i1, . . . , is(m+1)−2, 0, 0, is(m+1)+1 , . . . , id).
If this is an admissible point of Cm then we can concatenate the line Lm+1 to the sequence of
lines corresponding to it (see Figure 4). Now the proportion of lines in L(m + 1) being at least
Denjoy Theorem for commuting diffeomorphisms 11
1− 1/Am+1, the proportion of those lines which project on Cm into an admissible point is at least
equal to
− (1− Pm) = Pm −
By projecting in the (s(m+1)+1)-direction, this obviously implies that the proportion of admissible
points in Cm+1 is also greater than or equal to Pm − 1/Am+1, thus finishing the proof. �
Observe that a sequence of lines Lm as above determines a finite path (starting at the origin) of
points (x1(n), . . . , xd(n)) having non negative integer coordinates such that the distance between
two consecutive ones is equal to 1. Moreover, if we denote by n(M0) the length of this path plus 1,
the corresponding sum
n(N0)−1
τα(n)
x1(n),...,xd(n)
is bounded by
(1 + xs(m),m)
1−τs(m)
i 6=s(m)
(1 + xi,m)
τs(m)
1−τs(m)
s(m),m
j 6=s(m)
τs(m)
, (7)
where α(n) equals the unique index in {1, . . . , d} for which |xα(n)(n+ 1)− xα(n)(n)| = 1.
Now let us define Am=2
εmτs(m)/2A, where A is a large enough constant so that
m≥0 1/Am<1,
and let us consider any choice of the xk,m’s so that Xk,m ≃ 2
mτk . For such a choice we have
j 6=k
= X−εk,m ·
j 6=k
≃ 2−εmτk ·
j 6=k
(2mτk )τj
(2mτj )τk
= 2−εmτk , (8)
where ε := 1 − τ1 − · · · − τd > 0. Therefore, for each M0 ∈ N the preceding lemma provides us a
sequence of lines Lm, m ∈ {0, . . . ,M0}, such that Lm+1 intersects Lm for each m < M0, and such
that the corresponding expression (7) is bounded from above by
2εmτs(m)/2A ·
j 6=k
≤ AC ′
2−εmτs(m)/2 ≤ AC ′
2−εmτ
′/2 =: S < ∞, (9)
where τ ′ := min{τ1, . . . , τd} and C
′ is a constant (independent of M0) giving an upper bound for
the quotient between the left and the right hand expressions in (8).
With all this information in mind we can proceed to the proof of Theorem A in the case d > 2 in
the very same way as in the (second proof for the) case d = 2. Indeed, assume that fk, k ∈ {1, . . . , d},
are circle diffeomorphisms as in the statement of the theorem which are not conjugate to rotations,
and let I be a maximal open wandering interval for the dynamics (i.e. a connected component
of the complement of the minimal invariant Cantor set). Clearly, we can apply all our previous
discussion to the multi-indexed sequence (ℓi1,...,id) defined by ℓi1,...,id = |f
1 · · · f
(I)|. In particular,
for each M0 ∈ N we can find a finite path so that the sum (6) is bounded by the number S > 0
defined by (9) (which is independent of M0). Each such a path induces canonically a finite sequence
of compositions by the fk’s and their inverses. Therefore, in order to apply Lemma 1.1 to get a
contradiction, we need to verify that some of such sequences contains a (non trivial) element hn
which sends I into its L-neighborhood for L := |I|/2 exp(2τCS), where τ := max{τ1, . . . , τd} and
Denjoy Theorem for commuting diffeomorphisms 12
C := max{C1, . . . , Cd, C
1, . . . , C
d}, with Ck (resp. C
k) being the τk-Hölder constant of the function
log(f ′k) (resp. log((f
)′). To ensure this last property let U be the L-neighborhood of I, and let
N ∈ N be such that, given any wandering interval, among the first N iterates of f1, as well as for
f2, . . . , fd, at least one of them sends this interval inside U . If we take M0 large enough so that the
number of points with integer coordinates in LM0 which are contained in RM0 \RM0−1 exceeds N ,
then one can easily see that the associated sequence of compositions contains the desired element
hn. This finishes the proof of Theorem A.
3 Proof of Theorem B
The strategy for the proof of Theorem B is well known. We prescribe the rotation numbers
ρ1, . . . , ρd (which are supposed to be independent over the rationals), we fix a point p ∈ S
1, and for
each (i1, . . . , id) ∈ Z
d we replace the point Ri1ρ1 · · ·R
(p) by an interval Ii1,...,id of length ℓi1,...,id in
such a way that the total sum of the ℓi1,...,id ’s is finite. Doing this we obtain a new circle on which the
rotations Rρk induce nice homeomorphisms if we extend them apropiately to the intervals Ii1,...,id
(outside these intervals the induced homeomorphisms are canonically defined). More precisely, as
it is well explained in [4, 7, 10, 16], if there exists a constant C ′ > 0 so that for all (i1, . . . , id) ∈ Z
and all k ∈ {1, . . . , d} one has
ℓi1,...,1+ik,...,id
ℓi1,...,ik,...,id
i1,...,ik,...,id
≤ C ′, (10)
then one can perform the extension to the intervals Ii1,...,id in such a way the resulting maps fk,
k∈{1, . . . , d}, are respectively C1+τk diffeomorphisms and commute, and moreover their derivatives
are identically equal to 1 on the invariant minimal Cantor set.2 Indeed, one possible extension is
given by fk(x) = (ϕIi1,...,ik,...,id
)−1 ◦ϕIi1,...,1+ik,...,id
(x), where x belongs to the interior of the interval
Ii1...,ik,...,id. Here, ϕI:]a, b[→ R denotes the map
ϕI(x) =
It turns out that a good choice for the lengths is
ℓi1,...,id =
1 + |i1|1/τ1 + · · · |id|
Indeed, on the one hand, if we decompose the sum of the ℓi1,...,id ’s according to the biggest |ij |
we obtain
(i1,...,id)∈Z
ℓi1,...,id ≤ 1 +
|ij |
1/τj ≤ |ik|
for all j ∈ {1, . . . , d}
|ik| ≥ 1
1 + |i1|1/τ1 + · · · |id|
2Condition (10) is also necessary under these requirements. Indeed, there must exist a point in Ii1,...,ik,...,id for
which the derivative of the corresponding map fk equals ℓi1,...,1+ik,...,id/ℓi1,...,ik,...,id . Since the derivative of fk at
the end points of Ii1,...,ik,...,id is assumed to be equal to 1, condition (10) holds for C
′ being the τk-Hölder constant
of the derivative of fk.
Denjoy Theorem for commuting diffeomorphisms 13
and therefore, for some constant C > 0, this sum is bounded by
card{(i1, . . . , id) : |ij |
1/τj ≤ n1/τk for all j∈{1, . . . , d}, ik = n}
1 + n1/τk
≤ 1 + C
n1/τk
j 6=k
nτj/τk = 1 + C
j 6=k τj)/τk
n1/τk
= 1 + C
n(1−τk−ε)/τk
n1/τk
= 1 + C
n1+ε/τk
where ε := 1− (τ1 + · · ·+ τd). (Remark that, since ε > 0, the last infinite sum converges.)
On the other hand, the left hand expression in (10) is equal to
F (i1, . . . , id) :=
|1 + ik|
1/τk − |ik|
1 + |i1|1/τ1 + · · ·+ |1 + ik|
1/τk + · · ·+ |id|
1 + |i1|
1/τ1 + · · · + |ik|
1/τk + · · ·+ |id|
In order to obtain an upper bound for this expression first note that, if ik ≥ 0, then
F (i1, . . . , ik, . . . , id) ≤ F (i1, . . . ,−1− ik, . . . , id).
Therefore, we can restrict to the case where ik < 0. For this case, denoting B = 1 +
j 6=k |ij |
and a = |ik| we have
F (i1, . . . , id) =
a1/τk − (a− 1)1/τk
B + (a− 1)1/τk
B + a1/τk
a1/τk − (a− 1)1/τk
B + (a− 1)1/τk
)1−τk
B + a1/τk
B + (a− 1)1/τk
Both factors in the last expression are decreasing in B. Thus, since B ≥ 1,
F (i1, . . . , id) ≤
a1/τk − (a− 1)1/τk
1 + (a− 1)1/τk
)1−τk
1 + a1/τk
1 + (a− 1)1/τk
Now note that a ≥ 1. For a = 1 the right hand expression above equals 2τk . If a > 1 then the Mean
Value Theorem gives the estimate a1/τk − (a − 1)1/τk ≤ a
/τk, and therefore the preceding
expression is bounded from above by
((a− 1)1/τk )1−τk
a1/τk
(a− 1)1/τk
· 2 =
21/τk
We have then shown that for any (i1, . . . , id) ∈ Z
d one has
F (i1, . . . , id) ≤
21/τk .
In other words, if τ ′ = min{τ1, . . . , τd} then inequality (10) holds for each (i1, . . . , id) ∈ Z
d and
every k ∈ {1, . . . , d} for the constant C ′ = 21/τ
/τ ′, and this finishes the proof of Theorem B.
Denjoy Theorem for commuting diffeomorphisms 14
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Victor Kleptsyn
Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Suisse (Victor.Kleptsyn@math.unige.ch)
Andrés Navas
Universidad de Santiago de Chile, Alameda 3363, Santiago, Chile (andnavas@uchile.cl)
