Unique ergodicity of asynchronous rotations, and application
Résumé.
The main result of this paper is an analogue for a continuous family of tori of KroneckerWeyl’s unique ergodicity of irrational rotations. We show that the notion corresponding in this setup to irrationality, namely asynchronicity, is satisfied in some homogeneous dynamical systems. This is used to prove the ergodicity of naturals lifts of invariant measures.
Nous étudions sur une famille continue de tores les rotations dites asynchrones, analogues aux rotations irrationnelles sur les tores classiques. Le résultat principal est l’unique ergodicité de ces rotations sur un monoïde adapté. Nous prouvons que la condition d’asynchronicité est vérifiée dans une famille d’exemples issue de la dynamique homogène, ce qui nous permet de déduire l’ergodicité de relevés de certaines transformations dans des fibrés en tores.
1991 Mathematics Subject Classification:
37A171. Introduction
1.1. Motivations
The original motivation of this study is the inquiry of ergodic properties of torus extension of homogeneous dynamical systems. Such dynamics have drawn some attention recently  see for example [MR2060024], [MR2811599], [MR3332893] for unipotent actions, and [MR1870070] and [MR2753608] for diagonal actions.
As an informal example, consider a diagonal element () with positive diagonal entries, acting by left multiplication on the homogeneous space , and let be a probability invariant by and ergodic on . The main interesting cases for our purpose occur when the measure is not algebraic. This dynamical system is a factor of the action of , where is arbitrary, by left multiplication on . This latter space is a torus bundle above . Amongst the possible invariant measures that projects onto , there is a particular one, denoted by , which decomposes into the Haar measure of tori on each fiber. It is natural to ask about its ergodicity with respect to .
Following the classical Hopf argument (see [MR2261076]), one is naturally led to inquire about ergodic properties of the strong stable foliation of . It turns out that this foliation contains the orbits of another action, namely the multiplication by on , where is an eigenvector for associated to an eigenvalue . This action is an unipotent action, but since it is ”vertical” (in the sense trivial in the factor ), Ratner’s theory yield in this case no more information than KroneckerWeyl’s uniform distribution on the torus.
To visualize this action of on each fiber, one may think of it as a rotation by a fixed vector on a varying torus depending on the basepoint. Here, we will prefer to think of it as the rotation by a varying vector depending on the basepoint , on a fixed torus .
One may hope in this situation that is irrational for almost every . It turns out that under appropriate assumptions, the rotations defined by satisfy a stronger property, namely asynchronicity.
As we will see shortly, such rotations on torus bundle above a measured space like enjoy strong ergodic properties, enabling us in this setting to prove a unique ergodicity result. In some sense, this can be considered as a weak analogue of Furstenberg’s unique ergodicity of horocyclic flow.
Finally, we will return to the question of the ergodicity of with respect to , and related mixing properties.
These kind of fiberwise system were also investigated independently by Damien Thomine [Thomine], using another point of view.
1.2. Asynchronous rotations
Let be a standard probability space without atoms, and let be a torus , for some integer .
We think of a measurable map as the data, for each , of a rotation adding the angle in a torus indexed by . Despite what the above motivational example might suggest, in this abstract setting, the case of rotations on a family of circles above a probability space is already interesting, and contains most of the difficulties.
The set of such measurable maps , where we identify two maps if they coincide almost everywhere, is naturally an abelian group under pointwise addition of functions. We denote by this group, by a slight abuse of notation. We would like to study the
translation by in , but as it lacks a nice topology, we consider a compactification of the group , which will be a monoid, as follows.
Let be the space of probability measures on which project to on the first factor.
To an element , we can associate the probability measure on , supported by its graph, which is the pushforward of by the map . This defines an embedding of into . It is not hard to see that the group law on correspond to a fiberwise convolution product on , which turns into an abelian monoid, with neutral , where is the zero map.
The space is equipped naturally with a weak* topology, for which it is a compact metric space. A tricky fact is that the convolution product is not continuous of the two variables, but is of each variable separately. A more detailed description of these objects, and explanations of the implied claims, are given in Section 2.
There is a particular element in , the measure . It satisfies the relation: , .
We are interested in studying the dynamics of translation on the monoid . Unsurprisingly, we now need a kind of irrationality condition.
Definition.
The angle map is said to be asynchronous if the image measure gives zero mass to any translate of any proper closed subgroup of . Equivalently, for any nontrivial character , has no atoms.
Intuitively, for , this means that one looks at an action by rotation on a family of circles indexed by , by angles , which are different from one another if picked randomly following the probability . For , it means that for almost every couple , and do not belong to the same coset modulo any closed, strict subgroup of .
Theorem 1.
The following are equivalent.

The angle map is asynchronous.

The closure contains .

The convolution action of on is uniquely ergodic.
If these are true, the only invariant probability measure is the Dirac measure .
The fact that the invariant measure is a Dirac measure implies (see Proposition 3.3) that there exists a subset of the integers , of natural density , such that for any ,
The question whether is an attracting point of the dynamic, that is if
or if this fails along some subsequence of zero density, is more delicate, and its answer depends on .
If equipped with Lebesgue measure, , and is a map with nonvanishing derivative, then is an attracting point (Proposition 3.1), and there is no exceptional subsequence. The regularity condition is not optimal, as Thomine obtained similar results for maps [Thomine].
However, for an angle map which is only measurable, the convolution action of might behave more like an intermittent map with the neutral fixed point . An example of this phenomenon is the following. Again, let endowed with the Lebesgue measure . Let be the (probability) Hausdorff measure of dimension on the usual Cantor set , viewed as a subset of by identifying and . Define by
Then , and . Alternatively, can be defined as the reciprocal, outside of dyadic rationals, of the usual devil’s staircase, modulo .
Since does not have any atom, is an asynchronous angle map. We claim that the sequence does not intersect a fixed neighborhood of . Indeed, since is invariant by the multiplication on the circle, the graph of is contained in , so the measure is supported on , a proper compact subset of . This forbids to be close to . Still, by Theorem 1, subsequences like are scarce, as the points tend to for a subset of of density one.
1.3. Main example, and a related ergodicity result
As hinted in the motivational paragraph, asynchronous rotations occur naturally in the context of homogeneous dynamics on torus bundle.
More precisely, let be a connected, semisimple algebraic linear group defined over , the group of its points and its integer points. By the Borel  HarishChandra Theorem, is a lattice in . We will consider invariant measures on under some elements , under the following assumptions.
Definition.
An element is said triangularizable with positive eigenvalues if for every finite dimensional representation of defined over , has only real, positive eigenvalues.
It is the case, for example, when is the real split form of , meaning the real rank equals the complex rank, and if is the exponential of a nonzero element of a Cartan subalgebra. It also happens when is unipotent, but this case is less interesting for our purpose, since by Ratner’s Theory, invariant ergodic measures are algebraic. This hypothesis implicitly rules out the case where is the real compact form of , as it cannot contain such element .
Definition.
Let be a probability measure on , invariant and ergodic under the action of . Such a measure is said to be nonconcentrated if for every closed algebraic, strict subgroup containing , and every such that is closed, then .
We now consider a fiber bundle over the probability space , whose fibers are tori.
Let a representation defined over on a finitedimensional space endowed with the structure . We will always assume that , and that is irreducible over . The semidirect product is endowed with the group law
Up to replacing with a subgroup of finite index in a way such that , the set is a subgroup of , and the map
is a torus bundle. Indeed, the lattice of (for the action of multiplication on the left) stabilizing a point is precisely , thus the fiber of over is the torus . It will be convenient to have measurable coordinates where this fiber bundle is a direct product.
Let be a measurable fundamental domain for the action of , and put the restriction to of the invariant lift of . As previously, we denote by the dimensional torus . The map
is a measurable bijection, that we will use as an identification between and , the subscript indicating the coordinates we are using. Likewise, we will identify with and with .
For , the action of by multiplication on on the left, can be read in the coordinates as the map
i.e. it is a rotation by an angle map , with
We prove:
Theorem 2.
Assume that is triangularizable with positive eigenvalues, that is an invariant, nonconcentrated, ergodic probability on , that is irreducible over , and . Assume also that is an eigenvector for . Then the angle map
is asynchronous.
Using the identification of with , we still denote by the measure on such that , whose disintegration along each fiber of are the Haar measures on each tori.
Direct application of Theorem 1 gives:
Corollary 1.1.
We assume the same hypotheses as Theorem 2. Let be the set of probabilities on projecting onto . The action on induced by the left multiplication by on is uniquely ergodic, with invariant measure .
Now choose any . The action of left multiplication by on admits the action of on as a factor. A natural question is if the measure , which is invariant, is ergodic with respect to this action.
If is any eigenvector of , multiplication by is in some sense moving in some part of the stable, unstable or neutral direction (depending on the eigenvalue) of the action of . Theorem 2 and Hopf’s argument allows us to prove the following ergodicity result:
Theorem 3.
Assume that is triangularizable with positive eigenvalues, that is an invariant, nonconcentrated, ergodic probability on , and that is irreducible over , of dimension . Choose , then the action by left multiplication by on is ergodic with respect to the invariant measure .
If we assume moreover that is not unipotent, then the action of on is weakly mixing if and only if the action of on is weakly mixing, and the same property holds for strong mixing.
1.4. Plan of the paper
In Section 2, we collect some facts about the topology of .
In Section 4, we prove that in the algebraic setting, the smallest algebraic subgroup of containing the elements induced by Poincaré recurrence of tha action, is itself.
This result (Theorem 4), which is the main ingredient of the proof of Theorem 2, relies crucially on the nonconcentration of .
1.5. Acknowledgements
I would like to thank to JeanPierre Conze, Serge Cantat, Sébastien Gouëzel, Barbara Schapira and Damien Thomine for their feedback and comments on the subject.
2. The space
2.1. Topology of
We recall that be a standard measurable space means that is can be endowed with a complete, separable distance , such that is the algebra of its Borel sets. The facts about standard probability spaces we will use are summarized in
[MR1450400, Chapter 1.1]. Choosing such a distance on defines a topology on the space of probability measures on , and hence on .
Although the weak* topology of the space of measures on depends in a strong way on the choice of topology on , it turns out that:
Lemma 2.1.
The topology induced on does not depends on the choice of topology on .
Proof.
Let two complete, separable metric space endowed with probabilities , with a map an isomorphism such that . The topologies induced on measures on are generated by the open sets:
where is continuous with compact support for the relevant topology on .
Denote by the map , . To show that is a homeomorphism, it is sufficient by symmetry to show its continuity.
We fix a neighborhood , and wish to show that its preimage contains some neighborhood of the initial point , for some .
The map from to is measurable. By Lusin’s Theorem, for every , there is a compact set , such that , on which is continuous. Let be continuous, by TietzeUrysohn’s Theorem, there exists a continuous function which extends the continuous map . Moreover, since is bounded, can be chosen such that . If ,
because and the same holds for . If we choose such that , we have:
and therefore, provided that ,
As any neighborhood of contains finite intersections of sets of the form , this implies that is continuous, as required. ∎
A corollary of this discussion is that we can assume for example that and is the Lebesgue measure on this interval, endowed with its usual topology. Since in this case, the set of probability measures on is a compact, separable metric space, it follows that is also compact, separable and metric.
2.2. Graphs and measures
For a measurable map , we define the graph measure of as the direct image of by the map . Two measurables maps define the same graph measure if and only if they are equal almost everywhere.
Let be the set of graph measures, this is a subset of .
2.3. Disintegration along
Any can be disintegrated as a family of measures , such that for any continuous testfunction with compact support ,
Moreover, the map is measurable, and uniquely defined modulo zero sets. See e.g. [MR603625, Th. 5.8].
2.4. Convolution product
For two measures in , we define the fiberwise convolution product of by
where a continuous testfunction with compact support. Equivalently, is the usual convolution product of and .
The following Lemma, whose proof is left to the reader, summarizes elementary properties of this fiberwise convolution product.
Lemma 2.2.
The following holds.


is the neutral element of the commutative monoid , where is the map almost everywhere zero.

The set of invertible elements for is , the set of graph measures.
Remark that, if , , then one can check by hand (or see e.g. Proposition 3.1) that tends to as , but
so the fiberwise convolution product is not continuous. However, we have:
Lemma 2.3.
For any , the convolution map
is continuous.
Proof.
It is sufficient to check that the preimage by of any set of the form , for any , , , contains a set of the form for some and some . Let be such a neighborhood of , and let such that . As the map is measurable, again by Lusin’s Theorem, it is continuous on a set of measure . Define
It follows from the continuity of and the continuity of that this is a continuous map on , and moreover bounded by . Thus it can be extended to a bounded continuous map, say , on , still bounded by . Notice that for any ,
Let , then
By the choice of ,this implies that , as announced. ∎
3. Asynchronous maps
3.1. A simple example
As stated in the introduction, if has enough regularity properties, it turns out that is the limit point of the dynamic of on . This result will not be used in the sequel.
Proposition 3.1.
Assume is a map, such that does not vanish. Then for all , when .
Proof.
By continuity of , it is sufficient to check that tends to as tends to infinity. To do so, compute the FourierStieltjes coefficients
If , this coefficient is or , depending on whether or not. If , we can write
and integration by parts immediately shows that when with fixed. ∎
3.2. Proof of
Assume that is asynchronous. We may, and will, assume that endowed with its Haar probability measure . The space is then a dimensional torus, and to check that some is close to , it is sufficient to show that for a finite set of the nontrivial FourierStieljes coefficients of are close to zero.
Lemma 3.2.
For any nontrivial character of , we have
Proof.
For an integer , let be the character of , . Any character of can be written uniquely as a product , for some and every , . We have
If , then since . In this case, we have
so the statement is trivial.
If , then by assumption for almost every , so
Therefore, Lebesgue’s dominated convergence Theorem applies and we obtain the desired result. ∎
Let be a finite subset of nontrivial characters, and . By the previous Lemma, we have
so there exists such that for all , , meaning that is close to .
3.3. Proof of
Let us check that implies that is uniquely ergodic. Let be a sequence such that converges weakly to when . Let , then by onesided continuity of convolution,
Let be any invariant measure on , be a continuous function. Then
by invariance of . The Lebesgue dominated convergence Theorem implies
which means that is the Dirac measure at , as required.
3.4. Proof of
Assume that the convolution action of is uniquely ergodic. As is a fixed point, the invariant measure is necessarily , and thus as the invariant measure is a Dirac mass, there exist a subsequence such that tends to as .
Let be any nontrivial character of . Assume that has an atom. In this case, there would be a set of positive measure on which is a constant, say . Thus is a constant on , namely , and will have an atom of mass . Note that
But tends to , namely the Lebesgue measure on , which cannot be a limit of measures having a atom of fixed mass. This is a contradiction.
3.5. Sets of natural density one
Proposition 3.3.
Assume is asynchronous. Then there exists a set of full natural density such that for all ,
Proof.
We consider the measure on ,
As any weak limit of is invariant and is compact, by unique ergodicity of , converges to when goes to . This implies that for any neighborhood of , the proportion of outside goes to zero as .
Let be a decreasing basis of neigborhood of , and
Let be an integer such that for all ,
where is the uniform probability on . We can modify the sequence to be strictly increasing, and choose . Let be the subset
Notice that since the sets are decreasing with , if ,
Thus, for such that , we have
This proves that is a set of natural density one. By construction, we have
By continuity of the convolution with , the latter limit holds for the sequence with the same set . ∎
4. On the smallest algebraic group containing return elements
The following Theorem, which will be a crucial ingredient of the proof of Theorem 2, might be of independent interest.
Theorem 4.
Let be the group of real points of an algebraic group defined over , without nontrivial characters, be its integer points, be triangularizable with positive eigenvalues, and an invariant measure on . We assume that the measure is ergodic and nonconcentrated. Let be a fundamental domain for , and denote by the lift to of . Let be a subset of positive measure. Define
the set of elements of associated to return times in . Then the smallest algebraic subgroup of containing is .
To prove this, let be the smallest algebraic subgroup of containing . Our aim is to show that . This will be done in the following sequence of Lemmata.
4.1. Closure of
Lemma 4.1.
The set is closed
Proof.
Notice that is defined over , since consists of integer points. We claim that the the nontrivial characters of are of order . Indeed, if is such a character defined over , the image by of the subgroup generated by consists of rational with bounded denominators, and is a multiplicative subgroup, so . Therefore, is contained in , an algebraic group defined over . By definition of , , so as required. In particular, is of finite volume, by the Theorem of Borel and HarishChandra [MR0244260, Corollaire 13.2]. By [MR0244260, Proposition 8.1], this also implies that is a closed subset of , where is the connected component of the identity of , in the Zariski topology (a subgroup of finite index). This implies that is closed. ∎
4.2. Reduction step
Lemma 4.2.
To prove Theorem 4, we can (and will) assume that for all and such that , then .
Proof.
Consider the subset
Clearly, is a subset of of the same measure, and . So it is sufficient to prove the statement of Theorem 4 for instead of , and satisfies the above property. ∎
4.3. Invariance of
Lemma 4.3.
For almost every , .
Proof.
By a Theorem of Chevalley [MR1102012, Thm 5.1], there exists a finite dimensional representation of such that is the stabilizer of a line , that is . By Poincaré recurrence Theorem, for almost every , there exists a sequence and such that and . Fix such an .
By Lemma 4.2, we known that . It follows that , so since , we have
(1) 
By assumption, is triangularizable with positive eigenvalues, so has only positive, real eigenvalues. We claim that (1) implies that is contained in one of its eigenspaces.
Let
be the JordanChevalley decomposition of , that is: and commutes, is nilpotent, diagonalizable (with positive, real eigenvalues). If is the nilpotent index of , for such that ,
Let , and be its decomposition along the eigenspaces of corresponding to the eigenvalues of . Then
As a function of , this is a combination of polynomials and powers of eigenvalues. If is the highest eigenvalue for which , and is the largest for which , then we have the asymptotic as ,
However, we know that projectively, , so is colinear to . As preserves the eigenspace of associated to ,
because by definition of , . This shows that is an eigenvector, and so is .
We have proved that is contained in an eigenspace of . So is stabilized by , meaning that , as required. ∎
4.4. Conclusion of the proof of Theorem 4
Lemma 4.4.
We have .
Proof.
By ergodicity of with respect to , for almost every , is dense in the support of . By the previous Lemma, we have also for almost every , , so
Consider a typical satisfying both of these properties. By Lemma 4.1, is a closed set. By density of the orbit of in the support of , this implies that , so
By assumption, is nonconcentrated, so necessarily. ∎
5. Proof of Theorem 2
The proof is by contradiction. We assume that is not asynchronous. By translate of a subspace of , we mean a set of the form , where , and