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Absolutely Singular Dynamical Foliations
Abstract
Let A
3 be the product of the automorphism of T
2 and of the identity on T
1. A small perturbation g of A
3 among volume preserving diffeomorphisms will have an invariant family of smooth circles Γ forming a continuous foliation of T
3. Corresponding to the vector bundle tangent to the circles Γ there is a “central” Lyapunov exponent of (g, volume), which is nonzero for an open set of ergodic g's. This surprising result of Shub and Wilkinson is complemented here by showing that the volume on T
3 has atomic conditional measures on the Γ's: there is a finite k such that almost every Γ carries k atoms of mass 1/k.
... In this work, we consider conditions in which atomic disintegration is in fact a monoatomic disintegration. One of the motivations of this work comes from the work of Ponce et al. (J Mod Dyn,8(1):93-107, 2014); they prove that there is a minimal foliation and a set of full volume which intersects each leaf in one point, but their argument uses some of the hyperbolic structure of the system. We generalize some of their techniques in which we eliminate the need for Markov partitions, which are structures inherited from Anosov diffeomorphisms. ...
... A novelty of the partially hyperbolic systems is that the central foliation (when it exists) does not have the "good behavior" exhibited by stable and unstable foliations. Ruelle and Wilkinson [8] give an example where the disintegration of the volume in the central direction is atomic. The example discussed in [8] is a skew product, and its central leaves are compact and one-dimensional (i.e., circles). ...
... Ruelle and Wilkinson [8] give an example where the disintegration of the volume in the central direction is atomic. The example discussed in [8] is a skew product, and its central leaves are compact and one-dimensional (i.e., circles). In [7], Ponce et al. exhibit an even more curious dynamic example: a partially hyperbolic system that preserves volume, with a one-dimensional minimal foliation (that is, every leaf is dense) and that there is a set of full volume that intersects each leaf of this foliation in a single point. ...
In this work, we consider conditions in which atomic disintegration is in fact a monoatomic disintegration. One of the motivations of this work comes from the work of Ponce et al. (J Mod Dyn, 8(1):93–107, 2014); they prove that there is a minimal foliation and a set of full volume which intersects each leaf in one point, but their argument uses some of the hyperbolic structure of the system. We generalize some of their techniques in which we eliminate the need for Markov partitions, which are structures inherited from Anosov diffeomorphisms. We prove some results on which atomic disintegration and some contraction hypothesis on the foliation implies monoatomic disintegration.
... By classical work on normal hyperbolicity [15], perturbations of (1) admit an invariant center foliation with leaves that are circles close to {(x, y) = constant} (which is the invariant center foliation for (1)). The diffeomorphisms studied in [24] are shown by Ruelle & Wilkinson [23] to possess a set of full Lebesgue measure that intersects almost every circle from the center foliation in k points for some finite integer k. The number k remained unspecified in their result. ...
... The number k remained unspecified in their result. We will show that the result in [23] is true with k = 1. We thus get robust examples of conservative diffeomorphisms on T 3 with a center foliation of circles and an invariant set of full Lebesgue measure that intersects almost every center leaf in a single point. ...
... The theorem below recalls the results of [23,24]. Note that the center Lyapunov exponent λ c in the formulation of the theorem is negative, the inverse diffeomorphisms possess a positive center Lyapunov exponent as in [24]. ...
Shub & Wilkinson and Ruelle & Wilkinson studied a class of volume preserving diffeomorphisms on the three dimensional torus that are stably ergodic. The diffeomorphisms are partially hyperbolic and admit an invariant central foliation of circles. The foliation is not absolutely continuous, in fact, Ruelle & Wilkinson established that the disintegration of volume along central leaves is atomic. We show that in such a class of volume preserving diffeomorphisms the disintegration of volume along central leaves is a single delta measure. We also formulate a general result for conservative three dimensional skew product like diffeomorphisms on circle bundles, providing conditions for delta measures as disintegrations of the smooth invariant measure.
... Shub, Wilkinson [43] constructed partially hyperbolic, stably ergodic (with respect to volume) diffeomorphisms whose center leaves are circles and whose center Lyapunov exponent is non-zero, and they observed that for such maps the center foliation can not be absolutely continuous. Indeed, in a related setting, Ruelle, Wilkinson [41] observed that the center foliation has atomic disintegration: the Rokhlin conditional measures of the volume measure along the leaves are supported on finitely many orbits. That is the case also in Katok's construction, as observed before, but it should be noted that in Katok's example the center Lyapunov exponent vanishes. ...
... • maps fixing their center leaves, including perturbations of time-one maps of hyperbolic flows [4]; • maps with circle center leaves [3], including perturbations of certain skewproducts, of the type considered in [41,43]. Moreover, the second alternative is often very rigid: for example, for perturbations of the time-one map of a hyperbolic flow, it implies that the perturbation is itself the time-one map of a smooth flow. ...
... The entropy formula (for partial entropy) gives that h µ (f, F u ) is equal to the sum τ uu + τ wu of the two positive Lyapunov exponents. On the other hand, Ruelle-Wilkinson [41] showed that every center leaf contains finitely many µ-generic points. Thus, h µ (f, F wu ) = 0 and so (4) fails in this case. ...
Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic measures. We add to this theory by constructing stable examples for which the disintegration is singular without being atomic. In the context of diffeomorphisms with mostly contracting center direction, for which upper leafwise absolute continuity is known to hold, we provide examples where the center foliation is not lower leafwise absolutely continuous.
... The idea of the proof goes back to [Kat80], who proved that if all the LEs of a diffeomorphism with respect to an ergodic measure are strictly negative, then the measure is supported on a periodic orbit. A fiber version of Katok's result, meaning the corresponding result for skew products when all the LEs of the fiber maps are strictly negative, is proved in [RW01]. Singularities aside, our setup fits this setting, as both of the exponents of {T (x,y) } are strictly negative. ...
... Singularities aside, our setup fits this setting, as both of the exponents of {T (x,y) } are strictly negative. Our proof follows that in [RW01] nearly verbatim. The presence of singularities is immaterial because the proof uses Lyapunov charts, and for as long as one works within Lyapunov charts, the singularities of H are not 'visible' by property (b) in section 5.3. ...
In this paper we present a rigorous analysis of a class of coupled dynamical systems in which two distinct types of components, one excitatory and the other inhibitory, interact with one another. These network models are finite in size but can be arbitrarily large. They are inspired by real biological networks, and possess features that are idealizations of those in biological systems. Individual components of the network are represented by simple, much studied dynamical systems. Complex dynamical patterns on the network level emerge as a result of the coupling among its constituent subsystems. Appealing to existing techniques in (nonuniform) hyperbolic theory, we study their Lyapunov exponents and entropy, and prove that large time network dynamics are governed by physical measures with the SRB property.
... The idea of the proof goes back to [Kat80], who proved that if all the Lyapunov exponents of a diffeomorphism with respect to an ergodic measure are strictly negative, then the measure is supported on a periodic orbit. A fiber version of Katok's result, meaning the corresponding result for skew products when all the Lyapunov exponents of the fiber maps are strictly negative, is proved in [RW01]. Singularities aside, our setup fits this setting, as both of the exponents of {T (x,y) } are strictly negative. ...
... Singularities aside, our setup fits this setting, as both of the exponents of {T (x,y) } are strictly negative. Our proof follows that in [RW01] nearly verbatim. The presence of singularities is immaterial because the proof uses Lyapunov charts, and for as long as one works within Lyapunov charts, the singularities of H are not "visible" by Property (ii) in Sect. ...
We present in this paper a rigorous analysis of a class of coupled dynamical systems in which two distinct types of components, one excitatory and the other inhibitory, interact with one another. These network models are finite in size but can be arbitrarily large. They are inspired by real biological networks, and possess features that are idealizations of those in biological systems. Individual components of the network are represented by simple, much studied dynamical systems. Complex dynamical patterns on the network level emerge as a result of the coupling among its constituent subsystems. Appealing to existing techniques in (nonuniform) hyperbolic theory, we study their Lyapunov exponents and entropy, and prove that large time network dynamics are governed by physical measures with the SRB property.
... In some situations, this behaviour may be somewhat unexpected. Ruelle and Wilkinson [22] provide the first dynamical example of a pathological (meaning atomic) behaviour for a foliation. They exhibit a partially hyperbolic skew product f for which the compact center foliation F c has atomic disintegration with respect to volume measure: that is, there exists a set with full volume measure which intersects each center leave in finite points. ...
... They exhibit a partially hyperbolic skew product f for which the compact center foliation F c has atomic disintegration with respect to volume measure: that is, there exists a set with full volume measure which intersects each center leave in finite points. Later, Ponce, Tahzibi and Varão [20] have extended the ideas from [22] to provide a minimal foliation (that is all leaves are dense) such that volume has monoatomic disintegration: there exists a set of full volume which intersects each leaf in just one point. ...
In the context of locally constant skew-products over the shift with circle fiber maps we introduce the notion of measures with periodic repetitive pattern, inspired by \cite{GorIlyKleNal:05} and which includes the non-hyperbolic measures they construct. We prove that these measures have atomic disintegration along the central fibers.
... Although these are two extreme behaviors among, a priori, many possibilities for the disintegration of a measure, recent results have indicated that this dichotomy is more frequent than one would at first expect. In [29] D. Ruelle and A. Wilkinson proved that for certain skew product type of partially hyperbolic dynamics, if the fiberwise Lyapunov exponent is negative then the disintegration of the preserved measure along the fibers is atomic. Later A. Homburg [17] proved that some examples treated in [29] one can actually prove that the disintegration is composed by only one dirac measures. ...
... In [29] D. Ruelle and A. Wilkinson proved that for certain skew product type of partially hyperbolic dynamics, if the fiberwise Lyapunov exponent is negative then the disintegration of the preserved measure along the fibers is atomic. Later A. Homburg [17] proved that some examples treated in [29] one can actually prove that the disintegration is composed by only one dirac measures. A. Avila, M. Viana and A. Wilkinson [4] proved that for C 1 -volume preserving perturbations of the time-1 map of geodesic flows on negatively curved surfaces, the disintegration of the volume measure along the center foliation is either atomic or absolutely continuous and that in the latter case the perturbation should be itself the time-1 map of an Anosov flow. ...
Given a Borel action over a Lebesgue space X we show that if preserves an invariant system of packing regular metrics along a Borel lamination , then the ergodic measures preserved by the action are rigid in the sense that the system of conditional measures with respect to the partition are induced by the given invariant metric system or are supported in a countable number of boundaries of balls. The argument we employ does not require any structure on G other then second-countability and no hyperbolicity on the action as well. Our main result is interesting on its own, but to exemplify its strength and usefulness we show some applications in the context of cocycles over hyperbolic maps and to certain partially hyperbolic maps.
... Further results that give a good description of skews products from an ergodic point of view are due to Ruelle and Wilkinson [14] who consider a general measurable skew-product with an invertible and ergodic map in the base (which is a general probability space) and C 1+α diffeomorphisms on the fiber (which is a general Riemannian compact manifold). They prove that any ergodic measure for the skew-product which projects to the ergodic measure in the base and which has only negative fiberwise Lyapunov exponents admits an atomic fiber disintegration. ...
... One novelty of this paper is on one hand precisely that we do not a priori assume uniform hyperbolicity of the fiber dynamics. Moreover, comparing with [14], the fact that the fibers are one-dimensional, allows us to require C 1 regularity only and enables us to improve the conclusion about the atomic disintegration and to show the existence of a (bi-)graph structure. On the other hand that we do not assume that fiber maps are orientation preserving and hence, in general, do not have invariant "simple" graphs. ...
We study step skew-products over a finite-state shift (base) space whose fiber maps are injective maps on the unit interval. We show that certain invariant sets have a multi-graph structure and can be written graphs of one, two or more functions defined on the base. In particular, this applies to any hyperbolic set and to the support of any ergodic hyperbolic measure. Moreover, within the class of step skew-products whose interval maps are 'absorbing', open and densely the phase space decomposes into attracting and repelling double-strips such that their attractors and repellers are graphs of one single-valued or bi-valued continuous function almost everywhere, respectively.
... In addition, we show that the presence of a positive Lyapunov exponent in the central foliation is associated to the foliation non being absolutely continuous. Such a pathology of the central foliation was already discovered in other examples, e.g., volume preserving partially hyperbolic maps [37,39], but here emerges in a totally natural and robust manner for systems whose invariant measure is not previously known and it is not constant. ...
... This means that, despite each leaf being individually smooth, the foliation as a whole is very wild. This situation is strange but known to happen, see the papers of Ruelle, Shub and Wilkinson [37,39] where they presented an open set of volume preserving partially hyperbolic systems with non absolutely continuous central foliation for a perturbation of the product of an Anosov map by an identity map on the circle. This behaviour was later observed in many other partially hyperbolic systems, see [22,[42][43][44][45]47]. ...
We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.
... 可微映射，通过建立其在不稳定子丛的可积 性、稳定指数的刚性以及拓扑共轭的存在性之间的联系，本文给出了这类非结构稳定的系统在拓扑共 轭意义下的分类，同时研究了其同伦类的半共轭刚性。 关键词 一致双曲性 部分双曲性 Lyapunov 指数 可积性 刚性 MSC (2020) 主题分类 37C05, 37C15, 37D20 1 引言 动力系统中的刚性问题关注两个或多个系统之间的某种" 弱等价" 关系是否蕴含着某种" 强等价" 关系。就一致双曲系统与部分双曲系统而言，其刚性现象在数十年来被持续挖掘研究，比如：上同调 方程解的存在性及其正则度的提升 [13, 14, 43-46, 77]、同伦系统之间拓扑半共轭或拓扑共轭的存在性 [17, 18]、周期数据蕴含拓扑共轭的光滑性 [12, 22, 24-26, 29]、体积测度意义下 Lyapunov 指数刻画光滑 共轭 [4, 27, 67] 等等。 在研究具备双曲性系统的刚性、遍历性与结构稳定性等问题时， (不变的)叶状结构能将系统的动 力学行为简化、将维数降低、将问题约化，起着至关重要的作用。 (部分)双曲系统与非一致双曲系统 的叶状结构已然成为了相对独立的课题 [6,9,23,39,53,59,60,66]。而叶状结构的刚性问题也被广泛且 ...
... In this case, three different scenario can occur: there exist only non-compact leaves (DA -derived from Anosov -diffeomorphisms), there exist simultaneously compact and non-compact leaves (time-1 maps of Anosov flows), or there exist only compact central leaves. The latter, and in some sense easiest, of these casescompact central leaves -is still extremely rich (see, for instance, the pathological behaviors of the central foliations in [24,27]). On the other hand, using ingredients of one-dimensional dynamics, in this case one often has a very precise picture of the dynamics (see, for instance, [26,19]). ...
We study transitive step skew-product maps modeled over a complete shift of k, , symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents. We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves. Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.
... Many subsequent results showed that for volume preserving diffeomorphisms the non-absolute continuity of the center foliation is indeed a quite general phenomena, for instance see [71,74,73,34,5,6,80]. There is a different situation for diffeomorphisms which do not preserve the volume, see Viana and Yang [79]. ...
In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism L with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to L. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation f of with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.
... In contrast, the center foliation may exhibit a new type of behavior, it may have atomic disintegration: there is a set of full volume which intersects each center leaf in a finite number of points. At first this might sound as a pathological behavior, but it turns out that this is in fact a common behavior for the center foliation [2,16,19]. ...
In this work we completely classify conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate sceneario for our context since it is not possible to obtain an analogous result of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular a metric condition on the center foliation implies ergodicity and center foliation.
... Ruelle-Wilkinson and Pesin-Hirayama generalized the Mañé argument, we can state these results in a unique theorem as following: Theorem 1.8. [9], [15] Consider a dynamically coherent partially hyperbolic diffeomorphism f whose center leaves are fibers of a (continuous) fiber bundle. Assume that the all center Lyapunov exponents are negative (or positive) then the conditional measures of µ on the leaves of the center foliation are atomic with p, p ≥ 1, atoms of equal weight on each leaf. ...
In this paper we focused our study on Derived From Anosov diffeomorphisms (DA diffeomorphisms ) of the torus it is, an absolute partially hyperbolic diffeomorphism on homotopic to an Anosov linear automorphism of the We can prove that if is a volume preserving DA diffeomorphism homotopic to linear Anosov A, such that the center Lyapunov exponent satisfies with x belongs to a positive volume set, then the center foliation of f is non absolutely continuous. We construct a new open class U of non Anosov and volume preserving DA diffeomorphisms, satisfying the property for almost everywhere Particularly for every the center foliation of f is non absolutely continuous.
... One of the first to study the behavior of the center foliation was R. Mañé, in a letter (unpublished) to M. Shub, they relate the absolutely continuous of compact center foliations in which the Lyapunov exponents are non zero. These ideas were very useful in the study on absolutely continuous of compact center foliations (see [29] and [15]). The non absolutely continuous of the non compact center foliations is also very common, in [10] show there are open sets in PH r m (T 3 ), r ≥ 2, of diffeomorphisms with one dimensional non compact center foliation and non absolutely continuous. ...
In this paper we are considering partially hyperbolic diffeomorphims of the torus, with We prove, under some conditions, that if the all center Lyapunov exponents of the linearization A, of a \mbox{DA-diffeomorphism} f, are positive and the center foliation of f is absolutely continuous, then the sum of the center Lyapunov exponents of f is bounded by the sum of the center Lyapunov exponents of A. After, we construct a open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each f in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of f is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of f is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.
... We will also need a result of Ruelle-Wilkinson [69] about invertible cocycles with negative Lyapunov exponents. ...
In this paper we establish several results on the dimensional properties of invariant measures and sets associated to random walks and group actions by circle diffeomorphisms. Our main results include the exact dimensionality and a dimension formula of stationary measures, variational principles for dimensions in various settings and estimates of the Hausdorff dimensions of exceptional minimal sets. We also prove an approximation theorem for random walks on the circle which is analogous to the results of Katok, Avila-Crovisier-Wilkinson, Morris-Shmerkin. The proofs of our results are based on a combination of techniques including a new structure theorem for smooth random walks on circle, a dynamical generalization of the critical exponent of Fuchsian groups and some novel arguments inspired by the study of fractal geometry, hyperbolic geometry and holomorphic dynamics.
... Pathological invariant foliations, as foliations with atomic decomposition, seem to be ubiquitous in partially hyperbolic dynamics, and have been intensively studied by many authors. See Shub and Wilkinson [52], Ruelle and Wilkinson [50], Hirayama and Pesin [28], Homburg [29], Gogolev and Tahzibi [24] and Avila, Viana and Wilkinson [2]. Examples similar to Katok's example as those in this section are quite special cases. ...
Often topological classes of one-dimensional dynamical systems are finite codimension smooth manifolds. We describe a method to prove this sort of statement that we believe can be applied in many settings. In this work we will implement it for piecewise expanding maps. The most important step will be the identification of infinitesimal deformations with primitives of Birkhoff sums (up to addition of a Lipschitz function), that allows us to use the ergodic properties of piecewise expanding maps to study the regularity of infinitesimal deformations.
... [7] is a generalization of [59] to the skew-product setting; very high rates of expansion and contraction for g are imposed. [30] builds on results from [49], and studies volume preserving perturbations of some skew-product systems (e.g. A × Id T ). ...
We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.
... [7] is a generalization of [59] to the skew-product setting; very high rates of expansion and contraction for g are imposed. [30] builds on results from [49], and studies volume preserving perturbations of some skew-product systems (e.g. A × Id T ). ...
We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.
... The authors established the regularity of the center exponent within a specific family of volume preserving partially hyperbolic diffeomorphisms of the three-torus, they showed that the second derivative is nonzero, and in conclusion they constructed open sets of such diffeomorphisms which are stably ergodic, nonuniformly hyperbolic, and with pathological center foliations. These ideas were pushed further in [64], while in [62] the regularity of the exponents and formulas for the derivatives were obtained at linear automorphisms of the n-torus. ...
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in [0,1]), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time one map of a geodesic flow of a surface of negative curvature the metric entropy of the volume is Lipschitz with respect to the parameter. At the time one map of a geodesic flow on a manifold of constant negative curvature the topological entropy is differentiable with respect to the parameter, and we give a formula for the derivative. Under some regularity conditions, the critical points of the Lyapunov exponent function are non-flat (the second derivative is nonzero for some families). Also, again under some regularity conditions, the criticality of the Lyapunov exponent function implies some rigidity of the map, in the sense that the volume decomposes as a product along the two complimentary foliations. In particular for area preserving Anosov diffeomorphisms, the only critical points are the maps smoothly conjugated to the linear map, corresponding to the global extrema.
... This time the reason is not "for simplicity", but because in the case of coupled flows, even if the coupling has short range and is weak, there seems to be no detailed and constructive general theory of the SRB distributions, because no simple conditions are known that, via perturbation techniques, yield hyperbolicity of the flow and allow studying its properties. (For a glimpse on the kind of complications which arise when studying flows consult[91,92].) Instead, at least in the case of coupled maps, the theory is quite well understood[11,36,93,94], as in the example in Eq. (B.1) below, at small coupling ε. ...
A review on the fluctuation relation, fluctuation theorem and related topics.
... Proposition 28 (cf. [77] and the references therein). Let F be an f −invariant, n−dimensional foliation of M with C 2 leaves, and let E = T F. Let λ ± be the largest and smallest Lyapunov exponents for the cocycle (Df | E , ν). ...
We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth, ergodic perturbations of certain algebraic systems -- including the discretized geodesic flows over hyperbolic manifolds of dimension at least 3 and linear toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle -- if the abelianization of the smooth centralizer has sufficiently high rank, then the centralizer contains a smooth flow. In dimension 3, we obtain a global dichotomy: for an ergodic partially hyperbolic diffeomorphism f that preserves an orientable foliation by circles, either the centralizer is virtually trivial, or it contains a smooth flow (in which case, up to a finite cover, f is a smooth isometric extension of an area-preserving Anosov diffeomorphism). At the heart of this work are two very different rigidity phenomena. The first phenomenon was discovered in [2,3]: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is either equivalent to Lebesgue or atomic. The other phenomenon is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation [25]. We introduce a variety of techniques in the study of higher rank, abelian partially hyperbolic actions: most importantly, we demonstrate a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat measure rigidity for circle extensions of Anosov diffeomorphisms and apply normal form theory to upgrade regularity of the centralizer.
... Shub and Wilkinson in 2000 [17] considered the automorphism of T 3 , given by A 3 = A 2 0 0 1 , where A 2 = 2 1 1 1 , and they proved that arbitrarily close to A 3 , there exists a C 1 -open set U ⊂ Dif f 2 µ (T 3 ) such that for each g ∈ U , the central foliation F c g is not absolutely continuous. Furthermore, in the same year, Ruelle and Wilkinson [15] proved that if g ∈ U , then the foliation F c g are absolutely singular, this mean that the conditional measures defined by the Rokhlin disintegration are atomic. Later in 2003, Baraviera and Bonatti [3] considered a compact Riemannian manifold endowed with a C 2volume form ω, a C 1 Anosov flow X : R × M → M that preserves the volume ω, and f the time-one map of X (that is a C 1 partially hyperbolic diffeomorphism), then for all g inside an open set C 1 -close to f , F c g and any leaf L c of F c g , the set of points of L c having positive Lyapunov exponents has Lebesgue measure 0 in L c , and this implies that F c g is non-absolutely continuous with respect to Lebesgue for any ω-preserving g close to f such that ...
In this work, we consider a specific space of foliations with leaves and H\"older holonomies of the square , with some topology and we show that a generic such foliation is non-absolutely continuous, furthermore, the conditional measures defined by Rokhlin disintegration are Dirac measures on the leaves. This space of foliations is motivated by the foliations that appear in hyperbolic systems and partially hyperbolic systems.
... If a family of fixed points p t of f t , which depends smoothly on t ∈ I (it is in fact a center leaf) has the property that the derivative Dγ t (p t ) has different eigenvalues for different values of t ∈ I, then Katok shows that the center foliation F c is not absolutely continuous, and furthermore there exists a full volume subset of T 2 × I which intersects every center leaf in a unique point. Many subsequent results showed that for volume preserving diffeomorphisms the non-absolute continuity of the center foliation is indeed a quite general phenomena, for instance see [55,57,56,25,5,6,62]. There is a different situation for diffeomorphisms which don't preserve the volume, for a counter example see Viana and Yang,[61]. ...
In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism L with simple real spectrum, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to L. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation f of with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.
... One of the first to study the behavior of the center foliation was R. Mañé, in a letter (unpublished) to M. Shub, they relate the absolutely continuous of compact center foliations in which the Lyapunov exponents are non zero. These ideas were very useful in the study on absolutely continuous of compact center foliations (see [29] and [15]). The non absolutely continuous of the non compact center foliations is also very common, in [10] show there are open sets in PH r m (T 3 ), r ≥ 2, of diffeomorphisms with one dimensional non compact center foliation and non absolutely continuous. ...
In this paper we are considering partially hyperbolic diffeomorphims of the torus, with We prove, under some conditions, that if the all center Lyapunov exponents of the linearization A, of a \mbox{DA-diffeomorphism} f, are positive and the center foliation of f is absolutely continuous, then the sum of the center Lyapunov exponents of f is bounded by the sum of the center Lyapunov exponents of A. After, we construct a open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each f in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of f is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of f is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.
... In contrast, the center foliation may exhibit a new type of behavior, it may have atomic disintegration: there is a set of full volume which intersects each center leaf in a finite number of points. At first this might sound as a pathological behavior, but it turns out that this is in fact a common behavior for the center foliation [2,16,19]. ...
In this work we completely classify conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate sceneario for our context since it is not possible to obtain an analogous result of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular a metric condition on the center foliation implies ergodicity and center foliation.
... In this case, three different scenario can occur: there exist only non-compact leaves (DA -derived from Anosov -diffeomorphisms), there exist simultaneously compact and non-compact leaves (time-1 maps of Anosov flows), or there exist only compact central leaves. The latter, and in some sense easiest, of these casescompact central leaves -is still extremely rich (see, for instance, the pathological behaviors of the central foliations in [24,27]). On the other hand, using ingredients of one-dimensional dynamics, in this case one often has a very precise picture of the dynamics (see, for instance, [26,19]). ...
We study transitive step skew-product maps modeled over a complete shift of k , k \geq 2 , symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents.
We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves.
Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the \text{weak}⁎ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.
... In the compact center foliation setting, Ruelle and Wilkinson [15] and Hirayama and Pesin [8] generalized the Mañé's argument. Assuming all center Lyapunov exponent are negative (or positive) they establish that the Rokhlin decomposition of a volume form along to center leaves is atomic, with p, p ≥ 1, atoms of equal weight on each leaf. ...
In this paper we focused our study on Derived From Anosov diffeomorphisms (DA
diffeomorphisms ) of the torus it is, an absolute partially
hyperbolic diffeomorphism on homotopic to an Anosov linear
automorphism of the We can prove that if is a volume preserving DA diffeomorphism homotopic
to linear Anosov A, such that the center Lyapunov exponent satisfies
with x belongs to a positive volume set,
then the center foliation of f is non absolutely continuous. We construct a
new open class U of non Anosov and volume preserving DA diffeomorphisms,
satisfying the property for almost
everywhere Particularly for every the center
foliation of f is non absolutely continuous.
... The foliation W c is continuous (indeed, it is Hölder continuous) but is not absolutely continuous (see [229]). Moreover, there exists a set E of full measure and an integer k > 1 such that E intersects almost every leaf W c (x) at exactly k points (see [219]; the example in Section 10.2 is of this type). ...
... This lemma is a corollary of the results of Ruelle and Wilkinson in [110]. Intuitively one has that in the unstable Pesin manifold of almost every point intersected with a center the conditional measure must have an atom (the manifolds are exponentially contacted for the past) Then, Lemma 3.4.9 ...
We address the problem of existence and uniqueness (finite- ness) of ergodic equilibrium states for a natural class of partially hyperbolic dynamics homotopic to Anosov. We propose to study the disintegration of equilibrium states along central foliation as a tool to develop the theory of equilibrium states for partially hyperbolic dynamics.
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in [ 0 , 1 ] [0,1] ), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map.
We also obtain several applications. Near the time one map of a geodesic flow of a surface of negative curvature the metric entropy of the volume is Lipschitz with respect to the parameter. At the time one map of a geodesic flow on a manifold of constant negative curvature the topological entropy is differentiable with respect to the parameter, and we give a formula for the derivative. Under some regularity conditions, the critical points of the Lyapunov exponent function are nonflat (the second derivative is nonzero for some families). Also, again under some regularity conditions, the criticality of the Lyapunov exponent function implies some rigidity of the map, in the sense that the volume decomposes as a product along the two complementary foliations. In particular for area preserving Anosov diffeomorphisms, the only critical points are the maps smoothly conjugated to the linear map, corresponding to the global extrema.
Given a foliation, we say that it displays “extreme disintegration behavior” if either it has atomic disintegration or if it is leafwise absolutely continuous and its conditional measures are uniformly equivalent to the leaf volume, which we call UDB property. Both concepts are related to the decomposition of volume with respect to the foliation. We relate these behaviors to the measure-theoretical regularity of the holonomies, by proving that a foliation with atomic disintegration has holonomies taking full volume sets to zero volume sets, and we characterize the UBD property with the holonomies having uniformly bounded Jacobians. Both extreme phenomena appear in invariant foliations for dynamical systems.
We show that each non‐hyperbolic ergodic measure of a partially hyperbolic diffeomorphism on which is homotopic to Anosov admits a full measure subset which intersects each center leaf in at most two points.
We show that each non-hyperbolic ergodic measure of a partially hyperbolic diffeomorphism on which is homotopic to Anosov admits a full measure subset which intersects each center leaf in at most two points.
We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact.
A review on the fluctuation relation, fluctuation theorem and related topics
We study step skew-products over a finite-state shift (base) space whose fiber maps are C1 injective maps on the unit interval. We show that certain invariant sets have a multi-graph structure and can be written graphs of one, two or more functions defined on the base. In particular, this applies to any hyperbolic set and to the support of any ergodic hyperbolic measure. Moreover, within the class of step skew-products whose interval maps are “absorbing”, open and densely the phase space decomposes into attracting and repelling double-strips such that their attractors and repellers are graphs of one single-valued or bi-valued continuous function almost everywhere, respectively.
We address the problem of existence and uniqueness (finite- ness) of ergodic equilibrium states for a natural class of partially hyperbolic dynamics homotopic to Anosov. We propose to study the disintegration of equilibrium states along central foliation as a tool to develop the theory of equilibrium states for partially hyperbolic dynamics.
In this paper we mainly address the problem of disintegration of Lebesgue measure along the central foliation of volume-preserving diffeomorphisms isotopic to hyperbolic automorphisms of 3-torus. We prove that atomic disintegration of the Lebesgue measure (ergodic case) along the central foliation has the peculiarity of being mono-atomic (one atom per leaf). This implies the measurability of the central foliation. As a corollary we pro-vide open and nonempty subset of partially hyperbolic diffeomorphisms with minimal yet measurable central foliation.
We consider a minimal action of a finitely generated semigroup by homeomorphisms of the circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov’s theorem and its corollaries.
We study the synchronization properties of the random double rotations on
tori. We give a criterion that show when synchronization is present in the case
of random double rotations on the circle and prove that it is always absent in
dimensions two and higher.
In this chapter we study basic connections between compositions of independent random transformations and corresponding Markov chains together with some applications.
Ergodic theory of dynamical systems i.e., the qualitative analysis of iterations of a single transformation is nowadays a well developed theory. In 1945 S. Ulam and J. von Neumann in their short note [44] suggested to study ergodic theorems for the more general situation when one applies in turn different transformations chosen at random. Their program was fulfilled by S. Kakutani [23] in 1951. ‘Both papers considered the case of transformations with a common invariant measure. Recently Ohno [38] noticed that this condition was excessive. Ergodic theorems are just the beginning of ergodic theory. Among further major developments are the notions of entropy and characteristic exponents.
In this paper we first give a formulation of SRB (Sinai–Ruelle–Bowen) property for invariant measures of stationary random dynamical systems and then prove that this property is sufficient and necessary for a formula of Pesin's type relating entropy and Lyapunov exponents of such dynamical systems. This result is a random version of the main result in Part I of Ledrappier and Young's celebrated paper [11].
The ergodic theory of uniformly hyperbolic, or “Axiom A”, diffeomorphisms has been studied extensively, beginning with the pioneering work of Anosov, Sinai, Ruelle and Bowen. While uniformly hyperbolic systems enjoy strong mixing properties, they are not dense among C 1 diffeomorphisms. Using the concept of Lyapunov exponents, Pesin introduced a weaker form of hyperbolicity, which he termed nonuniform hyperbolicity. Nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones. Our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms. As a curious by-product of our construction, we prove that a pathological feature of central foliations – the complete failure of absolute continuity – can exist in a C 1 -open set of volume-preserving diffeomorphisms.
this paper we construct new examples of diffeomorphisms with robust statistical properties. Our interest in these examples arose from the insight they might give into a collection of natural, optimistic and still unresolved problems in the global theory of dynamical systems. For a compact C
Fubini foiled: Katok's paradoxical example in measure theory Pathological foliations and removable zero exponents
- J Milnor
- Sw
- M Shub
- A Wilkinson
- Sw
- M Shub
- A Wilkinson
Milnor, J.: Fubini foiled: Katok's paradoxical example in measure theory. Math. Intelligencer 19, no. 2, 30–32 (1997) [SW1] Shub, M. and Wilkinson, A.: Pathological foliations and removable zero exponents. Inv. Math. 139, 495–508 (2000) [SW2] Shub, M. and Wilkinson, A.: A stably Bernoullian diffeomorphism that is not Anosov. Preprint Communicated by Ya. G. Sinai