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Rigidity of stable Lyapunov exponents and integrability for Anosov maps
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Abstract
Let f be a non-invertible irreducible Anosov map on d-torus. We show that if the stable bundle of f is one-dimensional, then f has the integrable unstable bundle, if and only if, every periodic point of f admits the same Lyapunov exponent on the stable bundle with its linearization. For higher-dimensional stable bundle case, we get the same result on the assumption that f is a -perturbation of a linear Anosov map with real simple Lyapunov spectrum on the stable bundle. In both cases, this implies if f is topologically conjugate to its linearization, then the conjugacy is smooth on the stable bundle.
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We show that in the absence of periodic centre annuli, a partially hyperbolic surface endomorphism is dynamically coherent and leaf conjugate to its linearisation. We proceed to characterise the dynamics in the presence of periodic centre annuli. This completes a classification of partially hyperbolic surface endomorphisms.
In this paper, we prove that every specially absolutely partially hyperbolic endomorphism f on is dynamically coherent. The key point is that we rule out Reeb components in the unstable foliation of f . Moreover, we exhibit a specially partially hyperbolic endomorphism on , which preserves a Reeb foliation as unstable foliation and is not dynamically coherent.
We prove several limit theorems for a simple class of partially hyperbolic fast–slow systems. We start with some well know results on averaging, then we give a substantial refinement of known large (and moderate) deviation results and conclude with a completely new result (a local limit theorem) on the distribution of the process determined by the fluctuations around the average. The method of proof is based on a mixture of standard pairs and transfer operators that we expect to be applicable in a much wider generality.
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.
In this paper we give a classification of special endomorphisms of nil-manifolds: Let be a covering map of a nil-manifold and denote by the nil-endomorphism which is homotopic to f. If f is a special TA-map, then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.
We consider a class of partially hyperbolic systems on described by maps where are expanding maps of the circle. For sufficiently small and generic in an open set, we precisely classify the SRB measures for and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate.
Despite the invertible setting, Anosov endomorphisms may have infinitely many
unstable directions. Here we prove, under transitivity assumption, that an
Anosov endomorphism on a closed manifold M, is either special (that is, every
has only one unstable direction) or for a typical point in M there
are infinitely many unstable directions. Other result of this work is the semi
rigidity of the unstable Lyapunov exponent of a codimension one
Anosov endomorphism and close to a linear endomorphism of
for In the appendix we give a proof for ergodicity of
conservative Anosov endomorphism.
We establish a strong form of local rigidity for hyperbolic automorphisms of
the 3-torus with real spectrum. Namely, let
be a hyperbolic automorphism of the 3-torus with real spectrum and let f be a
small perturbation of L. Then f is smoothly () conjugate to
L if and only if obstructions to conjugacy given by the eigenvalues at
periodic points of f vanish. By combining our result and a local rigidity
result of Kalinin and Sadovskaya for conformal automorphisms this completes the
local rigidity program for hyperbolic automorphisms in dimension 3. Our work
extends de la Llave-Marco-Moriy\'on 2-dimensional local rigidity theory.
We show that partially hyperbolic diffeomorphisms of the 3-torus are dynamically coherent.
We give a new proof of the fact that the eigenvalues at correspondig periodic orbits form a complete set of invariants for the smooth conjugacy of low dimensional Anosov systems. We also show that, if a homeomorphism conjugating two smooth low dimensional Anosov systems is absolutely continuous , then it is as smooth as the maps. We furthermore prove generalizations of thse facts for non-uniformly hyperbolic systems as well as extensions and counterexaples in higher dimensions. 1. Introduction The main purpose of this paper is to present a new proof of the following Theorem 1.1. The new methods we use, allow us to some generalizations, which have not appeared before among them Theorem 1.2 and Theorem 1.3 Theorem 1.1. Let f ,g be two C k k = 2; 3; : : : ; 1;! Anosov diffeomorphisms of a compact two dimensional manifold M (resp. oe t , / t two Anosov flows of a three dimensional manifold) and h a homeomorphism of M satisfying: (1:1) h ffi f = g ffi h ( resp. h ffi oe t = / t ffi h...
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We study the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L4 is irreducible over ℚ. We consider a volume-preserving C1-small perturbation f of L. We show that if Lyapunov exponents of f with respect to the volume are the same as Lyapunov exponents of L, then f is C1+Hölder conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.
We prove that a class of weakly partially hyperbolic endomorphisms on are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on which does not admit a centre foliation.
In this paper we introduce the notion of "robust special Anosov endomorphisms", and show that Anosov endomorphisms of tori which are not neither an Anosov diffeomorphism
nor an expanding map, are not robust special.
We give three equivalent conditions for non-accessibility of an Anosov diffeomorphism on the 3-torus with a partially hyperbolic splitting. Since accessibility is an open property, this gives a negative answer to Hammerlindl’s question about homology boundedness of strong unstable foliation.
We shall prove the following Theorem. Let Fs and Fu be two continuous transverse foliations with uniformly smooth leaves, of some manifold. If f is uniformly smooth along the leaves of Fs and Fu, then f is smooth.
A partially hyperbolic diffeomorphism is dynamically coherent if its center, center-stable and center-unstable invariant distributions are integrable, i.e. tangent to foliations. Dynamical coherence is a key assumption in the theory of stable ergodicity. The main result: a partially hyperbolic diffeomorphism f\colon M\to M is dynamically coherent if the strong stable and unstable foliations are quasi-isometric in the universal cover \widetilde{M}, i.e. for any two points in the same leaf, the distance between them in \widetilde{M} is bounded from below by a linear function of the distance along the leaf.
In this paper we show that a little hyperbolicity goes a long way toward guaranteeing stable ergodicity, and in fact may be necessary for it. Our main theorem may be interpreted as saying that the same phenomenon producing chaotic behavior (i.e., some hyperbolicity) also leads to robust statistical behavior. Examples to which our theory applies include translations on certain homogeneous spaces and the time-one map of the geodesic flow for a manifold of constant negative curvature.
We consider an irreducible Anosov automorphism L of a torus T^d such that no
three eigenvalues have the same modulus. We show that L is locally rigid, that
is, L is C^1 conjugate to any C^1-small perturbation f with the same periodic
data. We also prove that toral automorphisms satisfying these assumptions are
generic in SL(d,Z). Examples constructed in the Appendix by Rafael de la Llave
show importance of the assumption on the eigenvalues.
Let L be a hyperbolic automorphism of , . We study the smooth conjugacy problem in a small -neighborhood of L. The main result establishes regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are -close to an irreducible linear hyperbolic automorphism L with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on with the same constant periodic eigenvalue data that are only H\"older conjugate. We show that these examples exhaust all possible ways to perturb conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of -close to the original example. Comment: Theorem B as stated was wrong in version 1. In this version the statement and the proof are corrected. A number of improvements was made thanks to the referee reports
We consider dynamical systems generated by partially hyperbolic surface endomorphisms of class C^r with one-dimensional strongly unstable subbundle. As the main result, we prove that such a dynamical system generically admits finitely many ergodic physical measures whose union of basins of attraction has total Lebesgue measure, provided that r>= 19.
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