Anosov Diffeomorphisms Research Articles

Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms

Abstract For a class of volume-preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of f is bounded above (respectively below) by the sum of the positive (respectively negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math.115(1) (2000), 157–193]. The results in this paper address a flexibility program by Bochi, Katok and Rodriguez Hertz [Flexibility of Lyapunov exponents. Ergod. Th. & Dynam. Sys.42(2) (2022), 554–591].

Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^{3}$

We show that within a C^{1} -neighborhood {\mathcal{U}} of the set of volume preserving Anosov diffeomorphisms on the three-torus {\mathbb{T}^3} which are strongly partially hyperbolic with expanding center, any f\in{\mathcal{U}}\cap{\operatorname{Diff}^2(\mathbb{T}^3)} satisfies the dichotomy: either the strong-stable and strong-unstable bundles E^{s} , E^{u} of f are jointly integrable, or any fully supported u -Gibbs measure of f is SRB.

Regularity with Respect to the Parameter of Lyapunov Exponents for Diffeomorphisms with Dominated Splitting

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.

On non-trivial hyperbolic sets and their bifurcations in families of diffeomorphisms of a two-dimensional torus.

We propose a simple model-two-parameter family of diffeomorphisms of a two-dimensional torus. Combining analytical and numerical methods, we find regions in the parameter plane such that each diffeomorphism of the family is hyperbolic and describe the structure of the corresponding hyperbolic sets. We also study bifurcations on the boundaries of these regions, which lead to the change of hyperbolicity type (from Anosov diffeomorphisms to DA-diffeomorphisms).

Transitive Centralizer and Fibered Partially Hyperbolic Systems

Abstract We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension $f_{0}\colon M\to M$ of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any $f\in \textrm{Diff}^{\infty }(M)$ sufficiently $C^{1}$-close to $f_{0}$ has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then $f$ is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we classify the centralizer of any partially hyperbolic diffeomorphism of a $3$-dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually ${{\mathbb{Z}}}\times{{\mathbb{T}}}$.

On the global product structure for uniformly quasiconformal Anosov diffeomorphisms

On the global product structure for uniformly quasiconformal Anosov diffeomorphisms

Homoclinic points of symplectic partially hyperbolic systems with 2D centre

We consider a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with area-preserving diffeomorphisms and prove that every hyperbolic periodic point has transverse homoclinic points.

Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus

Let f f and g g be two Anosov diffeomorphisms on T 3 \mathbb {T}^3 with three-subbundles partially hyperbolic splittings where the weak stable subbundles are considered as center subbundles. Assume that f f is conjugate to g g and the conjugacy preserves the strong stable foliation, then their center Lyapunov exponents of corresponding periodic points coincide. This is the converse of the main result of Gogolev and Guysinsky [Discrete Contin. Dyn. Syst. 22 (2008), pp. 183–200]. Moreover, we get the same result for partially hyperbolic diffeomorphisms derived from Anosov on T 3 \mathbb {T}^3 .

Joint integrability and spectral rigidity for Anosov diffeomorphisms

AbstractLet be an Anosov diffeomorphism whose linearization is irreducible. Assume that is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable subbundle and the unstable subbundle are jointly integrable, then is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization . Moreover, admits the finest dominated splitting in the weak stable subbundle with dimensions matching those for , and it has spectral rigidity along all these subbundles. In dimension 4, we also obtain a similar result by grouping the weak stable and unstable subbundles together as a center subbundle and assuming joint integrability of the strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism that is ‐close to an irreducible nonconformal automorphism , the extremal subbundles of are jointly integrable if and only if is smoothly conjugate to .

Eventual Fitting Shadowing Property for Hyperbolic Dynamical Systems

Let be a diffeomorphism map on a closed smooth manifold for dimension . We explain in this work any chain transitive set of generic diffeomorphism , if a diffeomorphism has another type of shadowing property is called, the eventual shadowing property on locally maximal chain transitive set, then is hyperbolic. In general, the eventual fitting shadowing property is not fulfilled in hyperbolic dynamical systems (satisfy in case L is Anosov diffeomorphism map) . In this paper, several concepts were presented. These concepts can be re-examined on other important spaces, and their impact on finding dynamical characteristics that may be employed in solving some mathematical problems.

Cartan actions of higher rank abelian groups and their classification

We study R k × Z ℓ \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.

A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics

AbstractFor smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gouëzel, Keller and Liverani [28, 33], which has been applied to deterministic systems with great success, may be adapted to study random systems that possess good mixing properties. As a consequence, we obtain general linear and higher-order response results, as well as the differentiability of the variance in quenched central limit theorems (CLTs), for random dynamical systems (RDSs) that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasize that our results apply to random dynamical systems over a general ergodic base map, and are obtained without resorting to infinite-dimensional multiplicative ergodic theory.

Hyperbolicity of maximal entropy measures for certain maps isotopic to Anosov diffeomorphisms

We prove the hyperbolicity of ergodic maximal entropy measures for a class of partially hyperbolic diffeomorphisms of Td, which have a compact two-dimensional center foliation.

Global rigidity of higher rank lattice actions with dominated splitting

AbstractLet $\alpha $ be a $C^{\infty }$ volume-preserving action on a closed n-manifold M by a lattice $\Gamma $ in $\mathrm {SL}(n,\mathbb {R})$ , $n\ge 3$ . Assume that there is an element $\gamma \in \Gamma $ such that $\alpha (\gamma )$ admits a dominated splitting. We prove that the manifold M is diffeomorphic to the torus ${{\mathbb T}^{n}={\mathbb R}^{n}/{\mathbb Z}^{n}}$ and $\alpha $ is smoothly conjugate to an affine action. Anosov diffeomorphisms and partial hyperbolic diffeomorphisms admit a dominated splitting. We obtained a topological global rigidity when $\alpha $ is $C^{1}$ . We also prove similar theorems for actions on $2n$ -manifolds by lattices in $\textrm {Sp}(2n,{\mathbb R})$ with $n\ge 2$ and $\mathrm {SO}(n,n)$ with $n\ge 5$ .

On closed manifolds admitting an Anosov diffeomorphism but no expanding map

A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where a nilmanifold is defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, which is conjectured to be the only type of manifolds admitting Anosov diffeomorphisms up to homeomorphism. The proof shows how to construct positive gradings from the eigenvalues of the Anosov diffeomorphism under some additional assumptions related to the rank, using the action of the Galois group on these algebraic units.

High pointwise emergence via saturated sets

In this article, we prove that the set of points whose pointwise emergence have same degree in with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

Structural stability for fibrewise Anosov diffeomorphisms on principal torus bundles

AbstractWe show that a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.

Globally Coupled Anosov Diffeomorphisms: Statistical Properties

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, h_varepsilon . Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map varepsilon mapsto h_varepsilon is Lipschitz continuous.

Topological structures on saturated sets, optimal orbits and equilibrium states

In this paper, we investigate topological complexity of saturated set from the viewpoints of upper capacity entropy and packing entropy. We obtain that if a topological dynamical system $ (X, T) $ satisfies almost product property, then for each nonempty compact convex subset $ K $ of invariant measures, the following two formulas hold for the saturated set $ G_K $:$ \begin{equation*} h_{top}^{UC}(T, G_{K}) = h_{top}(T, X)\ \mathrm{and}\ h_{top}^{P}(T, G_{K}) = \sup\{h(T, \mu):\mu\in K\}, \end{equation*} $where $ h_{top}^{UC}(T, G_{K}) $ is the upper capacity entropy of $ T $ on $ G_{K}, $ $ h_{top}^{P}(T, G_{K}) $ is the packing entropy of $ T $ on $ G_{K}, $ and $ h(T, \mu) $ is the Kolmogorov-Sinai entropy of $ \mu $. As applications, for a transitive Anosov diffeomorphism on a compact manifold, we use the above results to quantify topological complexity of optimal orbits and equilibrium states for both real and matrix valued potentials.

Formulae of volume growth for random Anosov diffeomorphisms

For $ C^2 $ random Anosov diffeomorphisms, we give a formula describing the volume growth by measure-theoretic fiber entropy and Lyapunov exponents. Furthermore, the topological fiber entropy coincides with the volume growth for $ C^2 $ fiber hyperbolic systems, which was for $ C^{\infty} $ random dynamical systems in [15] and $ C^2 $ Axiom A systems in [10]. We establish two tools, a volume lemma and a variational principle for random radii in $ C^2 $ random Anosov diffeomorphisms.