Anosov Diffeomorphisms Research Articles

Livšic measurable rigidity for 𝒞¹ generic volume-preserving Anosov systems

In this paper, we prove that for C 1 \mathcal {C}^1 generic volume-preserving Anosov diffeomorphisms of a compact connected Riemannian manifold, the Livšic measurable rigidity theorem holds. We also give a parallel result for C 1 \mathcal {C}^1 generic volume-preserving Anosov flows.

A new method for constructing Anosov Lie algebras

It is conjectured that every closed manifold admitting an Anosov diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov diffeomorphism. The main theorem of this article gives a general method for constructing Anosov diffeomorphisms on nilmanifolds. As a consequence, we give new examples which were overlooked in a corollary of the classification of low-dimensional nilmanifolds with Anosov diffeomorphisms and a correction to this statement is proven. This method also answers some open questions about the existence of Anosov diffeomorphisms which are minimal in some sense.

Countable Alphabet Random Subhifts of Finite Type with Weakly Positive Transfer Operator

We deal with countable alphabet locally compact random subshifts of finite type (the latter merely meaning that the symbol space is generated by an incidence matrix) under the absence of big images property and under the absence of uniform positivity of the transfer operator. We first establish the existence of random conformal measures along with good bounds for the iterates of the Perron–Frobenius operator. Then, using the technique of positive cones and proving a version of Bowen’s type contraction (see Equilibrium states and the ergodic theory of Anosov diffeomorphisms, 1975), we also establish a fairly complete thermodynamical formalism. This means that we prove the existence and uniqueness of fiberwise invariant measures (giving rise to a global invariant measure) equivalent to the fiberwise conformal measures. Furthermore, we establish the existence of a spectral gap for the transfer operators, which in the random context precisely means the exponential rate of convergence of the normalized iterated transfer operator. This latter property in a relatively straightforward way entails the exponential decay of correlations and the Central Limit Theorem.

Nilpotent sums of lie algebras, and applications

We consider the nilpotent sum operation for arbitrary finite-dimensional Lie algebras. Some properties of this operation resemble those of the available nilpotent product operation for groups. We apply our results to constructing nilmanifolds and Anosov diffeomorphisms on them.

Partially hyperbolic diffeomorphisms with a trapping property

We study partially hyperbolic diffeomorphisms satisfying atrapping property which makes them look as if they were Anosov atlarge scale. We show that, as expected, they share severalproperties with Anosov diffeomorphisms. We construct an expansivequotient of the dynamics and study some dynamical consequencesrelated to this quotient.

Open sets of diffeomorphisms with trivial centralizer in the C1 topology

On the torus of dimension 2, 3 or 4, we show that the subset of diffeomorphisms with trivial centralizer in the C1 topology has nonempty interior. We do this by developing two approaches, the fixed point and the odd prime periodic point, to obtain a trivial centralizer for an open neighbourhood of Anosov diffeomorphisms arbitrarily near certain irreducible hyperbolic toral automorphisms.

Hyperbolicity, transitivity and the two-sided limit shadowing property

We explore the notion of two-sided limit shadowing property introduced by Pilyugin \cite{P1}. Indeed, we characterize the $C^1$-interior of the set of diffeomorphisms with such a property on closed manifolds as the set of transitive Anosov diffeomorphisms. As a consequence we obtain that all codimention-one Anosov diffeomorphisms have the two-sided limit shadowing property. We also prove that every diffeomorphism $f$ with such a property on a closed manifold has neither sinks nor sources and is transitive Anosov (in the Axiom A case). In particular, no Morse-Smale diffeomorphism have the two-sided limit shadowing property. Finally, we prove that $C^1$-generic diffeomorphisms on closed manifolds with the two-sided limit shadowing property are transitive Anosov. All these results allow us to reduce the well-known conjecture about the transitivity of Anosov diffeomorphisms on closed manifolds to prove that the set of diffeomorphisms with the two-sided limit shadowing property coincides with the set of Anosov diffeomorphisms.

Topological properties of invariant sets for two-dimensional hyperbolic toral automorphisms

We study topological properties of invariant sets of Anosov diffeomorphisms with holes. Results related to cardinality, local maximality, entropy and dimension are presented. The main result states that the Hausdorff dimension of the invariant set can be computed by the entropy of the invariant set together with the hyperbolicity constants.

Polynomial global product structure

An Anosov diffeomorphism is topologically conjugate to an infranilmanifold automorphism if and only if it has polynomial Global Product Structure.

Center foliation: absolute continuity, disintegration and rigidity

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

Cohomology of fiber bunched cocycles over hyperbolic systems

We consider Hölder continuous fiber bunched $\text{GL}(d,\mathbb{R})$-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of a diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question of whether an Anosov diffeomorphism is smoothly conjugate to a $C^{1}$-small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of $\text{GL}(d,\mathbb{R})$, for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.

Smooth stabilizers for measures on the torus

For a dissipative Anosov diffeomorphism $f$ of the 2-torus, we give examples of $f$-invariant measures $\mu$ such that the group of $\mu$-preserving diffeomorphisms is virtually cyclic.

Ergodicity and partial hyperbolicity on the 3-torus

If a non-ergodic, partially hyperbolic diffeomorphism on the 3-torus is homotopic to an Anosov diffeomorphism A, it is topologically conjugate to A.

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of $$d$$ -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.

Foliations and conjugacy: Anosov structures in the plane

AbstractIn a non-compact setting, the notion of hyperbolicity, together with the associated structure of stable and unstable manifolds (for unbounded orbits), is highly dependent on the choice of metric used to define it. We consider the simplest version of this, the analogue for the plane of Anosov diffeomorphisms, studied earlier by White and Mendes. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations. We show that if the structure of stable and unstable manifolds is required to be preserved by these conjugacies, the number of distinct equivalence classes of Anosov diffeomorphisms in the plane becomes infinite.

On the exponential rate of convergence in the Birkhoff ergodic theorem

These quantities can be estimated by using estimates for the probabilities of large deviations, i.e., for the quantities pn = λ{|Anf − f∗| ≥ e} (see [1], [2]). It turns out that if, for some essentially bounded function f not only the upper bound, but also the exact exponential asymptotic formula for the probability of large deviations is known, then the same formula is valid also for the rate of convergence in the Birkhoff theorem (see the theorem below). As an example, we shall apply this result to transitive Anosov diffeomorphisms. Suppose that f ∈ L∞(Ω) andΔ = ‖f − f∗‖∞. We assume that the function f ≡ f∗ a.e.; otherwise, the estimate is trivial: Pn = 0 for all n ∈ N and e > 0. For all real e > 0 and d > 1, we set

Manifolds with higher homotopy which do not support Anosov diffeomorphisms

We show that various classes of closed manifolds with non-trivial higher homotopy groups do not support (transitive) Anosov diffeomorphisms. In particular, we show that a finite product of spheres at least one of which is even-dimensional does not support transitive Anosov diffeomorphisms.

The space of Anosov diffeomorphisms

We consider the space X L of Anosov diffeomorphisms homotopic to a fixed automorphism L of an infranilmanifold M. We show that if M is the 2-torus T 2 , then X L is homotopy equivalent to T 2 . In contrast, if the dimension of M is large enough, then we show that X L is rich in homotopy and has infinitely many connected components.

Anosov diffeomorphisms on real Bott manifolds

In this paper we study a relation between the Anosov relation of a Anosov diffeomorphism of an n−dimensional real Bott manifold M (A )= T k ×( 2)s M (B) and that of an (n −k)−dimensional real Bott manifold M (B).

Pointwise hyperbolicity implies uniform hyperbolicity

We provide a general mechanism for obtaining uniform information from pointwise data. For instance, a diffeomorphism of a compact Riemannian manifold with pointwise expanding and contracting continuous invariant cone families is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.