Related papers: Anosov Endomorphisms on the 2-torus: Regularity of…

In these expository notes, we give a proof of regularity of Anosov splitting for Anosov diffeomorphisms in a torus. We also generalize the idea to higher dimensions and to Anosov flows.

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…

In this paper we describe the tangent vectors of the stable and unstable manifold of a class of Anosov diffeomorphisms on the torus $\mathbb{T}^2$ using the method of formal series and derivative trees. We start with linear automorphism…

For a nonlinear Anosov diffeomorphism of the 2-torus, we present examples of measures so that the group of $\mu$-preserving diffeomorphisms is, up to zero-entropy transformations, cyclic. For families of equilibrium states $\mu$, we…

There exists a smooth foliation with 3 singular points on the two-dimensional torus such that any lifting of a leaf of this foliation on the universal covering of the torus is a dense subset of the covering.

A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two,…

We prove that if the stable foliation and the unstable foliation of an Anosov diffeomorphism on a connected compact manifold are $C^3$, then the diffeomorphism has fixed points. This is a partial positive answer to a Smale conjecture for…

Let $X_1^t$ and $X_2^t$ be volume preserving Anosov flows on a 3-dimensional manifold $M$. We prove that if $X_1^t$ and $X_2^t$ are $C^0$ conjugate then the conjugacy is, in fact, smooth, unless $M$ is a mapping torus of an Anosov…

We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over C and has a dense center foliation. For a small perturbation of L with a smooth center foliation we establish existence of a smooth leaf…

We give a new proof of a result by Fathi, which states that, to any homeomorphism of a closed surface which is isotopic to a pseudo-Anosov homeomorphism, we can associate a stable and an unstable invariant partition of the surface with…

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…

Within the smooth category, an intertwining is exhibited between the global rigidity of irreducible higher rank abelian Anosov actions on the n-torus and the classification of equilibrium-free flows on the n-torus that possess nontrivial…

We study cohomology of Holder continuous linear cocycles over a hyperbolic dynamical system and regularity of conjugacy between Anosov systems. For cocycles $A$ and $B$ with conjugate periodic data, we establish Holder cohomology under…

The original proof has a gap, and need extra hypothesis that the strong stable and strong unstable filiation both to be $C^1$. The argument is like the following: with the regularity, one can show that the weak-stable and weak-unstable…

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of…

We prove that a transitive uniformly $u$-quasiconformal Anosov diffeomorphism with a two-dimensional unstable distribution has a globally defined stable holonomy. As a corollary, we are able to remove an additional assumption in a theorem…

We prove the transitivity of real Anosov diffeomorphisms, which are Anosov diffeomorphisms where stable and unstable spaces decompose into a continuous sum of invariant one-dimensional sub-spaces with uniform contraction/expansion over the…

We study the existence of Anosov diffeomorphisms on complete open surfaces. We show that under the assumptions of density of periodic points and uniform geometry that such diffeomorphisms have a system of Margulis measures, which are a…

For groups of diffeomorphisms of $\T^2$ containing an Anosov diffeomorphism, we give a complete classification for polycyclic Abelian-by-Cyclic group actions on $\T^2$ up to both topological conjugacy and smooth conjugacy under mild…

We analyze a class of deformations of Anosov diffeomorphisms: these $C^0$-small, but $C^1$-macroscopic deformations break the topological conjugacy class but leave the high entropy dynamics unchanged. More precisely, there is a partial…