Holomorphic Anosov systems

Proceedings of the American Mathematical Society, 2019

We consider Anosov diffeomorphisms on T 3 such that the tangent bundle splits into three subbundles E s f ⊕ E wu f ⊕ E su f. We show that if f is C r , r ≥ 2, volume preserving, then f is C 1 conjugated with its linear part A if and only if the center foliation F wu f is absolutely continuous and the equality λ wu f (x) = λ wu A , between center Lyapunov exponents of f and A, holds for m a.e. x ∈ T 3. We also conclude rigidity of derived from Anosov diffeomorphism, assuming an strong absolute continuity property (Uniform bounded density property) of strong stable and strong unstable foliations.

2019

We obtain three main results about smooth group actions on surfaces. Our first theorem states that if a group of diffeomorphisms of a surface contains an Anosov diffeomorphism then the group contains a free subgroup or preserves one of the stable or unstable foliations up to finite index. We consider this result as a version of Tits alternative for diffeomorphism group. This theorem combined with various techniques including properties of Misiurewicz-Ziemian rotation sets, Herman-Yoccoz Theory of circle diffeomorphisms and the Ledrappier-Young entropy formula, etc, give us our second theorem, which is a global rigidity result about Abelian-by-Cyclic group actions on surfaces in the presence of an Anosov diffeomorphism. This gives a complete classification of Abelian-by-Cyclic group actions on the two-torus up to topological conjugacy and up to finite covers. Furthermore, the group structure combined with the theory of SRB measures and measures of maximal entropy yields the third mai...

2011

We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson-Varolin and by Peters; in particular, we prove (without using Oseledec's theory) that the stable manifold of any point where the negative Lyapunov exponents are well-defined is biholomorphic to a complex Euclidean space. Our approach is based on the solution of a linear control problem in spaces of subexponential sequences, and on careful estimates of the norm of hte conjugacy operator by a lower triangular matrix on the space of \textit{k}-homogeneous polynomial endomorphisms of \mathbb{C}^{d}.

Springer Proceedings in Mathematics, 2011

We present an infinite dimensional space of C 1+ smooth conjugacy classes of circle diffeomorphisms that are C 1+ fixed points of renormalization. We exhibit a one-to-one correspondence between these C 1+ fixed points of renormalization and C 1+ conjugacy classes of Anosov diffeomorphisms.

The Michigan Mathematical Journal, 2001

Journal of Mathematical Analysis and Applications

Given a C 2 − Anosov diffemorphism f : M → M, we prove that the jacobian condition J f n (p) = 1, for every point p such that f n (p) = p, implies transitivity. As application in the celebrated theory of Sinai-Ruelle-Bowen, this result allows us to state a classical theorem of Livsic-Sinai without directly assuming transitivity as a general hypothesis. A special consequence of our result is that every C 2 −Anosov diffeomorphism, for which every point is regular, is indeed transitive.

Journal of Geometric Analysis, 2006

We prove that a pseudoholomorphic diffeomorphism between two almost complex manifolds with boundaries satisfying some pseudoconvexity type conditions cannot map a pseudoholomorphic disc in the boundary to a single point. This can be viewed as an almost complex analogue of a well known theorem of J.E.Fornaess.

2009

In 1969, Hirsch posed the following problem: given a diffeomorphism f : N → N , and a compact invariant hyperbolic set Λ of f , describe the topology of Λ and the dynamics of f restricted to Λ. We solve the problem where Λ = M 3 is a closed 3-manifold: if M 3 is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions. The dynamics of the diffeomorphisms restricted to M 3 , called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions.

Siberian Mathematical Journal, 2004

This article is devoted to the algebraic approaches to Anosov diffeomorphisms. All examples of Anosov diffeomorphisms known so far are connected directly or indirectly with compact nilmanifolds. We consider some new necessary conditions for existence of these diffeomorphisms on nilmanifolds. We demonstrated the absence of Anosov diffeomorphisms on some classes of nilmanifolds. We also prove some results on Lie groups and lattices in them.

Annales de l’institut Fourier, 2011