Heber Enrich

A heteroclinic bifurcation of Anosov diffeomorphisms

We study some diffeomorphisms in the boundary of the set of Anosov diffeomorphisms mainly from the ergodic viewpoint. We prove that these diffeomorphisms, obtained by isotopy from an Anosov f : M 7→ M through a heteroclinic tangency, determine a manifold M of finite codimension in the set of C diffeomorphisms. We prove that any diffeomorphism F in M is conjugate to f ; moreover, there exists a unique SRB measure for F , and F is Bernoulli with respect to this measure. In particular, if the dimension of M is two, and μ is a volume element, we prove that the isotopy can be taken such that the measure is preserved. 0. Introduction 0.1. The understanding of the process of loss of hyperbolicity is an old problem. One way it is lost, starting from an Anosov diffeomorphism, is by creating a map which is derived from Anosov, where one of the eigenvalues at a fixed point (or two if they are complex conjugate) is pushed to the boundary of the unit circle. They maintain some of the topological and ergodic properties (see, among others, [L80], [C93], and [HY95]). There are other ways to arrive at the boundary B of the Anosov diffeomorphisms with local bifurcations; for instance, as a consequence of several general ideas, in [L80] an example in which the stable and unstable manifolds of a fixed point are modified until they become tangent at the fixed point is shown. The result is a diffeomorphism conjugate to an Anosov map; in particular, there appear two invariant foliations conjugate to the stable and unstable foliations of the Anosov map. In the same article, an example of a Kupka–Smale map in B conjugate to an Anosov map is shown, which therefore inherits many interesting properties that can be considered as obtained from an Anosov map through a global bifurcation. In the same vein, we study here some dynamical properties of a kind of diffeomorphism in B. ∗ To the memory of Professor R. Mane, who has been my advisor during the preparation of this article, a part of which has been presented as a doctoral thesis. † Also at IMPA/CNPq, Brazil.

The Pesin entropy formula for diffeomorphisms with dominated splitting

For any

Homoclinic tangencies near cascades of period doubling bifurcations

C^1

Equilibrium states and SRB-like measures of

diffeomorphism with dominated splitting, we consider a non-empty set of invariant measures that describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating sub-bundle. As a consequence, if those exponents are non-negative, and if the exponents on the dominated sub-bundle are non-positive, those measures satisfy the Pesin entropy formula.

C^1

Abstract We consider perturbations of the Feigenbaum map in n dimensions. In the analytic topology we prove that the maps that are accumulated by period doubling bifurcations are approximable with homoclinic tangencies. We also develop a n-dimensional Feigenbaum theory in the Cr topology, for r large enough. We apply this theory to extend the result of approximation with homoclinic tangencies for Cr maps.

-expanding maps of the circle

For any C 1 expanding map f of the circle we study the equilibrium states for the potential = logjf 0 j. We formulate a C 1 generalization of Pesin’s Entropy Formula that holds for all the SRB measures if they exist, and for all the (necessarily existing) SRB-like measures. In theC 1 -generic case Pesin’s Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non generic case for which no SRB measure exists.

Weak Pseudo-Physical Measures and Pesin's Entropy Formula for Anosov C1 diffeomorphisms

We consider C1 Anosov diffeomorphisms on a compact Riemannian manifold. We define the weak pseudo-physical measures, which include the physical measures when these latter exist. We prove that ergodic weak pseudo-physical measures do exist, and that the set of invariant probability measures that satisfy Pesins Entropy Formula is the weak*-closed convex hull of the ergodic weak pseudo-physical measures. In brief, we give in the C1-scenario of uniform hyperbolicity, a characterization of Pesins Entropy Formula in terms of physical-like properties.

The real analytic Feigenbaum–Coullet–Tresser attractor in the disc

We consider a real analytic diffeomorphism ψ0 on an n-dimensional disc 𝒟, n ≥ 2, exhibiting a Feigenbaum–Coullet–Tresser (FCT) attractor. We assume that in the C ω(𝒟) topology it is far from the standard FCT map φ0 fixed by the double renormalization. We prove that ψ0 persists along a codimension-one manifold ℳ ⊂ C ω(𝒟), and that it is the bifurcating map along any one-parameter family in C ω(𝒟) transversal to ℳ, from diffeomorphisms exhibiting sinks to those which exhibit chaos, filling a gap in the usually accepted proof of this assertion. The main tool in the proofs is a theorem of functional analysis, which we state and prove in this article, characterizing the existence of codimension-one submanifolds in any abstract functional Banach space.

Simultaneous continuation of infinitely many sinks at homoclinic bifurcations

We prove that the

SRB measures of certain almost hyperbolic diffeomorphisms with a tangency

C^3