Non-uniform hyperbolicity and universal bounds for S-unimodal maps

An S-unimodal map f is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the infimum of the Lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure. We prove that an S-unimodal map satisfies the Collet-Eckmann condition if and only if it has a uniform hyperbolic structure. The equivalence of several non-uniform hyperbolicity conditions follows. One consequence is that some renormalization of an S-unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the Collet-Eckmann condition is satisfied. The proof uses new universal bounds that hold for any S-unimodal map without periodic attractors. Oblatum 4-VII-1996 & 4-VII-1997     

Non-uniform hyperbolicity and existence of absolutely continuous invariant measures

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicityalongtheleavesimpliesexistenceofafinitenumberofergodicabsolutely continuous invariant probability measures which describe the asymptotics of almost every point. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain a partial extension of Keller’s theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly hyperbolic one dimensional maps.     

On non-uniform hyperbolicity assumptions in one-dimensional dynamics

We give an essentially equivalent formulation of the backward contracting property, defined by Juan Rivera-Letelier, in terms of expansion along the orbits of critical values, for complex polynomials of degree at least 2 which are at most finitely renormalizable and have only hyperbolic periodic points, as well as all C ³ interval maps with non-flat critical points.     

Prevalence of hyperbolicity for complex singular cycles

In this article we consider two kinds of complex singular cycles arising for vector fields defined on three-dimensional manifolds. We prove that, under some generic conditions, any one parameter family of vector fields passing through these cycles has the following property: Hyperbolicity is a prevalent phenomena.Dedicated to the memory of Professor R. Mañé.Partially Supported by Fondecyt 1941080 and Dirección de Investigación Universidad de Santiago de Chile     

Local hyperbolicity

The paper defines and studies a property of functions of several complex variables analytic at the origin which is called local hyperbolicity with respect to a given real direction. For homogeneous polynomials it reduces to ordinary hyperbolicity. Local hyperbolicity is also extended so as to correspond to the hyperbolicity of nonhomogeneous polynomials.     

Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics

Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale–Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.     

Normal hyperbolicity and linearisability

Summary It is well known that one can linearise a diffeomorphism near a compact invariant submanifold in the presence of 1-normal hyperbolicity. In this note we give a counterexample to a statement suggested by C. Pugh and M. Shub that one can weaken this normal hyperbolicity assumption.     

On the hyperbolicity constant in graphs

If X is a geodesic metric space and x₁,x₂,x₃∈X, a geodesic triangleT={x₁,x₂,x₃} is the union of the three geodesics [x₁x₂], [x₂x₃] and [x₃x₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if, for every geodesic triangle T in X, every side of T is contained in a δ-neighborhood of the union of the other two sides. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. . In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths , then , and if and only if G is isomorphic to C_m. Moreover, we prove the inequality for every graph, and we use this inequality in order to compute the precise value δ(G) for some common graphs.     

Aleksandrov surfaces and hyperbolicity

Aleksandrov surfaces are a generalization of two-dimensional Riemannian manifolds, and it is known that every open simply-connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). We prove a criterion for hyperbolicity of Aleksandrov surfaces which have nice tilings and where negative curvature dominates. We then apply this to generalize a result of Nevanlinna and give a partial answer for his conjecture about line complexes.

     

Pseudoconvexity and Gromov hyperbolicity

We give an estimate for the distance functions related to the Bergman, Carathéodory, and Kobayashi metrics on a bounded strictly pseudoconvex domain with C²-smooth boundary. Our formula relates the distance function on the domain with the Carnot- Carathéodory metric on the boundary. As a corollary we conclude that the domain equipped with the any of the standard invariant distances is hyperbolic in the sense of Gromov. When the boundary of the domain is C³-smooth, our estimate is exact up to a fixed additive term.     

Gromov hyperbolicity in Cartesian product graphs

If X is a geodesic metric space and x ₁, x ₂, x ₃ ∈ X, a geodesic triangle T = {x ₁, x ₂, x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) = inf{δ ≥ 0: X is δ-hyperbolic}. In this paper we characterize the product graphs G ₁ × G ₂ which are hyperbolic, in terms of G ₁ and G ₂: the product graph G ₁ × G ₂ is hyperbolic if and only if G ₁ is hyperbolic and G ₂ is bounded or G ₂ is hyperbolic and G ₁ is bounded. We also prove some sharp relations between the hyperbolicity constant of G ₁ × G ₂, δ(G ₁), δ(G ₂) and the diameters of G ₁ and G ₂ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the precise value of the hyperbolicity constant for many product graphs.     

On the Gromov hyperbolicity of the Kobayashi metric on strictly pseudoconvex regions in the almost complex case

We prove that every bounded strictly J-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We give various consequences of this fact, in particular concerning the dynamics of pseudo-holomorphic maps.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.