Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems

Abstract

Abstract. Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to "slow manifolds", which characterize the asymptotic convergence.Let {x i } i∈N be a regular orbit of a C 2 dynamical system f . Let S be a subset of its Lyapunov exponents. Assume that all the Lyapunov exponents in S are negative and that the sums of Lyapunov exponents in S do not agree with any Lyapunov exponent in the complement of S. Denote by E S x i the linear spaces spanned by the spaces associated to the Lyapunov exponents in S. We show that there are smooth manifolds W Sand. We establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non-resonance conditions. These systems of invariant manifolds are not, in general, a foliation.

Analytic reparametrization of semi-algebraic sets

“…These results have been applied in many questions of smooth dynamics. In the last several years more dynamical applications appeared in the study of the polynomial and rational dynamics in several complex variables, and in a more general context of the complexity of iterations [1,2,16,17,19,20,18,21,22,26,29,30,35,36,39,[43][44][45][46]51,52,64], as well as in the study of the behavior of discretized PDE's [42]. On the other hand, recently the C k -reparametrization theorem has been applied in the study of Anderson localization for Schrodinger operator on Z 2 with quasiperiodic potential [10,11].Yet another application appeared in counting rational points on and near algebraic varieties [47,54,49], see also [13,48].…”

Existence and Regularity of Invariant Graphs for Cocycles in Bundles: partial hyperbolicity case

“…item (1) in Theorem B, is well known in different settings including deterministic and random dynamical systems (see e.g. [HPS77,CL97,FdlLM06,LL10]). Reducing to the case that M consists of one element, it gives the smooth result about invariant ((strong) stable, center, pseudostable, etc) manifolds of an equilibrium (cf.…”

“…, consist of a finite sum of terms which can be explicitly calculated with the help of Faà-di Bruno formula (see e.g. [MR09,FdlLM06]); the non-constant factors are…”

The parameterization method for invariant manifolds III: overview and applications

“…The main difference is that in [dlLW97], the assumption on invertibility of the linearized operator used in the present paper is dropped, since the linearization of the evolution of an elliptic equation is a compact operator, hence, not invertible. The paper [FdlLM03] considers non-autonomous versions of non-resonant manifolds to orbits whose Lyapunov exponents are non-resonant. Indeed, the conditions of hyperbolicity in [FdlLM03] generalize the customary conditions in hyperbolicity-in which one allows some spread on the rate of expansion but requires uniformity-and those in non-uniform hyperbolic systems-in which one requires a single rate, but allows some non-uniformity along the orbit.…”