Uniform hyperbolicity revisited: Index of periodic points and equidimensional cycles

TL;DR

This paper investigates the structure of uniformly hyperbolic basic sets, showing the density of periodic points with simple Lyapunov spectra and exploring conditions under which all periodic points have robust simple spectra or can be approximated by equidimensional cycles, revealing mechanisms for coexistence of diverse periodic signatures.

Contribution

It proves the density of periodic points with simple Lyapunov spectra in basic sets and characterizes conditions for robust simple spectra or approximation by equidimensional cycles in three-dimensional manifolds.

Findings

Periodic points with simple Lyapunov spectrum are dense in basic sets.

Either all periodic points have robust simple spectra or can be approximated by equidimensional cycles.

Mechanism for coexistence of infinitely many periodic points with different signatures.

Abstract

In this paper we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r >= 1). In the case of the C1-topology we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1- robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the weak*-topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The later can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures.