Some results on perturbations to Lyapunov exponents

TL;DR

This paper investigates how small perturbations affect Lyapunov exponents, demonstrating the ability to eliminate zero exponents, distinguish all exponents, and exploring spectral properties in various dynamical contexts.

Contribution

It provides new results on perturbing zero Lyapunov exponents and analyzes spectral properties, including genericity and complex eigenvalues, in partially hyperbolic and conservative diffeomorphisms.

Findings

Zero Lyapunov exponents can be perturbed into nonzero ones.

Diffeomorphisms with non-simple spectrum are generically prevalent.

Ergodic conservative diffeomorphisms far from tangencies form a residual set.

Abstract

In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.