Abundance of observable Lyapunov irregular sets

TL;DR

This paper investigates the size of the set of points where Lyapunov exponents fail to exist, showing that in certain nonhyperbolic dynamical systems, this irregular set can have positive Lebesgue measure.

Contribution

It demonstrates that for specific surface diffeomorphisms with robust homoclinic tangency, the Lyapunov irregular set can have positive Lebesgue measure, extending previous examples.

Findings

Lyapunov irregular sets can have positive Lebesgue measure in certain dynamical systems.

Constructs examples where time averages both exist and do not exist on the irregular set.

Extends understanding of chaos and irregularity in nonhyperbolic dynamics.

Abstract

Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence is seldom discussed. In the present paper, we consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. The only known example with the Lyapunov irregular set of positive Lebesgue measure is a figure-8 attractor by the work of Ott and Yorke [OY2008], whose key mechanism (homoclinic loop) is easy to be broken by small perturbations. In this paper, we show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas [CV2001], as well as other several known nonhyperbolic dynamics, has the Lyapunov irregular set of positive Lebesgue measure. We can construct such positive Lebesgue measure sets both as the time averages exist and do not exist on it.