# Laypunov Irregular Points With Distributional Chaos

## TL;DR

This paper demonstrates that in certain dynamical systems with the exponential specification property, the set of points with divergent Lyapunov averages exhibits distributional chaos, contrasting with measure-zero results from classical theorems.

## Contribution

It establishes the presence of distributional chaos in Lyapunov-irregular sets for systems with exponential specification and Holder continuous cocycles, under conditions of multiple ergodic measures.

## Key findings

- Lyapunov-irregular set has distributional chaos of type 1
- Classical measure-zero results do not hold under specified conditions
- Distributional chaos occurs when multiple ergodic measures with different spectra exist

## Abstract

It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f with exponential specification property and a Holder continuous matrix cocycle A, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays distributional chaos of type 1.