Observable Lyapunov irregular sets for planar piecewise expanding maps

TL;DR

This paper constructs a family of 2D piecewise expanding maps with observable Lyapunov irregular sets and proves that such sets are not observable in real analytic cases, highlighting differences based on map regularity.

Contribution

It introduces a new family of maps with observable irregular sets and establishes non-observability results for analytic maps, advancing understanding of Lyapunov irregularity.

Findings

Constructed a family of maps with observable Lyapunov irregular sets.

Proved that Lyapunov irregular sets are not observable in real analytic maps.

Connected spectral analysis to irregular set observability.

Abstract

For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family $F_\sigma$ $(0<\sigma<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding maps such that each $F_\sigma$ has an observable (i.e. Lebesgue positive) Lyapunov irregular set. These maps are obtained by modifying the piecewise expanding map given in Tsujii (2000). In strong contrast to it, we also show that any Lyapunov irregular set of any 2-dimensional piecewise real analytic expanding map is not observable. This is based on the spectral analysis of piecewise expanding maps in Buzzi (2000).