Observable Lyapunov irregular sets for planar piecewise expanding maps

Yushi Nakano; Teruhiko Soma; Kodai Yamamoto

doi:10.3934/dcds.2023027

Article Contents

2023, Volume 43, Issue 7: 2737-2755. Doi: 10.3934/dcds.2023027

This issue Previous Article Partial regularity of the fractional Gelfand-Liouville system Next Article Propagation dynamics of a nonlocal reaction-diffusion system

Observable Lyapunov irregular sets for planar piecewise expanding maps

1.
Department of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratuka, Kanagawa, 259-1292, Japan
2.
Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan
3.
Joint Graduate School of Mathematics for Innovation, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

^*Corresponding author: Teruhiko Soma
^*Corresponding author: Teruhiko Soma

Received: June 2022

Revised: February 2023

Early access: March 2023

Published: July 2023

This work was partially supported by JSPS KAKENHI Grant Number 19K14575, 22K03342 and WISE program (MEXT) at Kyushu University

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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 [23]. 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 [5].

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Mathematics Subject Classification: Primary: 37C05, 37C30, 37C40.

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Figure 2.1. Splitting of $ D $

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Figure 3.1. The case of $ 1\leq k+1<\gamma^m-\gamma^{m-1} $. When $ k+1 = \gamma^m-\gamma^{m-1} $, $ R(k+1,m) $ is contained in $ D_{0,-} $

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Figure 3.2.

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Figure A.1. $ K_k $ is the $ \Delta(k) $-neighborhood of $ J_k $ in $ \mathbb{R} $

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