S-limit shadowing is generic for continuous Lebesgue measure-preserving circle maps

Jozef Bobok,Jernej Činč,Piotr Oprocha,Serge Troubetzkoy

doi:10.1017/etds.2021.112

Jozef Bobok, Jernej Činč + Show 2 more

Open Access

https://doi.org/10.1017/etds.2021.112

Copy DOI

Abstract

AbstractIn this paper we show that generic continuous Lebesgue measure-preserving circle maps have the s-limit shadowing property. In addition, we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well.

Highlights

The notion of shadowing is a classical notion in the theory of dynamical systems
The shadowing property ensures that computational errors do not accumulate
In the following sense: in systems with the shadowing property the approximate trajectories will reflect real dynamics up to some small error that is made at each iteration

Summary

Introduction

The notion of shadowing (or pseudo orbit tracing; see Definition 2.2) is a classical notion in the theory of dynamical systems. Mizera proved [19] that shadowing is a generic property in the class of continuous maps of the interval or circle These results were extended to many other one-dimensional spaces; see [12, 14, 18]. The particular setting that we are interested in this paper is the family of continuous Lebesgue measure-preserving maps of the unit circle Cλ (S1) endowed with the topology of uniform convergence, which makes it a complete space. We denote by C(S1) the class of continuous maps of the circle endowed with the topology of uniform convergence In this setting we obtain the following two new results.

Preliminaries

The proof