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Topological Transitivity of Nonautonomous Dynamical Systems

Michal Málek

TL;DR

This work extends the study of topological transitivity from autonomous to nonautonomous dynamical systems on compact metric spaces by translating classical $TT$ and $DO$ notions into nonautonomous analogues and embedding them within KS’s framework of $12$ conditions. It develops a detailed hierarchy of implications among $f_{1,n}$-image and $f_{1,n}$-preimage properties, and identifies key equivalences, notably $TT_{0} \Leftrightarrow DDO$ on general compact spaces and $D\omega \Rightarrow TT$ with $TT \Rightarrow TT_{0}$, while showing several non-implications via counterexamples. In spaces without isolated points, the authors prove a ten-condition equivalence among transitivity-type properties and establish $DDO \Rightarrow D\omega$, linking dense-orbit behavior to full omega-limit coverage. The results provide a cohesive foundation for transitivity in nonautonomous systems and clarify how KS-type conditions transfer to nonautonomous dynamics, including when uniform convergence or surjectivity fails to guarantee $TT$.

Abstract

This paper explores the concept of topological transitivity in nonautonomous dynamical systems, which are defined as sequences of continuous maps from a compact metric space to itself. It investigates various conditions (including intersection of any pair of open sets and existence of a dense orbit) that could be taken as definitions of the topological transitivity of a nonautonomous system, and addresses their relation both in the case of a general compact metric space and in the case where, in addition, the space has no isolated point. This provides the necessary basis for further investigation of transitivity of nonautonomous dynamical systems.

Topological Transitivity of Nonautonomous Dynamical Systems

TL;DR

This work extends the study of topological transitivity from autonomous to nonautonomous dynamical systems on compact metric spaces by translating classical and notions into nonautonomous analogues and embedding them within KS’s framework of conditions. It develops a detailed hierarchy of implications among -image and -preimage properties, and identifies key equivalences, notably on general compact spaces and with , while showing several non-implications via counterexamples. In spaces without isolated points, the authors prove a ten-condition equivalence among transitivity-type properties and establish , linking dense-orbit behavior to full omega-limit coverage. The results provide a cohesive foundation for transitivity in nonautonomous systems and clarify how KS-type conditions transfer to nonautonomous dynamics, including when uniform convergence or surjectivity fails to guarantee .

Abstract

This paper explores the concept of topological transitivity in nonautonomous dynamical systems, which are defined as sequences of continuous maps from a compact metric space to itself. It investigates various conditions (including intersection of any pair of open sets and existence of a dense orbit) that could be taken as definitions of the topological transitivity of a nonautonomous system, and addresses their relation both in the case of a general compact metric space and in the case where, in addition, the space has no isolated point. This provides the necessary basis for further investigation of transitivity of nonautonomous dynamical systems.
Paper Structure (4 sections, 5 theorems, 4 equations)

This paper contains 4 sections, 5 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Transitivity of nonautonomous dynamical system
  3. Case of compact metric spaces
  4. Spaces without an isolated point

Key Result

Theorem 1

Let $f_{1,\infty}$ be a nonautonomous dynamical system on $X$ possessing TT$_0$ property then it also has DO.

Theorems & Definitions (16)

  • Theorem 1: TT$_0$ $\Rightarrow$ DO
  • proof
  • Example 2: $\omega \not \Rightarrow$ TT$_0$
  • proof
  • Example 3: TT $\not\Rightarrow\omega$, TT$_0 \not\Rightarrow \omega$ TT
  • proof
  • Theorem 4: TT$_0 \Leftrightarrow$ DDO
  • proof
  • Theorem 5
  • proof
  • ...and 6 more