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On Transitivities for Skew Products

Nayan Adhikary, Anima Nagar

Abstract

The dual concepts of `universality' and `hypercyclicity' are better understood and studied as `topological transitivity'. In this article we consider transitivity properties of skew products, essentially with non-compact fibers. We study the `Universality Conditions' and `Hypercyclicity Criterion' associated with the dynamical properties of transitivity, weakly mixing and mixing for these skew products.

On Transitivities for Skew Products

Abstract

The dual concepts of `universality' and `hypercyclicity' are better understood and studied as `topological transitivity'. In this article we consider transitivity properties of skew products, essentially with non-compact fibers. We study the `Universality Conditions' and `Hypercyclicity Criterion' associated with the dynamical properties of transitivity, weakly mixing and mixing for these skew products.

Paper Structure

This paper contains 9 sections, 16 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological Dynamics.
  4. Measurable Dynamics.
  5. Linear Dynamics
  6. Topological Transitivity of Skew Products for (Semi)Flows
  7. Skew Products (semi)flows
  8. Skew Product (semi)cascades
  9. Transitivity properties of skew products for linear operators

Key Result

Theorem 2.5

Bes2linear chaos Let $X$ be a second countable metric space. Then a commuting sequence $(T_n)$ of self maps defined on $X$ is weakly mixing if and only if $(T_n)$ is hereditarily transitive. In fact, if $(T_n)$ is hereditarily transitive with respect to an increasing sequence $(n_k)$ of positive int

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 36 more