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Dynamical properties and some classes of non-porous subsets

Stefan Ivkovic, Serap Oztop, Seyyed Mohammad Tabatabaie

TL;DR

The paper investigates large, yet thin, non-sigma-porous sets in Lebesgue spaces and their connection to linear dynamics, specifically hypercyclicity. By constructing non-sigma-porous families Gamma_g via a general lemma and extending Bayart’s approach, the authors establish non-porosity results for sequences of weighted translation operators on general Lebesgue spaces, including discrete hypergroups and hypergroups, as well as for weighted compositions on L_infty and L^p spaces. Key contributions show that, under suitable conditions on the space and the weight, the set of non-hypercyclic vectors for operator sequences Lambda_n is not sigma-porous, with broad corollaries across discrete and continuous settings. This work clarifies the relationship between porosity and dynamical properties, offering tools to verify non-porosity for diverse operator families and potentially impacting ergodic theory and functional analysis in group- and hypergroup-based contexts.

Abstract

In this paper, we introduce several classes of non-σ-porous subsets of a general Lebesgue space. Also, we study some linear dynamics of operators and show that the set of all non-hypercyclic vectors of a sequence of weighted translation operators on Lp-spaces is not σ-porous.

Dynamical properties and some classes of non-porous subsets

TL;DR

The paper investigates large, yet thin, non-sigma-porous sets in Lebesgue spaces and their connection to linear dynamics, specifically hypercyclicity. By constructing non-sigma-porous families Gamma_g via a general lemma and extending Bayart’s approach, the authors establish non-porosity results for sequences of weighted translation operators on general Lebesgue spaces, including discrete hypergroups and hypergroups, as well as for weighted compositions on L_infty and L^p spaces. Key contributions show that, under suitable conditions on the space and the weight, the set of non-hypercyclic vectors for operator sequences Lambda_n is not sigma-porous, with broad corollaries across discrete and continuous settings. This work clarifies the relationship between porosity and dynamical properties, offering tools to verify non-porosity for diverse operator families and potentially impacting ergodic theory and functional analysis in group- and hypergroup-based contexts.

Abstract

In this paper, we introduce several classes of non-σ-porous subsets of a general Lebesgue space. Also, we study some linear dynamics of operators and show that the set of all non-hypercyclic vectors of a sequence of weighted translation operators on Lp-spaces is not σ-porous.
Paper Structure (3 sections, 19 theorems, 108 equations)

This paper contains 3 sections, 19 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Non-$\sigma$-porous subsets of Lebesgue spaces
  3. Applications

Key Result

Lemma 2.2

Let $\mathcal{F}$ be a non-empty family of non-empty closed subsets of a complete metric space $X$ such that for each $F\in\mathcal{F}$ and each $x\in X$ and $r>0$ with $B(x;r)\cap F\neq \varnothing$, there exists an element $J\in\mathcal{F}$ such that and $F\cap B(x;r)$ is not $\lambda$-porous at all elements of $J\cap B(x;r)$. Then, every set in $\mathcal{F}$ is not $\sigma$-$\lambda$-porous.

Theorems & Definitions (39)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • Theorem 2.7
  • proof
  • ...and 29 more