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Some examples of tame dynamical systems answering questions of Glasner and Megrelishvili

Alessandro Codenotti

TL;DR

The paper addresses Glasner and Megrelishvili's questions on tameness of dynamical actions on dendrites by producing two not-$\mathrm{tame}_1$ examples (one on a Wa\u{z}ewski dendrite and another on a novel dendrite with two ramification orders) and proving a universal bound $\beta(X,G)\le 2$ for dendritic actions. It then constructs, for every $\alpha<\omega_1$, metric tame systems with $\beta(X,G)=\alpha$, showing that arbitrary $\beta$-ranks can be realized in the tame setting and extending to connected cases via cones. The results separate dendrite actions from the broader tame class, provide rigid tame examples addressing prior questions, and lay groundwork for realizing higher beta-ranks while clarifying the limits of tameness on dendritic bases with precise combinatorial and topological control.

Abstract

Glasner and Megrelishvili proved that every continuous action of a topological group $G$ on a dendrite $X$ is tame. We produce two examples of an action on a dendrite which is not $\mathrm{tame}_1$, answering a question they raised. We then show that actions on dendrites have $β$-rank at most $2$ and produce examples of tame metric dynamical systems of $β$-rank $α$ for any $α<ω_1$, answering another question of Glasner and Megrelishvili.

Some examples of tame dynamical systems answering questions of Glasner and Megrelishvili

TL;DR

The paper addresses Glasner and Megrelishvili's questions on tameness of dynamical actions on dendrites by producing two not- examples (one on a Wa\u{z}ewski dendrite and another on a novel dendrite with two ramification orders) and proving a universal bound for dendritic actions. It then constructs, for every , metric tame systems with , showing that arbitrary -ranks can be realized in the tame setting and extending to connected cases via cones. The results separate dendrite actions from the broader tame class, provide rigid tame examples addressing prior questions, and lay groundwork for realizing higher beta-ranks while clarifying the limits of tameness on dendritic bases with precise combinatorial and topological control.

Abstract

Glasner and Megrelishvili proved that every continuous action of a topological group on a dendrite is tame. We produce two examples of an action on a dendrite which is not , answering a question they raised. We then show that actions on dendrites have -rank at most and produce examples of tame metric dynamical systems of -rank for any , answering another question of Glasner and Megrelishvili.
Paper Structure (6 sections, 18 theorems, 17 equations, 2 figures)

This paper contains 6 sections, 18 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Ważewski dendrites have actions that are not tame 1
  3. Another example of a dendrite with an action that is not tame 1
  4. Actions on dendrites cannot have high beta-rank
  5. Metric tame systems with arbitrary beta-rank
  6. Some open questions

Key Result

Proposition 2.1

Let $(X,G)$ be a proximal minimal system. Suppose that there is $a\in X$ and an uncountable set $\{b_i\}_{i\in I}\subseteq X\setminus\{a\}$ such that for every $i\in I$ the function $p_{a,b_i}$ defined by is an idempotent in $E(X,G)$. Then the idempotent $p_a$ with constant value $a$ is in $E(X,G)$ and it doesn't admit a countable neighbourhood basis. In particular $E(X,G)$ is not first countable

Figures (2)

  • Figure 1: $T_0$ and $T_1$, with the first few intervals of the form $(n/(n+1),(n+1)/(n+2))$. Points that will become of order $4$ during the construction are coloured in red, while points that will become of order $3$ are coloured in green.
  • Figure 2: (An approximation of) the stage $X_1$ of the construction. Points that have order $4$ or will become points of order $4$ in $X_2$ are coloured in red, while points that have order $3$ in $X_1$ or will become points of order $3$ in $X_2$ are coloured in green.

Theorems & Definitions (40)

  • Definition 1.1
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Definition 2.7
  • ...and 30 more