Some universality results for dynamical systems

TL;DR

This paper establishes universality results in topological and linear dynamical systems, showing the existence of universal models that factor all systems within certain classes, highlighting deep structural similarities.

Contribution

It introduces universal models for continuous and linear dynamical systems, demonstrating their ability to factor all systems in specified classes, a novel unifying perspective.

Findings

Existence of a dense, invariant set homeomorphic to Baire space for any continuous self-map.

Construction of a bounded linear operator factoring all operators with norm ≤ 1.

Universal self-map of Baire space factoring any sigma-compact family of continuous maps.

Abstract

We prove some "universality" results for topological dynamical systems. In particular, we show that for any continuous self-map $T$ of a perfect Polish space, one can find a dense, $T$-invariant set homeomorphic to the Baire space ${\mathbb N}^{\mathbb N}$; that there exists a bounded linear operator $U: \ell_1 \rightarrow \ell_1$ such that any linear operator $T$ from a separable Banach space into itself with $\Vert T\Vert\leq 1$ is a linear factor of $U$; and that given any $\sigma$-compact family ${\mathcal F}$ of continuous self-maps of a compact metric space, there is a continuous self-map $U_{\mathcal F}$ of ${\mathbb N}^{\mathbb N}$ such that each $T\in {\mathcal F}$ is a factor of $U_{\mathcal F}$.