Some universality results for dynamical systems

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 $U:{\ell }_1\to {\ell }_1$ such that any linear operator $T$ from a separable Banach space into itself with $\Vert T\Vert \le 1$ 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 $U_{\mathcal{F}}$.

arXiv comment15 pages