The topological entropy of Banach spaces

TL;DR

This paper explores the relationship between Banach spaces of real functions and topological entropy, revealing that certain subspaces contain functions with infinite entropy and constructing spaces with uniform entropy levels.

Contribution

It demonstrates that subspaces isometric to  contain functions with infinite entropy and constructs one-dimensional Banach spaces where all nonzero functions share the same topological entropy.

Findings

Subspaces of C([0,1]) isometric to  contain functions with infinite topological entropy

Constructed Banach spaces where all nonzero functions have a fixed topological entropy

Established links between Banach space structure and topological entropy values

Abstract

We investigate some properties of (universal) Banach spaces of real functions in the context of topological entropy. Among other things, we show that any subspace of $C([0,1])$ which is isometrically isomorphic to $\ell_1$ contains a functions with infinite topological entropy. Also, for any $t \in [0, \infty]$, we construct a (one-dimensional) Banach space in which any nonzero function has topological entropy equal to $t$.