Subspaces of interval maps related to the topological entropy

TL;DR

This paper studies the topological structure of spaces of continuous interval maps based on their entropy, revealing that certain subspaces are homeomorphic to Hilbert space and others are contractible, with dense subsets of piecewise monotone maps.

Contribution

It characterizes the topological and homotopy properties of subspaces of interval maps classified by entropy levels, including homeomorphism to Hilbert space and contractibility.

Findings

Spaces of maps with entropy ≥ a are homeomorphic to l2.

Spaces with entropy ≤ a are contractible.

Piecewise monotone maps form homotopy dense subsets.

Abstract

For $a\in [0,+\infty)$, the function space $E_{\geq a}$ ($E_{>a}$; $E_{\leq a}$; $E_{<a}$) of all continuous maps from $[0,1]$ to itself whose topological entropies are larger than or equal to $a$ (larger than $a$; smaller than or equal to $a$; smaller than $a$) with the supremum metric is investigated. It is shown that the spaces $E_{\geq a}$ and $E_{>a}$ are homeomorphic to the Hilbert space $l_2$ and the spaces $E_{\leq a}$ and $E_{<a}$ are contractible. Moreover, the subspaces of $E_{\leq a}$ and $E_{<a}$ consisting of all piecewise monotone maps are homotopy dense in them, respectively.