Subspaces of interval maps related to the topological entropy

Abstract

For $a\in [0,+\infty )$, the function space $E_{\ge a}$ ($E_{>a}$; $E_{\le a}$; $E_{<a}$) 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 $E_{\ge a}$ and $E_{>a}$ are homeomorphic to the Hilbert space $l_2$ and the spaces $E_{\le a}$ and $E_{<a}$ are contractible. Moreover, the subspaces of $E_{\le a}$ and $E_{<a}$ consisting of all piecewise monotone maps are homotopy dense in them, respectively.

arXiv comment12 pages, 1 figure, to appear in Topol. Methods Nonlinear Anal

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