Subspaces of interval maps related to the topological entropy

Xiaoxin Fan, Jian Li, Yini Yang, Zhongqiang Yang

DOI: http://dx.doi.org/10.12775/TMNA.2019.065

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.

Keywords


Interval maps; topological entropy; the Hilbert space $l_2$; homotopy dense; contractible

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