Subspaces of interval maps related to the topological entropy

Abstract

A. For a ∈ [0, +∞), the function space E ≥a (E > a ; E ≤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 ≥a and E > a are homeomorphic to the Hilbert space l 2 and the spaces E ≤a and E < a are contractible. Moreover, the subspaces of E ≤a and E < a consisting of all piecewise monotone maps are homotopy dense in them, respectively.2010 Mathematics Subject Classification. 37E05, 54F65, 54H20.

The topological structure of function space of transitive maps

“…Hence, it is natural to give some structural characteristics of those spaces using tools of infinite-dimensional topology. In [4], Fan et al showed that some subspaces of the space of continuous self-maps on a compact interval with the supremum metric related with the topological entropy are homeomorphic to 2 . We know that the transitive property is an important concept in topological dynamics.…”

“…since Cont(I) ≈ I is compact. The last two homeomorphisms can be found in [13, ] and [4]. Note the following well known fact: If the product X × K of a space X and a locally compact space K is homeomorphic to 2 , then X ≈ 2 .…”