The topological structure of function space of transitive maps

Abstract

Let $C(\mathbf{I})$ be the set of all continuous self-maps from $\mathbf{I}=[0,1]$ with the topology of uniformly convergence. A map $f\in C(\mathbf{I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in $\mathbf{I}$, there exists a positive integer $n$ such that $U\cap f^{-n}(V)\ne \varnothing .$ We note $T(\mathbf{I})$ and $\overline{T(\mathbf{I})}$ to be the sets of all transitive maps and its closure in the space $C(\mathbf{I})$. In this paper, we show that $T(\mathbf{I})$ and $\overline{T(\mathbf{I})}$ are homeomorphic to the separable Hilbert space ${\ell }_2$.

17 pages, 1 figure