The topological structure of function space of transitive maps

TL;DR

This paper proves that the set of transitive maps and its closure in the space of continuous functions on [0,1], equipped with the uniform topology, are topologically equivalent to the separable Hilbert space .

Contribution

It establishes that both the set of transitive maps and its closure are homeomorphic to , revealing their topological structure.

Findings

The set of transitive maps is homeomorphic to .

The closure of transitive maps is homeomorphic to .

Both sets are topologically equivalent to the separable Hilbert space.

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 $U\cap f^{-n}(V)\not=\emptyset.$ 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 $\ell_2$.