The topological structure of function space of transitive maps

Abstract

Let C(I) be the set of all continuous self-maps from I = [0, 1] with the topology of uniformly convergence. A map f ∈ C(I) is called a transitive map if for every pair of non-empty open sets U, V in I, there exists a positive integer n such that U ∩ f −n (V ) = ∅. We note T (I) and T (I) to be the sets of all transitive maps and its closure in the space C(I). In this paper, we show that T (I) and T (I) are homeomorphic to the separable Hilbert space 2 .