An example where topological entropy is continuous

Abstract:

Let ent denote topological entropy, and let ${C^r}({S^1},{S^1})$ denote the space of continuous functions of the circle to itself having r continuous derivatives with the ${C^r}$ (uniform) topology. Let ${f_0}$ denote a particular ${C^2}$ map of the circle (${f_0}$ is the first bifurcation point one comes to in a bifurcation from a full three shift to a map with finite nonwandering set). The main results of this paper are the following: Theorem A. The map ent: ${C^0}({S^1},{S^1}) \to R \cup \{ \infty \}$ is lower-semicontinuous at ${f_0}$. Theorem B. The map ent: ${C^2}({S^1},{S^1}) \to R$ is continuous at ${f_0}$. In proving these two theorems several general results on entropy of mappings of the circle are proved.