Entropy of general diffeomorphisms on line

Abstract

We study the continuity of topological entropy of general diffeomorphisms on line. First, we prove that the entropy map is continuous with respect to the strong $C^{0}$-topology on the union of uniformly topologically hyperbolic diffeomorphisms contained in $\text{Diff}_{0}^{r}(\mathbb{R})$ (whose first derivative is uniformly away from zero), which is a $C^{0}$-open and $C^{r}$-dense subset of $\text{Diff}_{0}^{r}(\mathbb{R})$, $r=1,2,\ldots ,\infty$, and $\unicode[STIX]{x1D714}$ (real analytic). Second, we give some examples where entropy map is not continuous. Finally, we prove some results on the continuity of entropy of general diffeomorphisms on the (real) line.