Entropy of diffeomorphisms of line (Q2357557)

For \(r \geq 1\), the author studies the set of \(C^r\)-diffeomorphisms of the line with bounded first derivative, denoted by \(\text{{Diff}}^r_b(\mathbb{R})\). One of the main results is that for \(r \geq 1\), there is a \(C^0\)-open and \(C^r\)-dense subset \(U\) of \(\text{{Diff}}^r_b(\mathbb{R})\) such that the topological entropy is continuous on \(U\) with respect to the strong \(C^0\) topology. As a consequence of this result, for any uniformly expanding or zero entropy map \(f \in U\), the topological entropy is locally constant at \(f\). Furthermore, the set of diffeomorphisms with robustly positive or robustly zero topological entropy is \(C^r\)-dense in \(\text{{Diff}}^r_b(\mathbb{R})\). Some other results in this paper are connected to examples of the author. It is shown for some open subset \(U\) of \(\text{{Diff}}^\infty_b(\mathbb{R})\) that for any \(f \in U\), the topological entropy is not locally constant at \(f\) with respect to the strong \(C^\infty\)-topology. Also, for some map \(f \in \text{{Diff}}^\infty_b(\mathbb{R})\), the topological entropy is not upper or lower semicontinuous at \(f\) with respect to the strong \(C^\infty\)-topology.