Dynamical Systems Seminar

Entropy of Diffeomorphisms of Line

Abstract:

For diffeomorphisms of line $R$, we set up the identity between their length growth rate and their entropy.  Then, we prove that there is $C^0$-open and $C^r$-dense subset $U$ of $\Diff^r (R)$ with bounded first-order derivative, $r=1,2,\cdots$, $+\infty$, such that the entropy map with respect to strong $C^0$-topology is continuous on $U$; moreover, for any $f \in U$, if it is uniformly expanding or $h(f)=0$, then the entropy map is locally constant at $f$.
 Also, we construct two examples:
 1. There exists open subset $U$ of $\Diff^{\infty} (R)$ such that the entropy map with respect to strong $C^{\infty}$-topology is not locally constant at every map in $U$.
 2. There exists $f \in \Diff^{\infty}(R)$ such that the entropy map with respect to strong $C^{\infty}$-topology is neither lower semi-continuous nor upper semi-continuous at $f$.