The infimum of Lipschitz constants in the conjugacy class of an interval map

Abstract:

How can we interpret the infimum of Lipschitz constants in the conjugacy class of an interval map? For a positive entropy map $f$, the exponential $\exp h(f)$ of the topological entropy gives a well-known lower bound. In the case of a countably piecewise monotone map that is topologically mixing and Markov, we characterize the infimum $\Lambda (f)$ of Lipschitz constants as the exponential of the Salama entropy of a certain reverse Markov chain associated with the map. Dynamically, this number represents the exponential growth rate of the number of iterated preimages of nearly any point; we show that it can be strictly larger than $\exp h(f)$. In addition we prove that if $f$ is piecewise monotone or $C^{\infty }$, these two quantities $\Lambda (f)$ and $\exp h(f)$ are equal.