The Infimum of Lipschitz Constants in the Conjugacy Class of an Interval Map

TL;DR

This paper investigates the minimal Lipschitz constants within conjugacy classes of interval maps, revealing equality with topological entropy for certain classes and characterizing the infimum via Salama entropy in specific cases.

Contribution

It establishes the equality of the infimum of Lipschitz constants and topological entropy for piecewise monotone and smooth maps, and characterizes the infimum using Salama entropy for mixing Markov maps.

Findings

Equality of Lipschitz infimum and topological entropy for piecewise monotone maps.

Strict inequality can occur for countably piecewise monotone maps.

Characterization of the infimum via Salama entropy in mixing Markov cases.

Abstract

How can we interpret the infimum of Lipschitz constants in a conjugacy class of interval maps? For positive entropy maps, the exponential of the topological entropy gives a well-known lower bound. We show that for piecewise monotone interval maps as well as for $C^{\infty}$ interval maps, these two quantities are equal, but for countably piecewise monotone maps, the inequality can be strict. Moreover, in the topologically mixing and Markov case, we characterize the infimum 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.