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

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 ∞ 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.

Constant slope, entropy, and horseshoes for a map on a tame graph

“…Even in the case G = [0, 1], we do not know the answer. This is a different issue than the infimum of Lipschitz constants addressed in [5].…”

“…Our proof techniques apply not only to constant-slope maps, but also to bounded-slope maps, leading us to the following theorem whose consequences we explore in §5. The interval version of the following theorem for bounded slope was recently proved in [5,Theorem 3.7]. THEOREM 3.3.…”

“…The question we want to address is: when is a continuous countably Markov and mixing map f of a tame graph G conjugate to a countably affine graph map g : G → G of constant slope e htop(f ) ? Our general strategy will follow that of [5] with some modification to adapt it to more general tame graphs.…”

Inequalities for entropy, Hausdorff dimension, and Lipschitz constants

“…Even without transitivity, if an interval map is either piecewise monotone or C ∞ smooth, then a modification of Parry's construction produces conjugate interval maps not of constant slope, but with Lipschitz constants (with respect to the Euclidean metric) arbitrarily close to the exponential of the entropy [2]. This gives equality of the HausLip constant and the entropy in these classes of interval maps as well.…”

“…This gives equality of the HausLip constant and the entropy in these classes of interval maps as well. The paper [2] also presents examples (non-smooth, countably many turning points) for which the infimum of Lipschitz constants among conjugate interval maps is strictly larger than the entropy. But all of those Lipschitz constants are computed with respect to the Euclidean metric.…”