Fair measure and fair entropy for non-Markov interval maps

Abstract

In a recent paper [8] the entropy of a special measure, the fair measure was introduced. The fair entropy is computed following backward trajectories in a way such that at each step every preimage can be chosen with equal probability. In this paper, we continue studying the fair measure and the fair entropy for non-invertible interval maps under the framework of thermodynamic formalism. We extend several results in [8] to the non-Markov setting, and we prove that for each symmetric tent map the fair entropy is equal to the topological entropy if and only if the slope is equal to [Formula: see text]. Moreover, we also show that the fair measure is usually an equilibrium state, which has its own interest in stochastic mechanics.