Abstract: 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 talk, 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 [RM] 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 2. Moreover, we also show that the fair
measure is usually an equilibrium state, which has its own interest in stochastic mechanics. If time permitted,
I will also dicuss some recent progresses on the higher regularity on the entropy functions and (Martin)
boundary problems related to the Fair measures. These are joint works with A. Rodrigues M. Denker and Gang Liao.