Recent advances on fair measures and fair entropy

In this talk I will discuss how to compute the entropy following backwards trajectories in a way that at each step every preimage can be chosen with equal probability introducing fair measure and fair entropy (joint with M. Misiurewicz). I will discuss some advances on the study of fair entropy for non-invertible interval maps under the framework of thermodynamic formalism, showing that the fair measure is usually an equilibrium state (joint with Y. Zhang). I will then talk about some recent results (joint with S. and Z. Roth) about transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps.