A direct proof of the existence of MME for finite horizon Sinai billiards

The Sinai billiard map T on the two-torus, i.e., the periodic Lorentz gaz, is a discontinuous map. Assuming finite horizon and bounded complexity, we prove that the Kolmogorov--Sinai entropy map associated with the billiard map T is upper semi-continuous, as well as the compactness of the set of T-invariant measures. In particular, for the potentials g ≡ 0 and g = -hₜₒₚ(Φ₁) τ, we recover the recent results of the existence of measures of maximal entropy (MME) for both the billiard map and flow Φₜ, due to Baladi and Demers for T, jointly with Carrand for Φₜ. For general finite horizon Sinai billiards, we provide bounds on the defect of upper semi-continuity of the entropy map and on the topological tail entropy. The bounded complexity condition is expected to hold generically.