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The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas,is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$for the topological entropy of $T$. We prove that $h_*$ is not smaller than thevalue given by the variational principle, and that it is equal to thedefinitions of Bowen using spanning or separating sets. Under a mild conditionof sparse recurrence to the singularities, we get more: First, using a transferoperator acting on a space of anisotropic distributions, we construct aninvariant probability measure $\mu_*$ of maximal entropy for $T$ (i.e.,$h_{\mu_*}(T)=h_*$), we show that $\mu_*$ has full support and is Bernoulli,and we prove that $\mu_*$ is the unique measure of maximal entropy, and that itis different from the smooth invariant measure except if all non grazingperiodic orbits have multiplier equal to $h_*$. Second, $h_*$ is equal to theBowen--Pesin--Pitskel topological entropy of the restriction of $T$ to anon-compact domain of continuity. Last, applying results of Lima and Matheus,as upgraded by Buzzi, the map $T$ has at least $C e^{nh_*}$ periodic points ofperiod $n$ for all large enough $n \in \mathbb{N}$.

On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps