The Sinai billiard map
T
T
on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition
h
∗
h_*
for the topological entropy of
T
T
. We prove that
h
∗
h_*
is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure
μ
∗
\mu _*
of maximal entropy for
T
T
(i.e.,
h
μ
∗
(
T
)
=
h
∗
h_{\mu _*}(T)=h_*
), we show that
μ
∗
\mu _*
has full support and is Bernoulli, and we prove that
μ
∗
\mu _*
is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to
h
∗
h_*
. Second,
h
∗
h_*
is equal to the Bowen–Pesin–Pitskel topological entropy of the restriction of
T
T
to a noncompact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map
T
T
has at least
C
e
n
h
∗
C e^{nh_*}
periodic points of period
n
n
for all
n
∈
N
n \in \mathbb {N}
.