  For a class of piecewise hyperbolic maps in two dimensions, we propose a
combinatorial definition of topological entropy by counting the maximal, open,
connected components of the phase space on which iterates of the map are
smooth. We prove that this quantity dominates the measure theoretic entropies
of all invariant probability measures of the system, and then construct an
invariant measure whose entropy equals the proposed topological entropy. We
prove that our measure is the unique measure of maximal entropy, that it is
ergodic, gives positive measure to every open set, and has exponential decay of
correlations against H\"older continuous functions. As a consequence, we also
prove a lower bound on the rate of growth of periodic orbits. The main tool
used in the paper is the construction of anisotropic Banach spaces of
distributions on which the relevant weighted transfer operator has a spectral
gap. We then construct our measure of maximal entropy by taking a product of
left and right maximal eigenvectors of this operator.
