Finiteness of measures of maximal entropy for smooth saddle surface endomorphisms

We show that ${\mathcal{C}}^{\mathrm{\infty }}$ local diffeomorphisms of closed surfaces whose topological entropy is larger than the logarithm of their degree admit a finite number of ergodic measures of maximal entropy. To do this, we construct families of rectangles, with a nice geometry, displaying a Markov property. We then analyze the behavior of the iterates of unstable curves intersecting these rectangles, using Yomdin theory.