Finitness of measured homoclinic classes with large Lyapunov exponents for ${\mathcal{C}}^2$ surface diffeomorphisms

We show the finiteness of homoclinic classes carrying measures with large Lyapunov exponents for ${\mathcal{C}}^2$ surface diffeomorphisms. As a consequence, we derive the finiteness of the set of ergodic measures of maximal entropy, in the case where the entropy of the system is large.