Finiteness of measured homoclinic classes with large Lyapunov exponents for <inline-formula><tex-math id="M1">$ \mathcal{C}^2 $</tex-math></inline-formula> surface diffeomorphisms

In this paper, 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.