Entropy formula for surface diffeomorphisms

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\mathrm{t}\mathrm{o}\mathrm{p}}(f)\ge \frac{{\lambda }^{+}(f)}{r}$ where ${\lambda }^{+}(f):={\lim }_{n\to +\mathrm{\infty }}\frac{1}{n}{\max }_{x\in M}\log \parallel D{f}_x^n\parallel$. We establish an equivalent formula for the topological entropy: $$h_{\mathrm{t}\mathrm{o}\mathrm{p}}(f)=\underset{n\to +\mathrm{\infty }}{\lim }\frac{1}{n}\log {\int }_M\parallel D{f}_x^n\parallel dx\mathrm{.}$$ We also characterize the topological entropy via the volume growth of curves and several applications are presented. Our approach builds on the key ideas developed in the works of Buzzi-Crovisier-Sarig (\emph{Invent. Math.}, 2022) and Burguet (\emph{Ann. Henri Poincaré}, 2024) concerning the continuity of the Lyapunov exponents.