A Ma\~n\'e-Manning formula for expanding measures for endomorphisms of $\mathbb P^k$

Let $k \ge 1$ be an integer and $f$ a holomorphic endomorphism of $\mathbb P^k (\mathbb C)$ of algebraic degree $d\geq 2$. We introduce a volume dimension for ergodic $f$-invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$, a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when $k=1$, but depends on the dynamics of $f$ to incorporate the possible failure of Koebe's theorem and the non-conformality of holomorphic endomorphisms for $k\geq 2$. If $\nu$ is an ergodic $f$-invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Ma\~n\'e-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of $\nu$. As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urba\'nski and McMullen in dimension 1 to any dimension $k\geq 1$. Our methods mainly rely on a theorem by Berteloot-Dupont-Molino, which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.