Multifractal Analysis of Equilibrium States of Endomorphisms of ${\mathbb{P}}^k$

Let $f$ be a holomorphic endomorphism of $\mathbb{C}{\mathbb{P}}^k$ of algebraic degree at least $2$ and let $X\subseteq \mathbb{C}{\mathbb{P}}^k$ be an uniformly expanding set. In this paper, we study multifractal analysis of equilibrium states of Hölder continuous functions for the non-conformal dynamical system $f:X\to X$. In lieu of Hausdorff dimensions, we use a new dimension theory (i.e., the volume dimension theory) to define various local dimension multifractal spectra and show that each of these spectra form a Legendre transform pair with the temperature function as in the conformal case. As an application of our main theorems, we also prove a conditional variational principle for such dimension multifractal spectra.