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arXiv · 2512.08696

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

Nathan Dalaklis·Yan Mary He
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Abstract

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 H\"older 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.

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Nathan Dalaklis, Yan Mary He. 2025-12-09. Multifractal Analysis of Equilibrium States of Endomorphisms of $\mathbb{P}^k$. https://arxiv.org/abs/2512.08696

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