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

TL;DR

This paper develops a new multifractal analysis framework for equilibrium states of holomorphic endomorphisms of complex projective space, introducing volume dimension theory and establishing Legendre transform relations with temperature functions.

Contribution

It introduces a novel volume dimension theory for multifractal analysis in non-conformal dynamics and connects it with thermodynamic formalism.

Findings

Established Legendre transform relations between multifractal spectra and temperature functions.

Proved a conditional variational principle for the new dimension spectra.

Extended multifractal analysis to non-conformal holomorphic endomorphisms.

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.