Multifractal Analysis of Equilibrium States of Endomorphisms of ℙ^k

2025/12/09 by Dalaklis, Nathan, He, Yan Mary
Mathematics · Physics and Astronomy · #28A80 #37A05 (Secondary) #37A50 #37D35 #37F80 (Primary) 28D20 #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · doi:10.48550/arxiv.2512.08696

openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28

Abstract

Let f be a holomorphic endomorphism of ℂℙ^k of algebraic degree at least 2 and let X ⊆ ℂℙ^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 → 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.

Citations

• L^q-spectra of measures on planar non-conformal attractors
• Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure
• Multifractal analysis for Bedford-McMullen carpets
• Multifractal analysis for conformal graph directed Markov systems
• Normalization of bundle holomorphic contractions and applications to dynamics
• On the variational principle for the topological entropy of certain non-compact sets
• Distribution of frequencies of digits via multifractal analysis
• Variational principles and mixed multifractal spectra
• Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension
• On the sum of digits of real numbers represented in the dyadic system.

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