Multifractal Analysis of Equilibrium States of Endomorphisms of $\mathbb{P}^k$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917135283912704 |
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| author | Dalaklis, Nathan He, Yan Mary |
| author_facet | Dalaklis, Nathan He, Yan Mary |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multifractal Analysis of Equilibrium States of Endomorphisms of $\mathbb{P}^k$ Dalaklis, Nathan He, Yan Mary Dynamical Systems 28A80, 37D35, 37F80 (Primary) 28D20, 37A50, 37A05 (Secondary) 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. |
| title | Multifractal Analysis of Equilibrium States of Endomorphisms of $\mathbb{P}^k$ |
| topic | Dynamical Systems 28A80, 37D35, 37F80 (Primary) 28D20, 37A50, 37A05 (Secondary) |
| url | https://arxiv.org/abs/2512.08696 |