Equality of Hölder exponents for distribution functions of Gibbs measures
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| Natura: | Preprint |
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2025
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| author | Allaart, Pieter Jaerisch, Johannes |
| author_facet | Allaart, Pieter Jaerisch, Johannes |
| contents | Pointwise Hölder exponents describe the degree of regularity of a function near a point. For a function $f:\mathbb{R}\to\mathbb{R}$, a number $α>0$ and a point $t_0\in\mathbb{R}$, write $f\in C^α(t_0)$ if there is a constant $C$ and a polynomial $P$ of degree less than $α$ such that \[ |f(t)-P(t-t_0)|\leq C|t-t_0|^α\qquad\mbox{for all $t\in\mathbb{R}$}. \] The pointwise Hölder exponent of $f$ at $t_0$ is the number \[ α_f(t_0):=\sup\{α>0: f\in C^α(t_0)\}. \] A simpler quantity, also frequently called pointwise Hölder exponent in the mathematical literature, is the number \[ \tildeα_f(t_0):=\sup\{α>0: f\in \tilde{C}^α(t_0)\}, \] where $f\in \tilde{C}^α(t_0)$ means that there is a constant $C>0$ such that $|f(t)-f(t_0)|\leq C|t-t_0|^α$ for all $t\in\mathbb{R}$. Clearly $α_f(t)\geq \tildeα_f(t)$, but strict inequality is possible and in fact common. In this paper we consider the case when $f=F_μ$ is the distribution function of a Gibbs measure $μ$ associated with an arbitrary Hölder continuous potential $ψ$ on a self-conformal set, and show that, under a very mild condition on $ψ$, $α_f(t)=\tildeα_f(t)$ for all $t$. As a consequence, we deduce that the pointwise Hölder spectrum of $f$ satisfies the multifractal formalism. As an application, we derive the pointwise Hölder spectrum of conjugacy maps between expanding piecewise $\mathcal{C}^{1+ε}$ maps of an interval. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equality of Hölder exponents for distribution functions of Gibbs measures Allaart, Pieter Jaerisch, Johannes Dynamical Systems Probability 37D35, 28A80 Pointwise Hölder exponents describe the degree of regularity of a function near a point. For a function $f:\mathbb{R}\to\mathbb{R}$, a number $α>0$ and a point $t_0\in\mathbb{R}$, write $f\in C^α(t_0)$ if there is a constant $C$ and a polynomial $P$ of degree less than $α$ such that \[ |f(t)-P(t-t_0)|\leq C|t-t_0|^α\qquad\mbox{for all $t\in\mathbb{R}$}. \] The pointwise Hölder exponent of $f$ at $t_0$ is the number \[ α_f(t_0):=\sup\{α>0: f\in C^α(t_0)\}. \] A simpler quantity, also frequently called pointwise Hölder exponent in the mathematical literature, is the number \[ \tildeα_f(t_0):=\sup\{α>0: f\in \tilde{C}^α(t_0)\}, \] where $f\in \tilde{C}^α(t_0)$ means that there is a constant $C>0$ such that $|f(t)-f(t_0)|\leq C|t-t_0|^α$ for all $t\in\mathbb{R}$. Clearly $α_f(t)\geq \tildeα_f(t)$, but strict inequality is possible and in fact common. In this paper we consider the case when $f=F_μ$ is the distribution function of a Gibbs measure $μ$ associated with an arbitrary Hölder continuous potential $ψ$ on a self-conformal set, and show that, under a very mild condition on $ψ$, $α_f(t)=\tildeα_f(t)$ for all $t$. As a consequence, we deduce that the pointwise Hölder spectrum of $f$ satisfies the multifractal formalism. As an application, we derive the pointwise Hölder spectrum of conjugacy maps between expanding piecewise $\mathcal{C}^{1+ε}$ maps of an interval. |
| title | Equality of Hölder exponents for distribution functions of Gibbs measures |
| topic | Dynamical Systems Probability 37D35, 28A80 |
| url | https://arxiv.org/abs/2509.11527 |