Equality of H\"older exponents for distribution functions of Gibbs measures

Abstract

Pointwise H\"older exponents describe the degree of regularity of a function near a point. For a function $f:\mathbb{R}\to\mathbb{R}$, a number $\alpha>0$ and a point $t_0\in\mathbb{R}$, write $f\in C^\alpha(t_0)$ if there exist a constant $C>0$, a number $h>0$ and a polynomial $P$ of degree less than $\alpha$ such that \[ |f(t)-P(t-t_0)|\leq C|t-t_0|^\alpha \qquad\mbox{for all $t\in (t_0-h,t_0+h)$}. \] The pointwise H\"older exponent of $f$ at $t_0$ is the number \[ \alpha_f(t_0):=\sup\{\alpha>0: f\in C^\alpha(t_0)\}. \] A simpler quantity, also frequently called pointwise H\"older exponent in the mathematical literature, is the number \[ \tilde{\alpha}_f(t_0):=\sup\{\alpha>0: f\in \tilde{C}^\alpha(t_0)\}, \] where $f\in \tilde{C}^\alpha(t_0)$ means that there exist $C>0$ and $h>0$ such that $|f(t)-f(t_0)|\leq C|t-t_0|^\alpha$ for all $t\in (t_0-h,t_0+h)$. Clearly $\alpha_f(t)\geq \tilde{\alpha}_f(t)$, but strict inequality is possible and in fact common. In this paper we consider the case when $f=F_\mu$ is the distribution function of a Gibbs measure $\mu$ associated with an arbitrary H\"older continuous potential $\psi$ on a self-conformal set, and show that, under a very mild condition on $\psi$, $\alpha_f(t)=\tilde{\alpha}_f(t)$ for all $t$. As a consequence, we deduce that the pointwise H\"older spectrum of $f$ satisfies the multifractal formalism. As an application, we derive the pointwise H\"older spectrum of conjugacy maps between expanding piecewise $\mathcal{C}^{1+\epsilon}$ maps of an interval.