Equality of Hölder exponents for distribution functions of Gibbs measures

Abstract

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 exist a constant $C>0$, a number $h>0$ and a polynomial $P$ of degree less than $α$ such that \[ |f(t)-P(t-t_0)|\leq C|t-t_0|^α\qquad\mbox{for all $t\in (t_0-h,t_0+h)$}. \] 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 exist $C>0$ and $h>0$ such that $|f(t)-f(t_0)|\leq C|t-t_0|^α$ for all $t\in (t_0-h,t_0+h)$. 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.