Equality of Hölder exponents for distribution functions of Gibbs measures

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 $\alpha >0$ and a point $t_0\in \mathbb{R}$, write $f\in C^{\alpha }(t_0)$ if there is a constant $C$ and a polynomial $P$ of degree less than $\alpha$ such that |f(t)-P(t-t_0)|\leq C|t-t_0|^α\qquadfor all $t\in\mathbb{R$}. The pointwise Hölder exponent of $f$ at $t_0$ is the number $${\alpha }_f(t_0):=\sup \{\alpha >0:f\in C^{\alpha }(t_0)\}\mathrm{.}$$ A simpler quantity, also frequently called pointwise Hölder 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 is a constant $C>0$ such that $\mathrm{\mid }f(t)-f(t_0)\mathrm{\mid }\le C\mathrm{\mid }t-t_0{\mathrm{\mid }}^{\alpha }$ for all $t\in \mathbb{R}$. Clearly ${\alpha }_f(t)\ge {\tilde{\alpha }}_f(t)$, 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 Hölder 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 Hölder spectrum of $f$ satisfies the multifractal formalism. As an application, we derive the pointwise Hölder spectrum of conjugacy maps between expanding piecewise ${\mathcal{C}}^{1+\epsilon }$ maps of an interval.