On a family of singular potentials: Parameter dependence of thermodynamic characteristics

TL;DR

This paper investigates how the thermodynamic and multifractal characteristics of a family of singular potentials depend on the position of their singularities, revealing continuity and semicontinuity properties across parameters.

Contribution

It provides a detailed analysis of the parameter dependence of pressure functions, spectra, and measures for singular potentials over the doubling map, including new continuity results.

Findings

Pressure functions are continuous in the parameter for non-negative t.

Pressure functions are lower semicontinuous but not continuous for negative t.

Continuity points for negative t are residual and have zero Hausdorff dimension.

Abstract

We consider the family of singular potentials $\psi_c = 2 \log(|\sin(\pi(x-c))|)$, $c\in \mathbb{T}$ over the doubling map and we examine the dependence of several thermodynamic and multifractal characteristics on the position of the singularity $c$. This includes the pressure functions $\mathcal P(t \psi_c)$, the Birkhoff spectrum of $\psi_c$, and the $L^q$ spectrum of the associated equilibrium measure $\mu_c$. For every $c \in \mathbb{T}$, it is known that $\mu_c$ is given by the diffraction measure of a generalized Thue--Morse sequence, with the classical Thue--Morse measure arising for $c = 0$. If $t\geqslant 0$, we show that $c \mapsto \mathcal{P}(t\psi_c)$ is continuous in $c$. If $t<0$, we prove that the function $c \mapsto \mathcal{P}(t\psi_c)$ is lower semicontinuous but not continuous. In this case, we show that the continuity points are precisely those values $c$ such that $\mathcal{P}(t\psi_c) = \infty$, which form a residual set of vanishing Hausdorff dimension in $\mathbb{T}$. We obtain similar statements about the parameter (semi-)continuity of the $L^q$ spectrum and the Birkhoff spectrum.