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

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.