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

We consider the family of singular potentials ${\psi }_c=2\log (\mathrm{\mid }\sin (\pi (x-c))\mathrm{\mid })$, $c\in \mathbb{T}$ 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 $t⩾0$, 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 $\mathcal{P}(t{\psi }_c)=\mathrm{\infty }$, 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.