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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.19001 |
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| _version_ | 1866914409335488512 |
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| author | Gohlke, Philipp Lamprinakis, Georgios Schmeling, Jörg |
| author_facet | Gohlke, Philipp Lamprinakis, Georgios Schmeling, Jörg |
| contents | We consider the family of singular potentials $ψ_c = 2 \log(|\sin(π(x-c))|)$, $c\in \mathbb{T}$ 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 $\mathcal P(t ψ_c)$, the Birkhoff spectrum of $ψ_c$, and the $L^q$ spectrum of the associated equilibrium measure $μ_c$. For every $c \in \mathbb{T}$, it is known that $μ_c$ 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 $c \mapsto \mathcal{P}(tψ_c)$ is continuous in $c$. If $t<0$, we prove that the function $c \mapsto \mathcal{P}(tψ_c)$ is lower semicontinuous but not continuous. In this case, we show that the continuity points are precisely those values $c$ such that $\mathcal{P}(tψ_c) = \infty$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19001 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a family of singular potentials: Parameter dependence of thermodynamic characteristics Gohlke, Philipp Lamprinakis, Georgios Schmeling, Jörg Dynamical Systems 37C45, 37D35 We consider the family of singular potentials $ψ_c = 2 \log(|\sin(π(x-c))|)$, $c\in \mathbb{T}$ 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 $\mathcal P(t ψ_c)$, the Birkhoff spectrum of $ψ_c$, and the $L^q$ spectrum of the associated equilibrium measure $μ_c$. For every $c \in \mathbb{T}$, it is known that $μ_c$ 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 $c \mapsto \mathcal{P}(tψ_c)$ is continuous in $c$. If $t<0$, we prove that the function $c \mapsto \mathcal{P}(tψ_c)$ is lower semicontinuous but not continuous. In this case, we show that the continuity points are precisely those values $c$ such that $\mathcal{P}(tψ_c) = \infty$, 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. |
| title | On a family of singular potentials: Parameter dependence of thermodynamic characteristics |
| topic | Dynamical Systems 37C45, 37D35 |
| url | https://arxiv.org/abs/2603.19001 |