Dimension of homogeneous iterated function systems with algebraic translations

• DOI: 10.48550/arxiv.2405.03124
• Publication Date: 2024-01-01
• Authors: Feng, De-Jun; Feng, Zhou
• Keywords: Mathematics - Classical Analysis and ODEs; 28A80, 42A85; Classical Analysis and ODEs (math.CA); FOS: Mathematics; Dynamical Systems (math.DS); Mathematics - Dynamical Systems

Let $ μ$ be the self-similar measure associated with a homogeneous iterated function system $ Φ= \{ λx + t_j \}_{j=1}^m $ on ${\Bbb R}$ and a probability vector $ (p_{j})_{j=1}^m$, where $0\neq λ\in (-1,1)$ and $t_j\in {\Bbb R}$. Recently by modifying the arguments of Varjú (2019), Rapaport and Varjú (2024) showed that if $t_1,\ldots, t_m$ are rational numbers and $0<λ<1$, then $$ \dim μ=\min\Big \{ 1, \; \frac{\sum_{j=1}^m p_{j}\log p_{j}}{ \log |λ| }\Big\}$$ unless $ Φ$ has exact overlaps. In this paper, we further show that the above equality holds in the case when $t_1,\ldots, t_m$ are algebraic numbers and $0<|λ|<1$. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varjú.