Dimension of homogeneous iterated function systems with algebraic translations

TL;DR

This paper extends the understanding of the dimension of self-similar measures for homogeneous iterated function systems with algebraic translation parameters, showing the dimension formula holds beyond rational translations.

Contribution

It generalizes previous results by proving the dimension formula for systems with algebraic translation parameters, not just rational ones.

Findings

Dimension formula holds for algebraic translations

Extension of previous rational case results

Uses adapted techniques from recent research

Abstract

Let $ \mu $ be the self-similar measure associated with a homogeneous iterated function system $ \Phi = \{ \lambda x + t_j \}_{j=1}^m $ on ${\Bbb R}$ and a probability vector $ (p_{j})_{j=1}^m$, where $0\neq \lambda\in (-1,1)$ and $t_j\in {\Bbb R}$. Recently by modifying the arguments of Varj\'u (2019), Rapaport and Varj\'u (2024) showed that if $t_1,\ldots, t_m$ are rational numbers and $0<\lambda<1$, then $$ \dim \mu =\min\Big \{ 1, \; \frac{\sum_{j=1}^m p_{j}\log p_{j}}{ \log |\lambda| }\Big\}$$ unless $ \Phi $ 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 $0<|\lambda|<1$. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varj\'u.