Random self-similar series over a rotation

TL;DR

The paper studies random self-similar series defined over an ergodic dynamical system, focusing on the law $P_X$ of $X(\omega)=\sum_{n\ge0} r_n(\omega)b(T^n\omega)$ that satisfies the self-similarity $X(\omega)=\varphi_{\omega}(X(T\omega))$. It extends classical self-similar measures to an ergodic setting, showing $P_X$ is always of pure type and giving explicit atomicity criteria while characterizing the generic case as continuous (but typically not Rajchman). In the Circle case, with irrational rotation, the image measure has box-counting dimension zero and is singular; the work also provides conditions under which $t_nX\bmod 1$ converges to Lebesgue along subsequences and proves a Pisot-type obstruction to Rajchman behavior. A general remark connects continuity of $P_X$ to fixed-point structure in more general dynamical systems, outlining a dichotomy between continuous and finitely-supported laws via minimal prefixes. These results contribute a robust framework for understanding the regularity and singularity of random self-similar measures beyond independence.

Abstract

We study the law of random self-similar series defined above an irrational rotation on the Circle. This provides a natural class of continuous singular non-Rajchman measures.

Theorems & Definitions (10)

• Lemma 1.1
• Theorem 2.1
• remark 1
• remark 2
• remark 3
• Lemma 2.2
• Definition 3.1
• remark 4
• Lemma 3.2
• remark 5