Abstract
We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the Lq-spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the Lq-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word Lq-spectrum of a random substitution and show that this coincides with the Lq-spectrum of the corresponding frequency measure for all q≥0. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.
| Original language | English |
|---|---|
| Article number | 63 |
| Number of pages | 44 |
| Journal | Communications in Mathematical Physics |
| Volume | 405 |
| Issue number | 3 |
| Early online date | 24 Feb 2024 |
| DOIs | |
| Publication status | Published - Mar 2024 |
Bibliographical note
Acknowledgments:The authors are grateful to Philipp Gohlke for his detailed comments on a draft version of this manuscript, which helped to remove some technical assumptions from Theorem D. They also thank Dan Rust and Tony Samuel for valuable input. AM thanks SFB1283 and the Universität Bielefeld for supporting a research visit during the summer of 2022, where some of the work on this project was undertaken. AM was supported by EPSRC DTP and the University of Birmingham. AR was supported by EPSRC Grant EP/V520123/1 and the Natural Sciences and Engineering Research Council of Canada. The authors thank the organisers of the Junior Ergodic Theory Meeting hosted at the ICMS in Edinburgh in March 2022, where this project began.