Abstract
We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsedà-Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM system has Hausdorff dimension strictly smaller than. In particular, we obtain singularity of these measures, answering partially a question of Alsedà and Misiurewicz [Random interval homeomorphisms. Publ. Mat. 58(suppl.) (2014), 15-36].
| Original language | English |
|---|---|
| Pages (from-to) | 1473-1488 |
| Number of pages | 16 |
| Journal | Ergodic Theory and Dynamical Systems |
| Volume | 44 |
| Issue number | 6 |
| DOIs | |
| State | Published - 4 Jun 2024 |
Bibliographical note
Publisher Copyright:© The Author(s), 2023. Published by Cambridge University Press.
Funding
Krzysztof Barański was supported by the National Science Centre, Poland, grant no. 2018/31/B/ST1/02495. Adam Śpiewak acknowledges support from the Israel Science Foundation, grant 911/19. A part of this work was done when the second author was visiting the Budapest University of Technology and Economics. We thank Balázs Bárány, Károly Simon and R. Dániel Prokaj for useful discussions, and the staff of the Institute of Mathematics of the Budapest University of Technology and Economics for their hospitality.
| Funders | Funder number |
|---|---|
| Israel Science Foundation | 911/19 |
| Narodowe Centrum Nauki | 2018/31/B/ST1/02495 |
Keywords
- interval homeomorphism
- random system
- stationary measure