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On the dimension of stationary measures for random piecewise affine interval homeomorphisms

Part of: Random dynamical systems Low-dimensional dynamical systems

Published online by Cambridge University Press:  04 August 2023

KRZYSZTOF BARAŃSKI  and
ADAM ŚPIEWAK Open the ORCID record for ADAM ŚPIEWAK [Opens in a new window]
KRZYSZTOF BARAŃSKI*
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
ADAM ŚPIEWAK
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland (e-mail: ad.spiewak@gmail.com)
*
e-mail: baranski@mimuw.edu.pl
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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 $1$. 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].

Keywords

random systemstationary measureinterval homeomorphism

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37E10: Maps of the circle 37H10: Generation, random and stochastic difference and differential equations 37H15: Multiplicative ergodic theory, Lyapunov exponents
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 16
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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