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interval homeomorphism

Ergodic Theory and Dynamical Systems (1)

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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
KRZYSZTOF BARAŃSKI, ADAM ŚPIEWAK
Journal:

Ergodic Theory and Dynamical Systems / Volume 44 / Issue 6 / June 2024

Published online by Cambridge University Press:

04 August 2023, pp. 1473-1488

Print publication:

June 2024
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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].