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Typical Behaviour of Random Interval Homeomorphisms

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Abstract

We consider the typical behaviour of random dynamical systems of order-preserving interval homeomorphisms with a positive Lyapunov exponent condition at the endpoints. Our study removes any requirement for continuous differentiability save the existence of finite derivatives of the homeomorphisms at the endpoints of the interval. We construct a suitable Baire space structure for this class of systems. Generically within this Baire space, we show that the stationary measure is singular with respect to the Lebesgue measure, but has full support on [0, 1]. This provides an answer to a question raised by Alsedà and Misiurewicz.

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Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. Note that differentiability at a point is not preserved under uniform limits, so we cannot use the usual supremum distance on [0, 1] for our space of functions.

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Authors and Affiliations

  1. Mathematical Institute, Silesian University in Opava, Na Rybnic̆ku 1, 74601, Opava, Czech Republic

    Jaroslav Bradík & Samuel Roth

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  1. Jaroslav Bradík
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  2. Samuel Roth
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Correspondence to Samuel Roth.

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Research was funded by institutional support for the development of research organizations (IČ47813059) and by Grant SGS 18/2019.

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Bradík, J., Roth, S. Typical Behaviour of Random Interval Homeomorphisms. Qual. Theory Dyn. Syst. 20, 73 (2021). https://doi.org/10.1007/s12346-021-00509-2

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  • DOI: https://doi.org/10.1007/s12346-021-00509-2

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