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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Notes
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.
References
Alsedà, L., Misiurewicz, M.: Random interval homeomorphisms. Publ. Mat. 58(suppl.), 15–36 (2014)
Barański, K., Śpiewak, A.: Singular stationary measures for random piecewise affine interval homeomorphisms. J. Dynam. Differ. Equ. 33(1), 345–393 (2021). https://doi.org/10.1007/s10884-019-09807-5
Czernous, W., Szarek, T.: Generic invariant measure for iterated systems of interval homeomorphism. Arch. Math. (Basel) 114(4), 445–455 (2020)
Czudek, K., Szarek, T.: Ergodicity and central limit theorem for random interval homeomorphisms. Isr. J. Math. 239(1), 75–98 (2020)
Gelfert, K., Stenflo, Ö.: Random iterations of homeomorphisms on the circle. Mod. Stoch. Theory Appl. 4(3), 253–271 (2017)
Gharaei, M., Homburg, A.J.: Random interval diffeomorphisms. Discrete Contin. Dyn. Syst. Ser. S 10(2), 241–272 (2017)
Lenz, D., Stollmann, P.: Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians. Duke Math. J. 131(2), 203–217 (2006)
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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