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Invariant densities for random systems of the interval

Part of:Measure-theoretic ergodic theoryErgodic theoryLow-dimensional dynamical systemsRandom dynamical systemsStochastic processes

Published online by Cambridge University Press:  29 December 2020

CHARLENE KALLE and
MARTA MAGGIONI
CHARLENE KALLE*
Affiliation:
Mathematisch Instituut, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands (e-mail: m.maggioni@math.leidenuniv.nl)
MARTA MAGGIONI
Affiliation:
Mathematisch Instituut, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands (e-mail: m.maggioni@math.leidenuniv.nl)
*
e-mail: kallecccj@math.leidenuniv.nl
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Abstract

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For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $-transformations and random Lüroth maps with a hole.

Keywords

absolutely continuous invariant measureβ-expansionsinterval mapLüroth expansionsrandom dynamics

MSC classification

Primary:37E05: Maps of the interval (piecewise continuous, continuous, smooth)28D05: Measure-preserving transformations37H05: Foundations, general theory of cocycles, algebraic ergodic theory
Secondary:37A05: Measure-preserving transformations37A10: One-parameter continuous families of measure-preserving transformations60G10: Stationary processes

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022, pp. 141 - 179
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Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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