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Invariant densities for random systems of the interval

Part of: Stochastic processes Measure-theoretic ergodic theory Low-dimensional dynamical systems Ergodic theory Random dynamical systems

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 transformations 37H05: Foundations, general theory of cocycles, algebraic ergodic theory
Secondary: 37A05: Measure-preserving transformations 37A10: One-parameter continuous families of measure-preserving transformations 60G10: Stationary processes
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
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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