On the number of invariant measures for random expanding maps in higher dimensions

Fawwaz Batayneh; Cecilia González-Tokman

doi:10.3934/dcds.2021100

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2021, Volume 41, Issue 12: 5887-5914. Doi: 10.3934/dcds.2021100 open access

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On the number of invariant measures for random expanding maps in higher dimensions

Fawwaz Batayneh^, and
Cecilia González-Tokman

School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

^* Corresponding author: Fawwaz Batayneh
^* Corresponding author: Fawwaz Batayneh

Received: June 2020

Revised: April 2021

Early access: June 2021

Published: December 2021

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Abstract

In [22], Jabłoński proved that a piecewise expanding $ C^{2} $ multidimensional Jabłoński map admits an absolutely continuous invariant probability measure (ACIP). In [6], Boyarsky and Lou extended this result to the case of i.i.d. compositions of the above maps, with an on average expanding condition. We generalize these results to the (quenched) setting of random Jabłoński maps, where the randomness is governed by an ergodic, invertible and measure preserving transformation. We prove that the skew product associated to this random dynamical system admits a finite number of ergodic ACIPs. Furthermore, we provide two different upper bounds on the number of mutually singular ergodic ACIPs, motivated by the works of Buzzi [9] in one dimension and Góra, Boyarsky and Proppe [19] in higher dimensions.

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Mathematics Subject Classification: Primary: 37H15, 37C30; Secondary: 37C83.

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Figure 1. $ I^{2} $ partitioned into $ 25 $ equal squares

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Figure 2. Bounds in (45) (solid) and (46) (dashed)

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