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

Abstract

<p style='text-indent:20px;'>In [<xref ref-type="bibr" rid="b22">22</xref>], Jabłoński proved that a piecewise expanding <inline-formula><tex-math id="M1">\begin{document}$ C^{2} $\end{document}</tex-math></inline-formula> multidimensional Jabłoński map admits an absolutely continuous invariant probability measure (ACIP). In [<xref ref-type="bibr" rid="b6">6</xref>], 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 [<xref ref-type="bibr" rid="b9">9</xref>] in one dimension and Góra, Boyarsky and Proppe [<xref ref-type="bibr" rid="b19">19</xref>] in higher dimensions.</p>