Doubling property of self-affine measures on carpets of Bedford and McMullen (Q2822156)

[The reviewer considers as most illustrative of the content of the paper to cite a part of the authors' introduction, with minor changes.]NEWLINENEWLINEIn the present paper, we study the doubling property of measures on carpets of Bedford and McMullen (recall that a Borel measure \(\mu\) on a metric space \(X\) is said to be doubling if there is a constant \(C\) such that \(0<\mu((B(x,2R))\leq C\mu(B(x,R))<\infty\) for all balls \(B(x,R)\subset X\) of positive radius \(R\)). Let \(\mathcal M\) be the family of such carpets, and let \(\mathbb {S}\in \mathcal M\) be a given carpet. We obtain an equivalent condition for a Borel measure to be doubling on \(\mathbb S\). After that, we focus on self-affine measures on \(\mathbb S\). These measures are seldom Ahlfors regular; for instance, if \(\mathbb S\) takes different values of Hausdorff and Assouad dimensions, it is impossible to support any Ahlfors regular measure. We therefore shall investigate the doubling property of self-affine measures on \(\mathbb S\). Our main results completely answer the following questions: {\parindent=0.7cm\begin{itemize}\item[(1)] When does a carpet \(\mathbb S\) carry a doubling self-affine measure? \item[(2)] Which self-affine measures are doubling on \(\mathbb S\) (when it carries doubling self-affine measures)? \item[(3)] What conditions guarantee that the uniform self-affine measure is doubling on \(\mathbb S\)? NEWLINENEWLINE\end{itemize}}