Actions of diagonal endomorphisms on conformally invariant measures on the 2-torus (Q2041661)

In this paper, the author starts by defining the \(p\)-fold map (for an integer \(p\geq 2\)) on the unit interval \(I\), \[ T_p(x)=p\cdot x\bmod 1. \] The paper studies ``higher dimensional analogues'' of a theorem by \textit{M. Hochman} and \textit{P. Shmerkin} [Invent. Math. 202, No. 1, 427--479 (2015; Zbl 1409.11054)], stating that \(\mu\)-almost every \(x\) is normal in base \(m\), where \(\mu\) is a \(T_p\)-invariant ergodic measure with positive entropy, and \(m\) and \(p\) are (multiplicatively) independent. Theorem~1.1, the main result, concerns almost sure equidistribution of \(T_{p_1}\times T_{p_2}\)-orbits for a certain measure. For the statement of this theorem, we need the definition of (lower) Hausdorff dimension of a probability measure \(\mu\) (on a metric space), \[ \dim\mu=\inf\{\dim_H A: \mu(A)>0\}. \] Furthermore, let \(\Pi_1\) and \(\Pi_2\) be the coordinate projections. We only reproduce Theorem~1.1, the notation used in Theorem~1.2 (concerning iterated function systems) needing somewhat too much preparation. Let \(m>p\geq 2\) be (multiplicatively) independent integers. Assume that \(\mu\) is an ergodic \(T_p\times T_p\)-invariant measure on the Borel sets of \([0,1]^2\) such that there exists a linear projection \(\pi:\mathbb R^2\rightarrow\mathbb R\) satisfying \(\dim\pi\mu<\dim\mu\) and \(\pi\not\in\{\Pi_1,\Pi_2\}\). The theorem states that \begin{itemize} \item[1.] \(\mu\)-almost all \((x,y)\) equidistribute for \(\lambda\times \Pi_2\mu\) under the map \(T_m\times T_p\); \item[2.] Let \(n\) be an integer (multiplicatively) independent from \(p\) such that \(m>n>p\). Then \(\mu\)-almost all \((x,y)\) equidistribute for \(\lambda\times \lambda\) under the map \(T_m\times T_n\). \end{itemize}