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

Abstract

Let $\nu$ be a probability measure that is ergodic under the endomorphism $(\times p, \times p)$ of the torus $\mathbb{T}^2$, such that $\dim \pi \mu < \dim \mu$ for some non-principal projection $\pi$. We show that, if both $m\neq n$ are independent of $p$, the $(\times m, \times n)$ orbits of $\nu$ typical points will equidistribute towards the Lebesgue measure. If $m>p$ then typically the $(\times m, \times p)$ orbits will equidistribute towards the product of the Lebesgue measure with the marginal of $\mu$ on the $y$-axis. We also prove results in the same spirit for certain self similar measures $\nu$. These are higher dimensional analogues of results due (among others) to Host, Lindenstrauss, and Hochman-Shmerkin.