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

Let $\nu$ be a probability measure that is ergodic under the endomorphism $(\times p,\times p)$ of the torus ${\mathbb{T}}^2$, such that $\dim \pi \mu < \dim \mu$forsomenon-principalprojection$\pi$\mathrm{.}Weshowthat,ifboth$m\neq n$areindependentof$p$,the$(\times m, \times n)$orbitsof$\nu$ 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.