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

Let ν be a probability measure that is ergodic under the endomorphism (× p, × p) of the torus T², such that dim πμ< dim μ for some non-principal projection π. We show that, if both m≠ n are independent of p, the (× m, × n) orbits of ν typical points will equidistribute towards the Lebesgue measure. If m> p then typically the (× m, × p) orbits will equidistribute towards the product of the Lebesgue measure with the marginal of μ on the y-axis. We also prove results in the same spirit for certain self similar measures. These are higher dimensional analogues of results due (among others) to Host, Lindenstrauss, and Hochman-Shmerkin.