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

2020/01/20 by Algom, Amir
#11K16 11A63 28A80 28D05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2001.07246

Abstract

Let ν be a probability measure that is ergodic under the endomorphism (× p, × p) of the torus \mathbbT², 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 ν. These are higher dimensional analogues of results due (among others) to Host, Lindenstrauss, and Hochman-Shmerkin.

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