2020/01/20 by Amir Algom, Algom, Amir
Computer Science · Mathematics · #11K16 11A63 28A80 28D05 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DS #msc:11A63 #msc:11K16 #msc:28A80 #msc:28D05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2001.07246
arxiv created 2020/01/20 · openalex publication_date 2020/01/20 · arxiv updated 2020/01/22 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
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.