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.
Bibliographical noteFunding Information:
Supported by ERC Grant 306494 and ISF Grant 1702/17.
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.
- Ergodic measures
- Self-similar measures
ASJC Scopus subject areas
- Mathematics (all)