Abstract
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.
Original language | English |
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Pages (from-to) | 545-564 |
Number of pages | 20 |
Journal | Monatshefte fur Mathematik |
Volume | 195 |
Issue number | 4 |
DOIs | |
State | Published - Aug 2021 |
Externally published | Yes |
Bibliographical note
Funding Information:Supported by ERC Grant 306494 and ISF Grant 1702/17.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.
Keywords
- Equidistribution
- Ergodic measures
- Self-similar measures
ASJC Scopus subject areas
- Mathematics (all)