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

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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 languageEnglish
Pages (from-to)545-564
Number of pages20
JournalMonatshefte fur Mathematik
Issue number4
StatePublished - Aug 2021
Externally publishedYes

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.


  • Equidistribution
  • Ergodic measures
  • Self-similar measures

ASJC Scopus subject areas

  • Mathematics (all)


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