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

Amir Algom

doi:10.1007/s00605-021-01595-9

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

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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
Pages (from-to)	545-564
Number of pages	20
Journal	Monatshefte fur Mathematik
Volume	195
Issue number	4
DOIs	https://doi.org/10.1007/s00605-021-01595-9
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)

Access to Document

10.1007/s00605-021-01595-9

Cite this

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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.",

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N2 - 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.

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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