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Simultaneous equidistributing and non-dense points for non-commuting toral automorphisms

Published online by Cambridge University Press:  11 May 2018

MANFRED EINSIEDLER  and
ALEX MAIER
MANFRED EINSIEDLER
Affiliation:
ETH Zürich, Departement Mathematik, Rämistrasse 101, 8092 Zürich, Switzerland email manfred.einsiedler@math.ethz.ch, alex.georg.maier@gmail.com
ALEX MAIER
Affiliation:
ETH Zürich, Departement Mathematik, Rämistrasse 101, 8092 Zürich, Switzerland email manfred.einsiedler@math.ethz.ch, alex.georg.maier@gmail.com
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Abstract

We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 1 , January 2020 , pp. 175 - 193
Copyright
© Cambridge University Press, 2018 

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References

Bergelson, V., Einsiedler, M. and Tseng, J.. Simultaneous dense and nondense orbits for commuting maps. Israel J. Math. 210(1) (2015), 23–45.10.1007/s11856-015-1244-yGoogle Scholar
Broderick, R., Fishman, L. and Kleinbock, D.. Schmidt’s game, fractals, and orbits of toral endomorphisms. Ergod. Th. & Dynam. Sys. 31(4) (2011), 1095–1107.10.1017/S0143385710000374Google Scholar
Chaika, J. and Eskin, A.. Every flat surface is Birkhoff and Oseledets generic in almost every direction. J. Mod. Dyn. 9 (2015), 1–23.Google Scholar
Dani, S. G.. On orbits of endomorphisms of tori and the Schmidt game. Ergod. Th. & Dynam. Sys. 8 (1988), 523–529.10.1017/S0143385700004673Google Scholar
Einsiedler, M. and Ward, T.. Functional Analysis, Spectral Theory, and Applications (Graduate Texts in Mathematics, 276) . Springer, Cham, 2017.10.1007/978-3-319-58540-6Google Scholar
Fishman, L.. Schmidt’s games, badly approximable matrices, and fractals. J. Number Theory 129 (2009), 2133–2153.Google Scholar
Fishman, L.. Schmidt’s game on fractals. Israel J. Math. 171 (2009), 77–92.10.1007/s11856-009-0041-xGoogle Scholar
Kleinbock, D. Y. and Margulis, G. A.. Bounded orbits of nonquasiunipotent flows on homogeneous spaces. Amer. Math. Soc. Transl. Ser. (2) 171 (1996), 141–172.Google Scholar
Lytle, B. and Maier, A.. Simultaneous dense and nondense orbits for noncommuting toral endomorphisms. Monatsh. Math. 185(3) (2018), 473–488.10.1007/s00605-018-1154-2Google Scholar
Schmidt, W. M.. On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 27–50.Google Scholar