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Effective equidistribution for generalized higher-step nilflows

Part of: Lie groups Ergodic theory

Published online by Cambridge University Press:  25 October 2021

MINSUNG KIM Open the ORCID record for MINSUNG KIM [Opens in a new window]
MINSUNG KIM*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
*
(e-mail: [email protected])
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Abstract

In this paper we prove bounds for ergodic averages for nilflows on general higher-step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at polynomial speed. We analyze the rate of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. Our main result follows from the technique of controlling scaling operators in irreducible representations and measure estimation on close return orbits on general nilmanifolds.

Keywords

nilflowscohomological equationsergodic averagescoadjoint orbits

MSC classification

Primary: 37A17: Homogeneous flows
Secondary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 37A25: Ergodicity, mixing, rates of mixing 37A45: Relations with number theory and harmonic analysis
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 12 , December 2022 , pp. 3656 - 3715
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Auslander, L., Green, L. and Hahn, F.. Flows on Homogeneous Spaces (Annals of Mathematics Studies, 53). Princeton University Press, Princeton, NJ, 1963, with the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg.CrossRefGoogle Scholar
Bourgain, J., Demeter, C. and Guth, L.. Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. 184 (2016), 633–682.CrossRefGoogle Scholar
Corwin, L. and Greenleaf, F. P.. Representations of Nilpotent Lie Groups and Their Applications. Part 1. Basic Theory and Examples (Cambridge Studies in Advanced Mathematics, 18). Cambridge University Press, Cambridge, 1990.Google Scholar
Dubrovin, B. A., Fomenko, A. T. and Novikov, S. P.. Modern Geometry—Methods and Applications. Part II: The Geometry and Topology of Manifolds (Graduate Texts in Mathematics, 104). Springer, New York, 1985.CrossRefGoogle Scholar
Forni, G.. Effective equidistribution of nilflows and bounds on Weyl sums. Dyn. Anal. Number Theory 437 (2016), 136–188.CrossRefGoogle Scholar
Flaminio, L. and Forni, G.. Equidistribution of nilflows and applications to theta sums. Ergod. Th. & Dynam. Sys. 26(2) (2006), 409–433.CrossRefGoogle Scholar
Flaminio, L. and Forni, G.. On the cohomological equation for nilflows. J. Mod. Dyn. 1(1) (2007), 37–60.CrossRefGoogle Scholar
Flaminio, L. and Forni, G.. On effective equidistribution for higher step nilflows. Preprint, 2014, arXiv:1407.3640.Google Scholar
Flaminio, L., Forni, G. and Tanis, J.. Effective equidistribution of twisted horocycle flows and horocycle maps. Geom. Funct. Anal. 26(5) (2016), 1359–1448.CrossRefGoogle Scholar
Gorodnik, A. and Spatzier, R.. Exponential mixing of nilmanifold automorphisms. J. Anal. Math. 123 (2014), 355–396.CrossRefGoogle Scholar
Green, B. and Tao, T.. The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. 175 (2012), 465–540.CrossRefGoogle Scholar
Humphreys, J.. Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics, 9). Springer, New York, 1972.CrossRefGoogle Scholar
Wooley, T. D.. Perturbations of Weyl sums. Int. Math. Res. Not. IMRN 2016(9) (2015), 2632–2646.CrossRefGoogle Scholar