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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: mkim16@mat.umk.pl)
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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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