Quantitative behavior of unipotent flows and an effective avoidance principle

We give an effective bound on how much time orbits of a unipotent group U on an arithmetic quotient G /Γ can stay near homogeneous subvarieties of G /Γ corresponding to ℚ-subgroups of G . In particular, we show that if such a U -orbit is moderately near a proper homogeneous subvariety of G /Γ for a...

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Bibliographic Details
Published inJournal d'analyse mathématique (Jerusalem) Vol. 153; no. 1; pp. 1 - 61
Main Authors Lindenstrauss, Elon, Margulis, Gregorii, Mohammadi, Amir, Shah, Nimish A.
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.09.2024
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Analysis
Dynamical Systems and Ergodic Theory
Functional Analysis
Mathematics
Mathematics and Statistics
Orbits
Partial Differential Equations
Subgroups
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Summary:We give an effective bound on how much time orbits of a unipotent group U on an arithmetic quotient G /Γ can stay near homogeneous subvarieties of G /Γ corresponding to ℚ-subgroups of G . In particular, we show that if such a U -orbit is moderately near a proper homogeneous subvariety of G /Γ for a long time, it is very near a different homogeneous subvariety. Our work builds upon the linearization method of Dani and Margulis. Our motivation in developing these bounds is in order to prove quantitative density statements about unipotent orbits, which we plan to pursue in a subsequent paper. New qualitative implications of our effective bounds are also given.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-023-0309-9