Effective mixing and counting in  Bruhat–Tits trees

Kwon, Sanghoon

doi:10.1017/etds.2016.28

Cited by 6 publications

(8 citation statements)

References 17 publications

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“…We remark that this theorem also follows from the main result of Kwon in [32] and from the work of Roblin [49, Chapitre 4, Corollaire 2]. Our proof relies on our previous result on the equidistribution of spheres (Theorem E) and is a relatively straightforward consequence thereof.…”

Section: Counting Lattice Pointsmentioning

confidence: 58%

“…As the subsequent proofs will show, we are naturally led to the study of the Markov chain M n which can simply be seen as the image by quotient map π of the simple random walk on the set of edges of the tree T . It came to our knowledge that this Markov chain was considered by Burger and Mozes [9] in the study of the notion of divergence groups in Aut(T ) and by Kwon [32] in the study of mixing properties of the discrete geodesic flow.…”

Section: The Markov Chainmentioning

confidence: 99%

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(Non)-escape of mass and equidistribution for horospherical actions on trees

2021

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Let G be a large group acting on a biregular tree T and $$\Gamma \le G$$ Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on $$G/\Gamma $$ G / Γ . In this article we show that there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on $$G/\Gamma $$ G / Γ . On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to $$\Gamma \backslash T$$ Γ \ T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on $$\Gamma \backslash T$$ Γ \ T . Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.

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Section: Counting Lattice Pointsmentioning

confidence: 58%

Section: The Markov Chainmentioning

confidence: 99%

(Non)-escape of mass and equidistribution for horospherical actions on trees

2021

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“…When Γ is geometrically finite, there is an error term in O(q −κ ) for some κ > 0 in this equidistribution claim evaluated on a locally constant function with compact support. The proof (see [1]) uses the mixing property of the square of the geodesic flow, and its exponential mixing property announced in [2] when Γ is geometrically finite.…”

Section: Abridged English Versionmentioning

confidence: 99%

“…Nous montrons d'ailleurs que l'image de cette mesure sur Γ\G X par l'application origine ℓ → ℓ(0) est un multiple de la mesure vol Γ\ \X sur Γ\V X. Le terme d'erreur lorsque Γ est géométriquement fini utilise la propriété de décroissance exponentielle des corrélations du mélange, annoncee dans [2].…”

Section: éQuidistribution De Perpendiculaires Communes Dans Des Arbresunclassified

Équidistribution non archimédienne et actions de groupes sur les arbres

Broise-Alamichel

Parkkonen

Paulin

2016

Comptes Rendus. Mathématique

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Nous donnons des résultats d'équidistribution d'éléments de corps de fonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps, dans leurs corps locaux complétés. Nous déduisons ces résultats de théorèmes d'équidistribution de perpendiculaires communes dans des quotients d'arbres par des réseaux de leur groupe d'automorphismes, démontrésà l'aide de propriétés ergodiques du flot géodésique discret.Abstract. Non-Archimedean equidistribution and group actions on trees. We give equidistribution results of elements of function fields over finite fields, and of quadratic irrationals over these fields, in their completed local fields. We deduce these results from equidistribution theorems of common perpendiculars in quotients of trees by lattices in their automorphism groups, proved by using ergodic properties of the discrete geodesic flow.

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“…It was claimed in Example 4.1 in [3] that geometrically finite groups have a WSG for any constant potential. However, that statement is misleading because a geometrically finite discrete group in Aut(T ) can never be full unless it is convex cocompact.…”

mentioning

confidence: 99%