Logarithm laws for flows on homogeneous spaces

DOI10.1007/S002220050350zbMATH Open0934.22016arXivmath/9812088OpenAlexW3099855963MaRDI QIDQ1818340

Author name not available (Why is that?)

Publication date: 17 April 2000

Published in: (Search for Journal in Brave)

Abstract: We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a sequence of subsets

A_{n}

of a homogeneous space

G / G a m m a

(

G

a semisimple Lie group,

G a m m a

a non-uniform lattice) and a sequence of elements

f_{n}

of

G

we prove that for almost all points

x

of the space, one has

f_{n} x i n A_{n}

for infinitely many

n

. The main tool is exponential decay of correlation coefficients of smooth functions on

G / G a m m a

. Besides the aforementioned application to geodesic flows, as a corollary we obtain a new proof of the classical Khinchin-Groshev theorem in simultaneous Diophantine approximation, and settle a related conjecture recently made by M. Skriganov.

Full work available at URL: https://arxiv.org/abs/math/9812088

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