A symmetric Random Walk defined by the time-one map of a geodesic flow

Pablo D. Carrasco; Túlio Vales

doi:10.3934/dcds.2020390

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2021, Volume 41, Issue 6: 2891-2905. Doi: 10.3934/dcds.2020390

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A symmetric Random Walk defined by the time-one map of a geodesic flow

Pablo D. Carrasco^, and
Túlio Vales

Av. Presidente Antônio Carlos 6627, Belo Horizonte-MG, BR31270-901

^* Corresponding author: Pablo D. Carrasco
^* Corresponding author: Pablo D. Carrasco

Received: July 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: June 2021

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Abstract

In this note we consider a symmetric Random Walk defined by a $ (f, f^{-1}) $ Kalikow type system, where $ f $ is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the Random Walk are deduced in these cases.

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Mathematics Subject Classification: Primary: 537C40, 37D30; Secondary: 60K37, 37H99.

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References

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