Effective Mixing and Counting in Bruhat-Tits Trees
Sanghoon Kwon

TL;DR
This paper proves exponential mixing of geodesic flows and provides effective counting formulas for orbits in Bruhat-Tits trees under certain spectral gap conditions, advancing understanding of dynamics on trees.
Contribution
It establishes exponential mixing and effective counting formulas for group actions on Bruhat-Tits trees with spectral gap assumptions, extending prior results to weighted settings.
Findings
Proved exponential mixing of geodesic translation map under non-arithmetic length spectrum.
Derived effective orbit counting formulas in Bruhat-Tits trees.
Established conditions for spectral gap and fullness of the group.
Abstract
Let be a locally finite tree, be a discrete subgroup of and be a -invariant potential. Suppose that the length spectrum of is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map with respect to the measure under the assumption that is full and has weighted spectral gap property. We also obtain the effective formula for the number of -orbits with weights in a Bruhat-Tits tree of an algebraic group.
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