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Effective mixing and counting in Bruhat–Tits trees
Published online by Cambridge University Press: 04 July 2016
Abstract
Let ${\mathcal{T}}$ be a locally finite tree,
$\unicode[STIX]{x1D6E4}$ be a discrete subgroup of
$\,\operatorname{Aut}\,({\mathcal{T}})$ and
$\widetilde{F}$ be a
$\unicode[STIX]{x1D6E4}$-invariant potential. Suppose that the length spectrum of
$\unicode[STIX]{x1D6E4}$ is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map
$\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}\rightarrow \unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}$ with respect to the measure
$m_{\unicode[STIX]{x1D6E4},F}^{\unicode[STIX]{x1D708}^{-},\unicode[STIX]{x1D708}^{+}}$ under the assumption that
$\unicode[STIX]{x1D6E4}$ is full and
$(\unicode[STIX]{x1D6E4},\widetilde{F})$ has a weighted spectral gap. We also obtain the effective formula for the number of
$\unicode[STIX]{x1D6E4}$-orbits with weights in a Bruhat–Tits tree
${\mathcal{T}}$ of an algebraic group.
- Type
- Research Article
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- Copyright
- © Cambridge University Press, 2016
References
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