# Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields

Abstract : Let $F$ be a non-discrete non-Archimedean locally compact field and $\mathcal{O}_F$ the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrm{Mat}(\mathbb{N}, F)$ of infinite matrices with enties in $F$ with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F) \times \mathrm{GL}(\infty,\mathcal{O}_F)$ and Theorem 1.6 that, for non-dyadic $F$, classifies ergodic probability measures on the space $\mathrm{Sym}(\mathbb{N}, F)$ of infinite symmetric matrices with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F)$.
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Alexander I. Bufetov, Yanqi Qiu. Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields. Compositio Mathematica, Foundation Compositio Mathematica, 2017, 153 (12), pp.2482-2533. ⟨10.1112/S0010437X17007412⟩. ⟨hal-01483629⟩

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