On the uniqueness of semistable embedding and domain of semistable attraction for probability measures on \(p\)-adic groups
Let \(G\) be a locally compact group, \(\Aut(G)\) be the group of topological automorphisms of \(G\). A probability measure \(\mu\) on \(G\) is said to be semistable if there exist \(\tau\in\Aut(G)\) and \(c\in (0, 1)\) such that \(\mu\) can be embedded in a continuous one-parameter semigroup \(\{\mu_t\}_{t\geq 0}\) of probability measures on \(G\) in such a way that \(\mu=\mu_1\) and \(\tau(\mu_t)=\mu_{ct}\) for all \(t\geq 0\). A measure \(\mu\) is said to be normal if \(\mu*\widetilde\mu= \widetilde\mu*\mu\), where \(\widetilde\mu(B)=\mu(B^{-1})\) for all Borel subsets \(B\) of \(G\). The main result of the article is Theorem 2.1: Any normal semistable measure on a \(p\)-adic Lie group has a unique semistable embedding. This implies the uniqueness of semistable embedding of any operator-semistable measure on a finite dimensional \(p\)-adic vector space. Theorem 2.1 can be considered as an analogue for \(p\)-adic Lie groups of the well known results by W. Hazod and E. Siebert about the uniqueness of semistable embedding for some real nilpotent Lie groups. In Section 3, for a unipotent \(p\)-adic algebraic group, the author compares the class of semistable measures and the class of measures whose domain of semistable attraction is nonempty.
scientific article; zbMATH DE number 5302206