### A Remark on Kitaoka formal power series attached to local densities.

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On classifie les orbites de $H$ sur l’immeuble de Bruhat-Tits de $G$ pour trois paires sphériques $(G,H)$ de groupes $p$-adiques classiques.

In this paper, we first give several properties with respect to subgroups of self-similar groups in the sense of iterated function system (IFS). We then prove that some subgroups of $p$-adic numbers ${\mathbb{Q}}_{p}$ are strong self-similar in the sense of IFS.

Let p be a prime number, and let [...] Q¯ p${\tilde{\overline{\mathbf{\text{Q}}}}}_{\mathbf{p}}$ be the completion of Q with respect to the pseudovaluation w which extends the p-adic valuation vp. In this paper our goal is to give a characterization of closed subfields of [...] Q¯ p ${\tilde{\overline{\mathbf{\text{Q}}}}}_{\mathbf{p}}$, the completion of Q with respect w, i.e. the spectral extension of the p-adic valuation vp on Q.

This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama et al. in 2008, but we also show that this system is in some sense isomorphic to some other rational base number systems by means of finite transducers. We identify the numbers with finite and eventually periodic representations and we also determine the number of representations of a given p-adic number.