On minimal decomposition of $p$-adic polynomial dynamical systems

Ai-Hua Fan (1) , Lingmin Liao (2, 3)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 Department of Mathematics [Wuhan]
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

A polynomial of degree $\ge 2$ with coefficients in the ring of $p$-adic numbers $\mathbb{Z}_p$ is studied as a dynamical system on $\mathbb{Z}_p$. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on $\mathbb{Z}_2$, we exhibit all its minimal subsystems.

Mots clés: $p$-adic dynamical system; minimal component; quadratic polynomial

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

Ai-Hua Fan, Lingmin Liao. On minimal decomposition of $p$-adic polynomial dynamical systems. Advances in Mathematics, 2011, 228, pp.2116--2144. ⟨hal-00529905v2⟩