On dynamics of power series over unramified extensions of \({\mathbb Q}_p\). (Q2782022)

Let \(K\) be an unramified extension of \(Q_p\), \(O\) its ring of integers, and \(M\) the maximal ideal of \(O\). Let \(\bar M\) be the integral closure of \(M\) in the algebraic closure of \(K\). A power series \(u\in O[[X]]\) is called stable, if \(u(0)=0\) and \(u'(0)\) is neither \(0\) nor a root of unity. The author considers systems of commuting power series and studies their dynamic properties.NEWLINENEWLINEThe main theorem states that if such a system \(S\) contains an invertible stable power series and an non-invertible stable series, then it contains also a non-invertible stable series \(f\) such that every stable series lying in \(S\) can be written as \(u\circ f_n\), where \(u\in S\) is stable, and \(f_n\) is the \(n\)-th iterate of \(f\) for some \(n\). This brings some support to Lubin's conjecture, which roughly states that if an invertible stable series commutes with a non-invertible stable series, then there must be a formal group around. The main result is then applied to the study of periodic points in \(\bar M\) of stable power series. In particular an explicit formula is obtained for the number of periodic points of order \(p^n\), counted with their multiplicities.