Optimal cycles in ultrametric dynamics and minimally ramified power series (Q2787624)

This work is concerned with ultrametric power series in one variable, with an indifferent fixed point at the origin whose multiplier \(\lambda\) is not a root of unity. The focus is on the structure of periodic points near the origin, within the broader question of local linearisability.NEWLINENEWLINEIn holomorphic dynamics, Yoccoz showed that a quadratic polynomial \(P_\lambda(z)=\lambda z+z^2\) is not linearizable precisely if every neighbourhood of the origin contains a periodic point other than the origin. Moreover, the distance of such a cycle to the origin is minimal for quadratic polynomials, when compared with cycles of the same period of normalised power series with the same linear term.NEWLINENEWLINEThe authors establish an analogous result for normalised power series over an arbitrary non-Archimedean field \(K\) with residue characteristic \(p\). They derive a lower bound for the distance from the origin of a periodic point of period \(qp^n\), and call a cycle \textit{optimal} if it attains such a bound. They show (under some conditions on \(q\) and \(\lambda\)) that there is a polynomial of degree at most \(2q+1\) that has an optimal cycle of period \(qp^n\) for any \(n\). Then they demonstrate the special role of quadratic polynomials, and the connection with ramification theory. Thus if \(p\) is odd, and \(K\) is an algebraically closed with characteristic \(p\), then the polynomial \(P_\lambda\) has a unique cycle of period \(qp^n\) for any \(n\) if the reduction of \(P_\lambda\) has the property of being minimally ramified, and no optimal cycle otherwise. In proving these results, the authors extend a theorem of \textit{S. Sen} [Ann. Math. (2) 90, 33--46 (1969; Zbl 0199.36301)] on wildly ramified field automorphisms.NEWLINENEWLINEThe proof of these results establishes a connection between the geometric location of periodic orbits of ultrametric power series and the lower ramification number of wildly ramified field automorphisms. The authors also give an extension of Sen's theorem on wildly ramified field automorphisms, and a characterization of minimally ramified power series in terms of an iterative residue.