Continuous automorphisms and an equivalence relation In K[[X]]

Abstract

Let K be an arbitrary commutative field and let R=K[[X]]R=K[[X]] be the ring of formal power series in one variable. Let GRGR be the set of all power series of the form u=Xvu=Xv, where vv is a unity in RR. Relative to the usual composition GRGR becomes a topological group with respect to the X−X−adic topology of RR. We also study an equivalence relation on RR. Let R=K[[X]]R=K[[X]] be the ring of formal power series in one variable over a fixed commutative field KK. We denote by ordf=min{i:ai≠0}ordf=min{i:ai≠0} for any f∈Rf∈R . It is well known that ordfordf is a valuation on RR and RR becomes a complete topological ring relative to the topology induced by this valuation. Let  GG={u∈R:ordu=1}GG={u∈R:ordu=1} and for u,v∈RGu,v∈RG we denote (uov)(X)=v(u(X))(uov)(X)=v(u(X)), a new element of RGRG.