On multiplicatively-additive iteration groups

Abstract

Define on the set G:=mathbb R^+times mathbb R the operation (t,a)*(s,b)=(ts,tb+a). (G,*) is a non-commutative group with the neutral element (1, 0). We consider a non-commutative translation equation F(eta ,F(xi ,x))=F(eta *xi ,x), eta , xi in G, xin I, F(1,0)=mathrm{id}, where I is an open interval and F:Gtimes Irightarrow I is a continuous mapping. This equation can be written in the form: F((t,a),F((s,b),x))=F((ts,tb+a),x), t,s in mathbb R^+, xin I. For t=1 the family {F(t,a)} defines an additive iteration group, however for a=0 it defines a multiplicative iteration group. We show that if F(t, 0) for some tne 1 has exactly one fixed point x_t, (F(t,0)-mathrm{id})(x_t-mathrm{id})ge 0 and for an a>0 F(1,a)>mathrm {id}, then there exists a unique homeomorphism varphi :Irightarrow mathbb R such that F((s,b),x)=varphi ^{-1}(svarphi (x)+b) for sin mathbb R^+ and bin mathbb R.