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- Title
On multiplicatively-additive iteration groups.
- Authors
Zdun, Marek Cezary
- Abstract
Define on the set G:=R+×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(η,F(ξ,x))=F(η∗ξ,x), η,ξ∈G, x∈I, F(1,0)=id, where I is an open interval and F:G×I→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∈R+, x∈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 t≠1 has exactly one fixed point xt, (F(t,0)-id)(xt-id)≥0 and for an a>0F(1,a)>id, then there exists a unique homeomorphism φ:I→R such that F((s,b),x)=φ-1(sφ(x)+b) for s∈R+ and b∈R.
- Subjects
ITERATIVE methods (Mathematics); NONCOMMUTATIVE function spaces; CARTOGRAPHY; HOMEOMORPHISMS; AFFINE geometry
- Publication
Aequationes Mathematicae, 2019, Vol 93, Issue 1, p205
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-018-0622-z