On multiplicatively-additive iteration groups

Define on the set G:=R+×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G:=\mathbb R^+\times \mathbb R$$\end{document} the operation (t,a)∗(s,b)=(ts,tb+a)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t,a)*(s,b)=(ts,tb+a)$$\end{document}. (G,∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G,*)$$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(\eta ,F(\xi ,x))=F(\eta *\xi ,x)$$\end{document}, η,ξ∈G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta , \xi \in G$$\end{document}, x∈I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in I$$\end{document}, F(1,0)=id\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(1,0)=\mathrm{id}$$\end{document}, where I is an open interval and F:G×I→I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F:G\times I\rightarrow I$$\end{document} is a continuous mapping. This equation can be written in the form: F((t,a),F((s,b),x))=F((ts,tb+a),x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F((t,a),F((s,b),x))=F((ts,tb+a),x)$$\end{document}, t,s∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t,s \in \mathbb R^+$$\end{document}, x∈I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in I$$\end{document}. For t=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1$$\end{document} the family {F(t,a)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{F(t,a)\}$$\end{document} defines an additive iteration group, however for a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document} it defines a multiplicative iteration group. We show that if F(t, 0) for some t≠1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\ne 1$$\end{document} has exactly one fixed point xt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_t$$\end{document}, (F(t,0)-id)(xt-id)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(F(t,0)-\mathrm{id})(x_t-\mathrm{id})\ge 0$$\end{document} and for an a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0 $$\end{document}F(1,a)>id\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(1,a)>\mathrm {id}$$\end{document}, then there exists a unique homeomorphism φ:I→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :I\rightarrow \mathbb R$$\end{document} such that F((s,b),x)=φ-1(sφ(x)+b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F((s,b),x)=\varphi ^{-1}(s\varphi (x)+b)$$\end{document} for s∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in \mathbb R^+$$\end{document} and b∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\in \mathbb R$$\end{document}.


Introduction
In this paper we deal with the continuous solutions of the functional equation F ((t, a), F ((s, b), x)) = F ((ts, tb + a), x), t, s > 0, a, b ∈ R, x ∈ I, (1) where I is an open non-empty interval.
An inspiration which led to consider the above equation were some models in economics connected to the description of a special order in a space of sequences of preference of consumption outcomes called "impatience" (see [2]). Arsen Kochov in a private correspondence presented an idea that every sequence of consumption outcomes can be represented by an increasing homeomorphism with one fixed point and the problem of the existence of impatience order can be reduced to the determination of suitable properties of semigroups G of increasing homeomorphisms of R onto R satisfying the conditions: (ii) if f has a fixed point p f and g has a fixed point p g , then p f > p g if and only if f • g > g • f .
The existence of the relation of impatience is related to the "affinization" of the above semigroups, that is to the problem of describing when these semigroups are conjugate to the family of affine functions. The aim of the present paper is to show a construction of large families of semigroups of homeomorhisms possessing at most one fixed point satisfying conditions (i) and (ii) which are conjugate to the family of all affine functions. Their construction relates to the determination of two parameter families of homeomorphisms F (t, a) : R → R satisfying Eq. (1).
Let ϕ : I → R be an increasing surjection. Define The parameters t and a are determined uniquely. Thus we may write a f instead of a and t f instead of t and define If ϕ is a homeomorphism then Realm ϕ with the operation of composition is a group and Realm * ϕ is a semigroup.
Define the family of functions where p ∈ R and s > 0. Every function l s p has a unique fixed point, namely x = p, if s = 1. Moreover, Hence l s p •l t q = l t q •l s p if and only if p = q. However for t, s ∈ (0, 1), l s p •l t q > l t q •l s p if and only if p > q. Thus the family {l s p , s ∈ (0, 1), p ∈ R} is a semigroup satisfying (i) and (ii). Realm * ϕ is conjugate to the family of affine functions {l s p , s ∈ (0, 1), p ∈ R}. Hence we get the following.
The second statement is a simple consequence of the fact that the set of affine functions {tx+a, t ∈ (0, 1), a, ∈ R} satisfies (i) and (ii) and is conjugate to the family Realm * ϕ.
Note that G is a non-commutative group and G * is a non-commutative semigroup and Note that for every homeomorphism ϕ : I → R andû ∈ G there exists a unique homeomorphism F (û, ·) ∈ Realmϕ such that Ind F (û, ·) =û. Hence the last relation implies that the function F : G × I → I satisfies the noncommutative translation equation Puttingû = (t, a) andv = (s, b), the last equation can be written in the following form

x).
For t = 1 the family {F (t, a)} defines an additive iteration group of homeomorphisms, however for a = 0 it defines a multiplicative iteration group of homeomorphisms. In fact, putting Hence {G t , t > 0} is a multiplicative iteration group and {H a , a ∈ R} is an additive iteration group. Thus Eq. (1) describes simultaneously multiplicative and additive iteration groups.

Definition. A family of continuous functions {F (t, a)
: where F satisfies (1) is said to be a multiplicatively-additive iteration group.
The main purpose of this paper is to investigate when a multiplicativelyadditive iteration group is conjugate to a group of affine functions.

Main results
In this section we deal with multiplicatively-additive iteration groups {F (t, a), t > 0, a ∈ R}, possessing the property that at least one function F (t, 0) has a unique fixed point and at least one function F (1, a) does not have a fixed point. We answer when these groups are conjugate to the family of all affine functions. More precisely, we show the following.
In order to prove the theorem we will first show a few lemmas. In the assumptions of these lemmas the function F by default is a non-constant continuous solution of (1) such that F (1, 0) = id.
Put Proof. Let F (s, 0, x 0 ) = x 0 for an x 0 ∈ I. Put This is a contradiction. It is the same when we assume that x 0 < x 1 .
Since the set { n 2 k , n, k ∈ N} is dense in R + the continuity of F with respect to t implies that F (t, 0, x 0 ) = x 0 for t > 0. Hence we infer that if F (s, 0) has a unique fixed point, then all functions F (t, 0), t > 0 have this property.

Vol. 93 (2019)
On multiplicatively-additive iteration groups 209 Lemma 2. If F (1, a) has no fixed points for some a, then for every b = 0 the function F (1, b) has no fixed points either.
Proof. Suppose that there exist x 0 ∈ I and b = 0 such that F ((1, b) Similarly as in the previous proof, by induction we get that F (1, b 2 n , x 0 ) = x 0 and F (1, ma 2 n , x 0 ) = x 0 for n ∈ N and m ∈ Z. By the density of the set { ma 2 n , m, n ∈ Z} in R and the continuity of F with respect to the second variable we get that F (1, a, x 0 ) = x 0 for all a ∈ R, which is a contradiction.
The assumption of continuity of F in Theorem 1 can be replaced by the condition that F is Lebesgue measurable with respect to the first and second variables and continuous with respect to the third one.
Let G t , t > 0 and H a , a ∈ R be the functions defined by (2). The weakened assumption on F says that all these functions are continuous and for every x ∈ I the functions t → G t (x) and a → H a (x) are Lebesgue measurable. Hence, it follows by (3) and (4), that {G t , t > 0} and {H a , a ∈ R} are measurable iteration groups. It is known that measurable iteration groups are continuous, that is the functions (t, x) → G t (x) and (a, x) → H a (x) are continuous (see [5, Th.1.1], [7]). It follows, by (1), the following equality F ((t, a), x) = H a (G t (x)) holds for t > 0, a ∈ R and x ∈ I. Thus F is continuous. The remaining part of the proof is the same. By Theorem 1 there exists a homeomorphism ϕ such that {F (t, a), 0 < t < 1, a ∈ R} = Realm ϕ. By Remark 1 we get our assumption.

Conjecture.
For every semigroup G of homeomorphisms with at most one fixed point satisfying (i) and (ii) there exists an increasing surjection ϕ such that G ⊂ Realm * ϕ.