## The structure of iteration groups of continuous functions.(English)Zbl 0801.39005

An iteration group $$G$$ on the open interval $$I$$ is a family of continuous functions $$f^ t : I \to I$$, $$t \in\mathbb{R}$$, such that $$f^ t \circ f^ s = f^{t + s}$$, $$t,s \in \mathbb{R}$$. Consider only iteration groups such that every $$f^ t$$ is either the identity mapping or has no fixed point. Then it is shown that either
(i) for every $$s,t$$ such that $$f^ s \neq Id$$ and $$f^ t \neq Id$$, there exist $$n,m \in \mathbb{Z}$$ such that $$| n | + | m | \neq 0$$ and $$f^{ns} \equiv f^{mt}$$ or
(ii) there exist $$s,t\in \mathbb{R}$$ such that for every $$n,m\in Z$$ with $$| n | + | m | \neq 0$$ and every $$x \in I$$ we have $$f^{ns} (x) \neq f^{mt} (x)$$.
In case (i) it is shown that there is a rational iteration group on $$I$$: $$g^ w : I \to I$$, $$w \in \mathbb{Q}$$ such that $$f^ u \circ f^ v = f^{u + v}$$, $$u,v \in \mathbb{Q}$$ and, an additive function $$\lambda : \mathbb{R} \to \mathbb{Q}$$ such that $$f^ t = g^{\lambda (t)}$$, $$t \in \mathbb{R}$$. In case (ii) a general expression is given for the group $$G$$ and this takes one of two forms, depending on whether the limit set $$L_ G =$$ {limit points of the orbit $$f^ t(x)$$, $$t \in \mathbb{R}$$ for fixed $$x\}$$ is a closed interval or not.
Reviewer: I.N.Baker (London)

### MSC:

 39B12 Iteration theory, iterative and composite equations 37C10 Dynamics induced by flows and semiflows

### Keywords:

iteration group; fixed point
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### References:

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