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

Zdun, Marek Cezary

Aequationes Math. 46, No. 1-2, 19-37 (1993).

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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