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Commuting functions and simultaneous Abel equations. (English) Zbl 0828.39006

Let \({\mathcal F}= \{f_t, t\in T\}\) be a family of pairwise commuting homeomorphisms of an interval \((a, b)\) onto itself, having no fixed points. Given \(f, g\in {\mathcal F}\) a sequence \(\{s_k(x), k\in \mathbb{N}\}\) of integers can be defined such that \(f^{s_k(x)+ 1}\leq g^k(x)\leq f^{s_k(x)}\) (if \(f<\text{Id}\); when \(f> \text{Id}\), the inequalities are reversed), where \(f^m\) stands for the \(m\)th iterate of \(f\). M. C. Zdun [Some remarks on the iterates of commuting functions, European Conference on Iteration Theory, Lisbon, 1991, J. P. Lampreia et al. (eds), World Scientific, Singapore, 336-342 (1992)] proved that there exists a finite and non-zero limit \(\nu(f, g):= \lim_{k\to \infty} (s_k(x)/k)\) which actually does not depend on \(x\).
In Theorem 1 conditions are given for the system of Abel equations \(\alpha(f_t(x))= \alpha(x)+ \nu(t)\), where \(f_t\in {\mathcal F}\), \(\nu(t):= \nu(f_{t_0}, t_t)\) with an arbitrarily fixed \(f_{t_0}\), to possess either a homeomorphic solution depending on an arbitrary function or a unique (up to an arbitrary constant) continuous solution \(\alpha: (a, b)\to \mathbb{R}\) (other possibilities are excluded). As a consequence the authors obtain conditions of a rational iteration group to be embeddable in a continuous iteration group.
A slight generalization of the Krylov-Bogolubov Theorem (Theorem 2) is also proved which is then used as a main tool in proving Theorem 1. The results generalize some due to M. C. Zdun [Aequationes Math. 36, No. 2/3, 153-164 (1988; Zbl 0662.39004); ibid. 38, No. 2/3, 163-177 (1989; Zbl 0686.39009)].

MSC:

39B12 Iteration theory, iterative and composite equations
37B99 Topological dynamics
26A18 Iteration of real functions in one variable
60A10 Probabilistic measure theory
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