On simultaneous Abel equations. (English) Zbl 0686.39009

Zdun, M. C.

Aequationes Math. 38, No. 2-3, 163-177 (1989).

The system (1) \(\phi [f(x)]=\phi (x)+1\quad \phi [g(x)]=\phi (x)+s\) is investigated, where x ranges over an open interval, and f and g are continuous commuting self-bijections of the interval. One would assume that then f and g are in some sense iterates of each other, and indeed the results are in that general direction. From the many results we (partially) quote: If \(\forall x\) \(f^ m(x)\neq g^ n(x)\) (m,n\(\in {\mathbb{Z}})\) whenever \(| m| +| n| \neq 0\) then there exists a unique s such that (1) has a continuous solution unique up to an additive constant. This solution is monotonic and s is irrational. The results are also applied to the problem of embedding two functions in a continuous iteration group.

Reviewer: Gy.Targonski

MSC:

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

Keywords:

simultaneous Abel equations; flows; iteration groups; system; continuous solution; continuous iteration group

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