On simultaneous Abel equations. (English) Zbl 0686.39009

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


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