On the structure of some flows on the unit circle. (English) Zbl 0891.39017

Bajger, Mariusz

Aequationes Math. 55, No. 1-2, 106-121 (1998).

Let \(\mathcal S\) be the unit circle of the complex plane and \(T:{\mathcal S}\rightarrow{\mathcal S}\) be continuous. A point \(z\in{\mathcal S}\) is a fixed point of \(T\) if \(T(z)=z\). The flow of \(T\) is the set of all iterates \(T^t\) of \(T\) (\(t\) positive, 0, negative, integer, rational or irrational). A flow \(T\) has the property \(\Omega\) (to paraphrase the author) if there exists a real \(\omega\) for which \((T^\omega)^n\) has no fixed point on \(\mathcal S\) for any nonzero integer \(n\). In general, a flow is strictly disjoint if \((T^t)^n\) for any real \(t\) can have a fixed point for a positive integer \(n\) only if \(T^t\) is the identity on \(\mathcal S\). A homeomorphism \(h\) of \(\mathcal S\) composed with (each element of) the flow of \(T\), then composed with the inverse of \(h\), forms a flow conjugate to that of \(T\). Roughly speaking, the following is offered as main result. Every strictly disjoint flow with property \(\Omega\) on \(\mathcal S\) consists either of the identity alone or is conjugate to the flow of rotations of \(\mathcal S\) or is conjugate to a flow of \(P\) satisfying the functional equation \(\phi(P^t(z))=\phi(z)c(t)\) for all real \(t\) and all \(z\in{\mathcal S}\), where \(c\) satisfies \(c(s+t)=c(s)c(t)\) for all real \(s,t\).

Reviewer: Johann Aczél (Waterloo/Ontario)

MSC:

37E10	Dynamical systems involving maps of the circle
39B12	Iteration theory, iterative and composite equations
37C10	Dynamics induced by flows and semiflows

Keywords:

functional equations; flows; fixed points; homeomorphisms; conjugates