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On the structure of some flows on the unit circle. (English) Zbl 0891.39017
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 $$s,t$$.

##### 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
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