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Fixed points of polynomial maps. I: Rotation subsets of the circles. (English) Zbl 0771.30027

Let \(A=\{\theta_ i\), \(0\leq i\leq n-1\}\) be a subset of \(S=\mathbb{R}\pmod 1\) such that \(0\leq\theta_ 0<\dots<\theta_{n-1}<1\). If for the map \(f:\theta\to d\theta\pmod 1\) we have \(f(\theta_ i)=\theta_{i+m}\), where \(m\) is an integer and \((i+m)\) is taken \(\bmod n\), then \(A\) is said to be a degree-\(d\) \({m\over n}\) rotation set with combinatorial rotation number \({m\over n}\). It is not assumed that \((m,n)=1\). Such sets are useful in polynomial dynamics. Examples are provided by the set of external arguments at a repelling fixed point of a polynomial of degree \(d\), introduced by A. Douady and J. H. Hubbard, Etude dynamique des polynômes complexes, Publ. Math. Orsay, 84-02, 1-75 (1984; Zbl 0552.30018) and ibid. 85-04, 1-138 (1984; Zbl 0571.30026). Suppose that \(m=kp\), \(n=kq\), \((p,q)=1\). An analysis shows that \(A\) is completely determined by its rotation number \({p \over q}\) and its deployment sequence \(0\leq s_ 1\leq\dots\leq s_{n-1}\), \(n=pq\), where \(s_ i\) is the number of \(\theta_ i\) in \([0,i/(d-1))\). Conversely, a sufficient condition is given for the rotation number and deployment sequence to correspond to an \(A\). [For a review of part II see the following review].
Reviewer: I.N.Baker (London)

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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