## 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

### Keywords:

rotation set; repelling fixed point; rotation number

### Citations:

Zbl 0552.30018; Zbl 0571.30026
Full Text: