Counting rotational subsets of the circle $\mathbb{R}/\mathbb{Z}$ under the angle multiplying map $t\mapsto dt$

A rotational set is a finite subset $A$ of the unit circle $T = R / Z$ such that the angle-multiplying map $σ_{d} : t \mapsto d t$ maps $A$ onto itself by a cyclic permutation of its elements. Each rotational set has a geometric rotation number $p / q$ . These sets were introduced by Lisa Goldberg to study the dynamics of complex polynomial maps. In this paper we provide a necessary and sufficient condition for a set to be $σ_{d}$ -rotational with rotation number $p / q$ . As applications of our condition, we recover two classical results and enumerate $σ_{d}$ -rotational sets with rotation number $p / q$ that consist of a given number of orbits.

Counting rotational subsets of the circle $R / Z$ under the angle multiplying map $t \mapsto d t$

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Counting rotational subsets of the circle $R / Z$ under the angle multiplying map $t \mapsto d t$

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Counting rotational subsets of the circle R/ZR/Z under the angle multiplying map t↦dtt↦dt

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Counting rotational subsets of the circle R/ZR/Z under the angle multiplying map t↦dtt↦dt

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Counting rotational subsets of the circle $R / Z$ under the angle multiplying map $t \mapsto d t$

讲解博客

AI智能校验

Counting rotational subsets of the circle $R / Z$ under the angle multiplying map $t \mapsto d t$

讲解博客

AI智能校验