Symbolic dynamics for angle-doubling on the circle III. Sturmian sequences and the quadratic map

Abstract

AbstractBy the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling maphon the circleT. In particular, a connected and locally connected Julia set can be considered as a topological factorT/ ≈ ofTwith respect to a specialh-invariant equivalence relation ≈ onT, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α →αfromTonto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point α, they gave a topological classification of the abstract Julia setsT/α. It turns out thatT/αcontains simple closed curves iff the point α has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.