TY - JOUR

T1 - Symbolic dynamics for angle-doubling on the circle III. Sturmian sequences and the quadratic map

AU - Keller, Karsten

PY - 1994/1/1

Y1 - 1994/1/1

N2 - By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling map h on the circle T. In particular, a connected and locally connected Julia set can be considered as a topological factor T/ ≈ of T with respect to a special h -invariant equivalence relation ≈ on T, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α → α from T onto 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 sets T/ α. It turns out that T/ α contains 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.

AB - By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling map h on the circle T. In particular, a connected and locally connected Julia set can be considered as a topological factor T/ ≈ of T with respect to a special h -invariant equivalence relation ≈ on T, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α → α from T onto 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 sets T/ α. It turns out that T/ α contains 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.

UR - http://www.scopus.com/inward/record.url?scp=0038849210&partnerID=8YFLogxK

U2 - 10.1017/S0143385700008154

DO - 10.1017/S0143385700008154

M3 - Journal articles

AN - SCOPUS:0038849210

SN - 0143-3857

VL - 14

SP - 787

EP - 805

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 4

ER -