Parabolic tools. (English) Zbl 1205.37057
External and internal rays in (super)attracting basins and corresponding puzzle pieces belong to the fundamental tools in polynomial dynamics. They have been used with great success by many researchers, including Douady and Hubbard and Yoccoz. The authors of this article introduce this terminology for rational functions with a certain type of parabolic dynamics. More precisely, for \(d>1\) and \(v_d:=\frac{d-1}{d+1}\) consider the Blaschke product \(P_d(z):= \frac{z^d + v_d}{1 + v_d z^d}\). The map \(P_d\) has the unit disk as a forward invariant parabolic basin of the double fixed point \(z=1\). In \(\mathbb{D}\), \(P_d\) has a single critical point at \(z=0\) of order \(d-1\). On the unit circle, \(P_d\) is conjugate to \(z\mapsto z^d\), and hence one can extend combinatorial concepts such as itineraries to the maps \(P_d\). This leads to a senseful concept of parabolic rays for these Blaschke products in analogy to rays for \(z\mapsto z^d\). Now, if \(f\) is a holomorphic map which has a fixed parabolic basin with a unique critical point of degree \(d-1\), then on this basin, the dynamics of \(f\) is conjugate to the dynamics of \(P_d\) on \(\mathbb{D}\). Hence one can carry over the introduced concepts to this more general setting.
For the special case of \(d=2\), Petersen and Roesch define parabolic puzzles using the idea of Yoccoz puzzle pieces for quadratic polynomials. While the introduction of rays is rather straightforward, the construction of parabolic puzzles requires a careful and extensive technical setup. As an application, the authors consider the set \(Per_1(1)\) of Möbius conjugacy classes of rational maps of degree \(2\) with parabolic fixed point of multiplier \(1\). They prove that the Julia set of any non-renormalizable map in the connectedness-locus of \(Per_1(1)\) is locally connected.
For the special case of \(d=2\), Petersen and Roesch define parabolic puzzles using the idea of Yoccoz puzzle pieces for quadratic polynomials. While the introduction of rays is rather straightforward, the construction of parabolic puzzles requires a careful and extensive technical setup. As an application, the authors consider the set \(Per_1(1)\) of Möbius conjugacy classes of rational maps of degree \(2\) with parabolic fixed point of multiplier \(1\). They prove that the Julia set of any non-renormalizable map in the connectedness-locus of \(Per_1(1)\) is locally connected.
Reviewer: Helena Mihaljevic-Brandt (Kiel)
MSC:
37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
37F20 | Combinatorics and topology in relation with holomorphic dynamical systems |
37F45 | Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) |
30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
30D40 | Cluster sets, prime ends, boundary behavior |
32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
Keywords:
parabolic rays; parabolic Yoccoz puzzle; quadratic rational maps; parabolic Mandelbrot set; Yoccoz’ combinatorial analytic invariant; Blaschke product; locally-connected Julia set; non-renormalizableReferences:
[1] | Douady A., Etude dynamique des polynômes complexes, I (1984) · Zbl 0552.30018 |
[2] | A. Douady and J.H. Hubbard, Etude dynamique des polynômes complexes, II, publications mathématiques d’Orsay, 85-01 (1985) |
[3] | Kiwi I., Contemp Math, 269, in: Laminations and Foliations in Dynamics, Geometry and Topology pp 111– (1998) |
[4] | Lehto O., Quasiconformal Mappings in the Plane (1973) · Zbl 0267.30016 |
[5] | Milnor J., ’Géométrie Complexe Et Systèmes Dynamiques’ pp 277– (2000) |
[6] | Petersen C.L., Ergod. Th. Dynam. Syst. 13 pp 785– (1993) |
[7] | Petersen C.L., Fields Am. Math. Soc. 53, in: The Yoccoz Combinatorial Analytic Invariant (2008) · Zbl 1171.37024 |
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