pilgrim. combinations of complex dynamical systems

[...]... combination procedures, e.g mating Here is a summary of the remainder of this Introduction §1.5 is a survey of known results regarding the combinatorics of complex dynamical systems I have tried to give as complete a bibliography as possible, as much of this material is unpublished and/or scattered Often, references are merely listed without further discussion of their contents I apologize for any omissions... is homotopic in S 2 − PF to an element of Γ By lifting homotopies, it is easily seen that this property depends only on the set [Γ ] of homotopy classes of elements of Γ in S 2 −PF We shall actually require a slightly stronger version of this definition, given in §1.8.3 Thurston linear map Let RΓ be the vector space of formal real linear combinations of elements of Γ Associated to an F -invariant multicurve... general theory of combinations and decompositions The main goal of this work is to provide a solution to Problem (5) above We shall give: • a combination procedure (Theorem 3.2), taking as input a list of data consisting of seven objects satisyfing fourteen axioms, and producing as output a well-defined branched mapping F of the sphere to itself; 1.4 Summary of this work 13 • an analysis of how the combinatorial... the process of microimplantation The inverse of tuning became known as renormalization and explains the presence of small copies of the Mandelbrot set inside itself Among rational maps, Douady and Hubbard noticed from computer experiments that a different combination procedure, now called mating , explained the dynamical structure of certain quadratic rational functions in terms of a pair of critically... aspects of the “dictionary” between rational maps and Kleinian groups as dynamical systems In particular, we propose to view the Canonical Decomposition Theorem as an analog of the JSJ decomposition of a closed irreducible three-manifold §1.7 discusses connections between the analysis of postcritically finite rational maps and other, non -dynamical topics (e.g geometric Galois theory; groups of intermediate... a ”coding map” of a one-sided shift on d symbols, and the equivalence relation determining the fibers is also a subshift of finite type (see [Fri] and also the chapter on semi-Markovian spaces in [CP]) In [Kam3], [Kam5] coding maps and the structure of the set of coding maps are investigated 1.5 Survey of previous results 17 1.5.2 Combinations and decompositions No general theory of combinations and... attracting cycle under iteration Equivalently: f is expanding on a neighborhood of its Julia set with respect to the Poincar´ metric on the complement of the postcritical set The condie tion of being hyperbolic is an open condition in the complex manifold Ratd of rational maps of a given degree; a connected component of the set of hyperbolic maps is called a hyperbolic component in parameter space It is... success and failure of combination theorems and the structure of parameter spaces, especially compactness properties of and tangencies between hyperbolic components Rees [Ree2] gives compactness results for certain real one-parameter subsets of hyperbolic components, independent of the fine combinatorial details of the map Petersen [Pet] relates the failure of mating to noncompactness of hyperbolic components... deformation space of N has compact closure in the space of all hyperbolic structures on M 5 The limit set of the fundamental group of N , regarded as a Kleinian group, is a Sierpinski carpet Alternatively, one might replace condition (3) with the following: the limit of any deformation of N corresponding to pinching a finite set of disjoint simple closed curves exists The equivalence of (1) and (2) follows... → S 2 The set of critical values of FA is the image on the sphere of the set of points of order at most two on the torus Since the endomorphism on the torus must preserve this set of four points, FA is postcritically finite 30 If e.g A = then FA is expanding with respect to the orbifold metric 02 inherited from the Euclidean metric on the torus Let γ be the curve which is the image of the line x = . Kong London Milan Paris Tokyo KevinM .Pilgrim Combinations of Complex Dynamical Systems 13 Author KevinM .Pilgrim Department of Mathematics Indiana University Bloomington, IN 47401, USA e-mail: pilgrim@ indiana.edu Cataloging-in-Publication. 79 6.1 Statement of Uniqueness of Decompositions Theorem 79 6.2 Proof of Uniqueness of Decomposition Theorem 79 7 Counting classes of annulus maps 83 7.1 Statement of Number of Classes of Annulus Maps. Uniqueness of combinations 59 4.1 Structure data and amalgamating data. 59 4.2 Combinatorial equivalence of sphere and annulus maps 60 4.3 Statement of Uniqueness of Combinations Theorem 61 4.4 Proof of