Xavier Buff, Adam L. Epstein, Sarah Koch, Daniel Meyer, Kevin Pilgrim, Mary Rees, Tan Lei (2012)
Annales de la faculté des sciences de Toulouse Mathématiques
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We survey known results about polynomial mating, and pose some open problems.
Carsten Lunde Petersen, Daniel Meyer (2012)
Annales de la faculté des sciences de Toulouse Mathématiques
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The different notions of matings of pairs of equal degree polynomials are introduced and are related to each other as well as known results on matings. The possible obstructions to matings are identified and related. Moreover the relations between the polynomials and their matings are discussed and proved. Finally holomorphic motion properties of slow-mating are proved.
Xavier Buff, Adam L. Epstein, Sarah Koch (2012)
Annales de la faculté des sciences de Toulouse Mathématiques
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One crucial tool for studying postcritically finite rational maps is Thurston’s topological characterization of rational maps. This theorem is proved by iterating a holomorphic endomorphism on a certain Teichmüller space. The graph of this endomorphism covers a correspondence on the level of moduli space. In favorable cases, this correspondence is the graph of a map, which can be used to study matings. We illustrate this by way of example: we study the mating of the basilica with itself. ...
Robert L. Devaney (2009)
Fundamenta Mathematicae
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We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of these rays crosses infinitely many other internal rays at many points. We then use this construction to show that there are infinitely many disjoint copies of the Mandelbrot set in the parameter planes for these maps.
Petracovici, Lia (2006)
Annales Academiae Scientiarum Fennicae. Mathematica
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Arnaud Chéritat (2012)
Annales de la faculté des sciences de Toulouse Mathématiques
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After giving an introduction to the procedure dubbed and quickly recalling known results about more classical notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree rational map. We give a conjectural...
John Hubbard (2012)
Annales de la faculté des sciences de Toulouse Mathématiques
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In the theory of rational maps an important role is played by . These are probably the best understood of all rational functions, but they are bizarre, and involve gluing dendrites together to get spheres carrying Peano curves. In the theory of Kleinian groups, there is a parallel construction, the construction of , that is central to Thurston’s hyperbolization theorem for 3-manifolds that fiber over the circle with pseudo-Anosov monodromy. It also involves gluing dendrites and Peano...