Dynatomic curve and core entropy for iteration of polynomials

Abstract : When studying dynamical systems generated by a family of polynomials, it arises naturally cyclotomic type algebraic curves containing periodic or preperiodic points. In the periodic case of the family fc(z) = zd + c, the first chapter of this thesis shows that all these curves are smooth and irreducible, generalizing the known results to the case d = 2. In the preperiodic case of the same family, the second chapter of this thesis shows, against all expected that these curves are in general reducible. In addition, there contains a characterization of irreducible components and their analytical and geometrical relationship. The second theme of this thesis a new topic developed by W. Thurston, it is core entropy of polynomials. Thurston gave an algorithm, without proof, for compute these entropies. The thesis contains a rigorous proof of this algorithm and new methods to study the variation of these entropies from several views. The last topic of this thesis gives a necessary and sufficient condition for a kind of rational map having a C1-arc in its Julia set.
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Submitted on : Thursday, October 17, 2013 - 11:54:12 AM
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Yan Gao. Dynatomic curve and core entropy for iteration of polynomials. Dynamical Systems [math.DS]. Université d'Angers, 2013. English. ⟨tel-00874123⟩

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