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Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle

Arnaud Chéritat (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

After giving an introduction to the procedure dubbed slow polynomial mating and quickly recalling known results about more classical notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree $3$ 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 $6$ rational map. We...

The $1 / 2$ -complex Bruno function and the Yoccoz function: a numerical study of the Marmi-Moussa-Yoccoz conjecture.

Carletti, Timoteo (2003)

Experimental Mathematics

The $3 n + 1$ -problem and holomorphic dynamics.

Letherman, Simon, Schleicher, Dierk, Wood, Reg (1999)

Experimental Mathematics

The arborification-coarborification transform : analytic, combinatorial, and algebraic aspects

Jean Ecalle, Bruno Vallet (2004)

Annales de la Faculté des sciences de Toulouse : Mathématiques

The Fatou and Julia sets of multivalued analytical functions.

Gumenyuk, P.A. (2002)

Sibirskij Matematicheskij Zhurnal

The generalized Julia set perturbed by composing additive and multiplicative noises.

Wang, Xingyuan, Jia, Ruihong, Sun, Yuanyuan (2009)

Discrete Dynamics in Nature and Society

The geometry of some natural conjugacies in $ℂ^{n}$ dynamics.

Robertson, John W. (2004)

International Journal of Mathematics and Mathematical Sciences

Topology and measure of buried points in Julia sets

Clinton P. Curry, John C. Mayer, E. D. Tymchatyn (2013)

Fundamenta Mathematicae

It is well-known that the set of buried points of a Julia set of a rational function (also called the residual Julia set) is topologically “fat” in the sense that it is a dense $G_{δ}$ if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points. Finally,...

Trees and the dynamics of polynomials

Laura G. DeMarco, Curtis T. McMullen (2008)

Annales scientifiques de l'École Normale Supérieure

In this paper we study branched coverings of metrized, simplicial trees $F : T \to T$ which arise from polynomial maps $f : ℂ \to ℂ$ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space $ℙ T_{D}$ compactifying the moduli space of polynomials of degree $D$ ; that $F$ records the asymptotic behavior of the multipliers of $f$ ; and that any meromorphic family of polynomials over $Δ^{*}$ can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial...

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