A combinatorial invariant for escape time Sierpiński rational maps

Mónica Moreno Rocha ¹

¹ Centro de Investigación en Matemáticas, A.C. 36240 Guanajuato, México

Fundamenta Mathematicae, Tome 222 (2013) no. 2, pp. 99-130

Résumé

An escape time Sierpiński map is a rational map drawn from the McMullen family $z \mapsto z^n+\lambda /z^n$ with escaping critical orbits and Julia set homeomorphic to the Sierpiński curve continuum. We address the problem of characterizing postcritically finite escape time Sierpiński maps in a combinatorial way. To accomplish this, we define a combinatorial model given by a planar tree whose vertices come with a pair of combinatorial data that encodes the dynamics of critical orbits. We show that each escape time Sierpiński map realizes a subgraph of the combinatorial tree and the combinatorial information is a complete conjugacy invariant.

Détail

DOI : 10.4064/fm222-2-1

Keywords: escape time sierpi ski map rational map drawn mcmullen family mapsto lambda escaping critical orbits julia set homeomorphic sierpi ski curve continuum address problem characterizing postcritically finite escape time sierpi ski maps combinatorial accomplish define combinatorial model given planar tree whose vertices come pair combinatorial encodes dynamics critical orbits each escape time sierpi ski map realizes subgraph combinatorial tree combinatorial information complete conjugacy invariant