Peter Makienko

On decomposable rational maps.

AbstractIfRisarationalmap,theMainResultisauniformizationTheoremforthespaceofdecompositionsoftheiteratesofR. Secondly,weshowthatFatouconjectureholdsfordecomposablerationalmaps. 1 Introduction This paper gives a dynamical approach to the algebraic problem of decom-position of rational maps. That is to describe the set of decompositions ofa rational map R, along with the decompositions of all its iterates R n . Wewant to link geometric structures with the decomposition of rational maps.To this end, we construct a space which describes the space of decompositionof the cyclic semigroup generated by R.We found that the fact that a map is decomposable impose dynamicalconsequences. In particular, we show using elementary arguments that theFatou conjecture is true for decomposable rational maps.We would like to thank M. Zieve for useful comments and discussions andfor kindly providing his example of a prime rational map which is virtuallyindecomposable. 0 ThisworkwaspartiallysupportedbyPAPIITprojectIN100409.

On dynamical Teichmüller spaces

Following ideas from a preprint of the second author, see (2), we investigate relations of dynamical Teichmuller spaces with dynamical objects. We also establish some connections with the theory of deformations of inverse limits and laminations in holomorphic dynamics, see (1).

Non-persistently recurrent points, qc-surgery and instability of rational maps with totally disconnected Julia sets

Let R be a rational map with a totally disconnected Julia set J(R). If the postcritical set on J(R) contains a non-persistently recurrent (or conical) point, then we show that the map R cannot be a structurally stable map. Introduction and statements Fatou’s problem of the density of hyperbolic maps in the space of rational maps is one of the principal problems in the field of holomorphic dynamics. Due to Mane, Sad and Sullivan [MSS], we can reformulate this problem in the following way: If the Julia set J(R) contains a critical point, then the rational map R is a structurally unstable map. For convenience we give the definition of the structural stability of a rational map. For other basic notations and definitions we refer to the book of Milnor [M]. Definition 1. Let Ratd be the space of all rational maps of degree d with the topology of coefficient convergence. A map R ∈ Ratd is called structurally stable if there exists a neighborhood U ⊂ Ratd of R such that: For any map R1 ∈ U there exists a quasiconformal map f : C → C conjugating R to R1. We give a condition, Assumptions “G” (see below), on a rational map with totally disconnected Julia set and with a critical point on J(R) to be unstable. In the pioneer paper [BH], Branner and Hubbard prove that the Lebesgue measure of the Julia set is zero if there exists only one critical point on J(R). Our result (see Theorem A below) restricted to the Branner–Hubbard case is weaker, but it can be applied for maps with two or more critical points on J(R). Let R be a rational map with a totally disconnected Julia set. Let us normalize R so that the point z = ∞ becomes the attractive fixed point. Let Pc(R) be a postcritical set of the map R and P (R) = Pc(R) ∩ J(R) be a postcritical set on the Julia set. Let S = C\ ⋃ n R −n(Pc(R)), then R : S → S is an unbranched autocovering. Definition 2. We say that a closed simple geodesic γ ⊂ S is linked with P (R) if the interior I(γ) of γ intersects P (R). Received by the editors June 13, 2005. 2000 Mathematics Subject Classification. Primary 37F45; Secondary 37F30. c ©2006 American Mathematical Society Reverts to public domain 28 years from publication

Semigroups of Mappings and Correspondences: Characters and Representations in Holomorphic Dynamical Systems

We use the theory of representation of semigroups to get algebraic characterizations of conjugacy of semigroups of endomorphisms. This text is a short version of the chapter of a monograph in preparation. Several results are already available on the paper Semigroup representations in holomorphic dynamics published in (Cabrera et al. Discrete Contin. Dyn. Syst. 33 (2013), no. 4, 1333–1349).Complimentary part is On Decomposable Rational Maps which is published in (Cabrera and Makienko, Conform Geom Dyn 15:21–218, 2011) and was prepared after these expository notes.