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**Views of parameter space: topographer and resident.**
*(English)*
Zbl 1054.37020

Astérisque 288. Paris: Société Mathématique de France (ISBN 2-85629-144-9/pbk). 418 p. (2003).

The aim of this book is to describe the topology of certain spaces of parameters \(V\), consisting in families of critically finite rational maps acting on the Riemann sphere \(\mathbb{C} P(1)\). We recall that a rational map is said critically finite if its critical points have finite forward orbits. Note that the dynamics of the maps of \(V\) is assumed to be constant on finite subset \(Z\) of \(\mathbb{C} P(1)\).

Actually, the author describes the topology of a larger space \(B\), consisting in critically finite branched coverings of \(\mathbb{C} P(1)\), with the same dynamics on \(Z\) as the elements of \(Y\). This topology is described by two theorems, called the topographer’s view and the resident’s view.

The topographer’s view is a geometrising theorem, close to the Thurston’s one on critically finite branched coverings of \(\mathbb{C} P(1)\) [On the combinatorics of iterated rational maps, Preprint, Princeton University, I.A.S. (1985), or A. Douady and J. H. Hubbard, Acta Math. 171, No. 2, 263–297 (1993; Zbl 0806.30027)].

Roughly speaking, the topographer’s view asserts that \(B\) is an increasing union of some topological spaces \(B_n\), where each \(B_n\) is homotopically equivalent to a finite ordered graph of topological spaces whose edges are tori. Moreover, the components of \(B_n\) correspond to some nodes of \(B_{n+1}\) (we let \(B_0 = V\)).

The resident’s view tells us how to pass from the graph \(B_n\) to the next graph \(B_{n+1}\). More precisely, it is a view of the fundamental group of the parameter space \(V\) : this group injects in the fundamental group of \(\mathbb{C} P(1)/\mathbb{Z}\). This injection induces a map from the universal cover of \(V\) into the universal cover of \(\mathbb{C} P(1)/\mathbb{Z}\), and this map gives informations on the variation of the dynamics on \(V\).

The proofs of the topographer’s view and the resident’s view follow the approach of Thurston for the preceding cited result. The idea is to iterate a pullback map on an appropriate Teichmüller space. The analytic properties of the Teichmüller distance are crucial.

Actually, the author describes the topology of a larger space \(B\), consisting in critically finite branched coverings of \(\mathbb{C} P(1)\), with the same dynamics on \(Z\) as the elements of \(Y\). This topology is described by two theorems, called the topographer’s view and the resident’s view.

The topographer’s view is a geometrising theorem, close to the Thurston’s one on critically finite branched coverings of \(\mathbb{C} P(1)\) [On the combinatorics of iterated rational maps, Preprint, Princeton University, I.A.S. (1985), or A. Douady and J. H. Hubbard, Acta Math. 171, No. 2, 263–297 (1993; Zbl 0806.30027)].

Roughly speaking, the topographer’s view asserts that \(B\) is an increasing union of some topological spaces \(B_n\), where each \(B_n\) is homotopically equivalent to a finite ordered graph of topological spaces whose edges are tori. Moreover, the components of \(B_n\) correspond to some nodes of \(B_{n+1}\) (we let \(B_0 = V\)).

The resident’s view tells us how to pass from the graph \(B_n\) to the next graph \(B_{n+1}\). More precisely, it is a view of the fundamental group of the parameter space \(V\) : this group injects in the fundamental group of \(\mathbb{C} P(1)/\mathbb{Z}\). This injection induces a map from the universal cover of \(V\) into the universal cover of \(\mathbb{C} P(1)/\mathbb{Z}\), and this map gives informations on the variation of the dynamics on \(V\).

The proofs of the topographer’s view and the resident’s view follow the approach of Thurston for the preceding cited result. The idea is to iterate a pullback map on an appropriate Teichmüller space. The analytic properties of the Teichmüller distance are crucial.

Reviewer: Christophe Dupont (Orsay)

### MSC:

37Fxx | Dynamical systems over complex numbers |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |

57N16 | Geometric structures on manifolds of high or arbitrary dimension |

30F60 | Teichmüller theory for Riemann surfaces |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |