By Régis petit, Michel Lambert

ISBN-10: 2702705731

ISBN-13: 9782702705735

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The theory of iteration of rational maps combines hyperbolic geometry and complex analysis, as the classical Teichmüller theory does. The tools that are used involve quasiconformal maps, the Teichmüller metric and quadratic differentials. f / of f , which is a space of rational maps equipped with a marking which is a quasiconformal conjugacy with f . The theory of Teichmüller spaces associated to holomorphic dynamical systems was developed by McMullen and Sullivan. A new ingredient in the theory, compared to the classical theory of surface homeomorphism actions on Teichmüller spaces, is the use of the action of the branched cover on the set of homotopy classes of essential multi-curves.

First of all, while in the case of surfaces of finite type there is one commonly used definition of the mapping class group, a definition which is purely topological (with some variations, regarding the actions on the boundary components, or the fact that the mapping classes considered preserve the punctures pointwise, and so on), in the case of surfaces of infinite type there are many possibilities. 8 The mapping class group of R, which is also called in Chapter 15 the Teichmüller modular group (because it depends on the choice of the Teichmüller space), is defined as the space of homotopy classes of quasiconformal homeomorphisms of R.

The author considers families of flat affine two-tori obtained as quotients of quadrilaterals glued along their boundaries by affine maps. He studies the dependence of such flat affine tori on the shape of the quadrilaterals and on the annuli that are used to define them. He shows that the flat affine tori that are obtained by gluing affine quadrilaterals along their sides are all homogeneous (that is, their groups of affine automorphisms act transitively on these spaces). He then Introduction to Teichmüller theory, old and new, IV 23 describes in detail a construction of affine tori with developing image in A2 f0g.