Annales scientifiques de l'É.N.S. 4 e série, tome 18, n o 2 (1985), p. 287-343 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1985, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

The mapping f will be called critically finite if Pf is a finite set. We will give in the appendix some examples of critically finite branched mappings, which bring out some of the difficulties in the proof of Thurston's Theorem.We will assume throughout this paper that f is a critically finite branched mapping, of degree d>l, and we set p=#Pf.Remark. The critical set of f'~ is usually larger than f~f for n>l. This is not true of Pf: we have Pf = P$~ for any n >~ 1.Clearly there exists a smallest function uf among functions u: S2-+N*U{oo} such that (1) u(x)--1 when z~Pf, and (2) u(x) is a multiple of u(y)degy f for each yef-l(x). We will say that the orbifold Of=(S 2, uf) of f is hyperbolic if its Euler characteristic satisfies x(Of) <0.Remark. We will see in Section 9 that x(Of)<~O for any critically finite branched mapping. Such orbifolds are usually hyperbolic: for instance, if p>/5, Of will clearly be hyperbolic. We will completely classify branched mappings with non-hyperbolic orbifold orbifold in Section 9.The theory of orbifolds is covered in IT1] and [T2]: we will not require any of this theory until Section 9. There is a natural definition of the universal covering space of an orbifold, and with this definition Of is hyperbolic if for any complex structure on Of (i.e., on $2), the universal covering space (gf is isomorphic to the disc. Two branched mappings f, g: $2-~ S 2 are equivalent iff there exist homeomorphisms 0, 0': (S 2, Pf)--+(S 2, Pg) such that the diagram

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A self-consistent, kinetic theory of ion–solvent interactions is developed within the framework of continuum mechanics. It is shown that the hydrodynamic coupling between viscous momentum transport and dielectric relaxation leads not only to a theory of ion mobility but also to a description of the dielectric properties of electrolyte solutions. The concept of kinetic polarization deficiency is introduced, whereby the static permittivity of a solution is reduced from that of the pure solvent by an amount proportional to the product of solvent dielectric relaxation time and low frequency conductivity of the solution. Furthermore, if the viscous and dielectric relaxation times are assumed to be comparable it is demonstrated that ’’deformation inertia’’ should make a significant contribution to the decrement Δε0. Ion mobility is calculated to first order in a coupling parameter which is inversely proportional to the fourth power of the ion radius, the limiting case of zero ion size is analyzed, and general aspects of ion migration are investigated with the aid of the principle of minimum dissipation. Given that the pure solvent has a dispersion characterized by a single Debye relaxation time τD, it is asserted that the solution will, as a consequence of dielectric friction, possess an infinite number of relaxation times extending from τD down to the longitudinal time τL=τD(ε∞/ε0).

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Curt McMullen proved that those Julia sets to which the tableau argument applies (these are precisely those which are Cantor sets) are of measure zero, and has kindly agreed to our including his result as Theorem 5.9 of our paper.Next we take up the problem of showing which patterns actually occur. The result is very satisfying: Theorems 8.2 and 9.1 say that every pattern does occur, and tells exactly how two polynomials are related if they have the same pattern.We could have proved the results of Chapter 5 without ever mentioning patterns, simply by working in the dynamical plane itself. But in Chapters 7, 8 and 9 we see what the drudgery in Chapters 2 and 3 has bought us: a complete abstract description of the escape locus: the space of cubic polynomials for which both critical points escape. In Chapter 7 we set up a parameter space for patterns which is itself a complex manifold, called the parapattern space. It is universal in the sense that it parametrizes patterns,

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