The field of complex dynamics in higher dimension was initiated in the 1920’s by Fa-tou. It was motivated by studies in Newton’s method, celestial mechanics and functional equations. Recently, new methods from pluripotential theory have been introduced to the subject. These techniques have produced many new interesting results. We give an introduction to this subject and a summary of the most relevant developments.

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Complex dynamics in higher dimensions

The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and 'elementary' introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices - on analysis, R-trees, and Berkovich's general theory of analytic spaces - are included to make the book as self-contained as possible. The authors first give a detailed description of the topological structure of the Berkovich projective line and then introduce the Hsia kernel, the fundamental kernel for potential theory. Using the theory of metrized graphs, they define a Laplacian operator on the Berkovich line and construct theories of capacities, harmonic and subharmonic functions, and Green's functions, all of which are strikingly similar to their classical complex counterparts. After developing a theory of multiplicities for rational functions, they give applications to non-Archimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical Fatou-Julia theory of rational iteration. They illustrate the theory with concrete examples and exposit Rivera-Letelier's results concerning rational dynamics over the field of p-adic complex numbers. They also establish Berkovich space versions of arithmetic results such as the Fekete-Szego theorem and Bilu's equidistribution theorem.

Potential Theory and Dynamics on the Berkovich Projective Line

Introduction Basic examples and definitions 1. Measure preserving endomorphisms 2. Compact metric spaces 3. Distance expanding maps 4. Thermodynamical formalism 5. Expanding repellers in manifolds and in the Riemann sphere, preliminaries 6. Cantor repellers in the line, Sullivan's scaling function, application in Feigenbaum universality 7. Fractal dimensions 8. Conformal expanding repellers 9. Sullivan's classification of conformal expanding repellers 10. Holomorphic maps with invariant probability measures of positive Lyapunov exponent 11. Conformal measures References Index.

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Conformal Fractals: Ergodic Theory Methods

The simplest holomorphic dynamical systems which display interesting behavior are the polynomial maps of C The dynamical study of these maps began with Fatou and Julia in the s and is currently a very active area of research If we are interested in studying invertible holomorphic dynamical systems then the simplest examples with interesting behavior are probably the polynomial di eomorphisms ofC These are maps f C C such that the coordinate functions of f and f are holomorphic polynomials For polynomial maps of C the algebraic degree of the polynomial is a useful dynamical invariant In particular the only dynamically interesting maps are those with degree d greater than one For polynomial di eomorphisms we can de ne the algebraic degree to be the maximum of the degrees of the coordinate functions This is not however a conjugacy invariant Friedland and Milnor FM gave an alternative de nition of a positive integer deg f which is more natural from a dynamical point of view If deg f then deg f coincides with the minimal algebraic degree of a di eomorphism in the conjugacy class of f As in the case of polynomial maps of C the polynomial di eomorphisms f with deg f are rather uninteresting We will make the standing assumption that deg f For a polynomial map of C the point at in nity is an attractor Thus the recurrent dynamics can take place only on the set K consisting of bounded orbits A normal families argument shows that there is no expansion on the interior of K so chaotic dynamics can occur only on J K This set is called the Julia set and plays a major role in the study of polynomial maps For di eomorphisms of C each of the objectsK and J has three analogs Correspond ing to the set K in one dimension we have the sets K resp K consisting of the points whose orbits are bounded in forward resp backward time and the set K K K consisting of points with bounded total orbits Each of these sets is invariant and K is compact As is in the one dimensional case recurrence can occur only on the set K Corresponding to the set J in dimension one we have the sets J K and the set J J J Each of these sets is invariant and J is compact A normal families argu ment shows that there is no forward instability in the interior of K and no backward instability in the interior of K Thus chaotic dynamics that is recurrent dynamics with instability in both forward and backward time can occur only on the set J The techniques that Fatou and Julia used in one dimension are based on Montel s theory of normal families and do not readily generalize to higher dimensions A di erent tool appears in the work of Brolin Br who made use of the theory of the logarithmic

Polynomial diffeomorphisms of C2. IV: The measure of maximal entropy and laminar currents.

Polynomial diffeomorphisms of C2. III: Ergodicity, exponents and entropy of the equilibrium measure

In this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.

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M. Ju. Ljubich

Papers

Entropy properties of rational endomorphisms of the Riemann sphere