Introduction. Mappings from an interval to itself provide the simplest possible examples of smooth dynamical systems. Such mappings have been widely studied in recent years since they occur in quite varied applications, and since their surprisingly complex behavior may provide a useful introduction to the study of higher dimensional situations. The present paper sets up an effective calculus for describing the qualitative behavior of the successive iterates of a piecewise monotone mapping. Let I be a closed interval of real numbers. By definition, a continuous mapping f from I to itself is p!ecewise monotone if I can be subdivided into finitely many subintervals I I, .... I Z on which f is alternately strictly increasing or strictly decreasing. Each such maximal interval on which f is monotone is called a lap of f, and £ = Z(f) points c I ..... cz_ I at which f called the turninq points of f. is the lap number. The separating has a local minimum or maximum are

The Real Analytic Theory of Teichmüller Space

On iterated maps of the interval

Let Fg 1, g> 1, be the mapping class group consisting of all isotopy classes of base-point and orientation preserving homeomorphisms of a closed, oriented surface F of genus g. Let )~(~1) be its Euler characteristic in the sense of Wall, that is Z(F~I)= [Fgl: F] l z(E/F), where F is any torsion free subgroup of finite index in F~ 1 and E is a contractible space on which F acts freely and properly discontinuously. An example of such a space is the Teichmiiller space ~-~1, and g(F~ ~) can be interpreted as the orbifold Euler characteristic of ~-~'/F~I =JOin, the moduli space of curves of genus g with base point. The purpose of this paper is to prove the following formula for )~(F~I):

background

http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01390325/fulltext.pdf

The Euler characteristic of the moduli space of curves

A principal ℝ
 +
 5
 -bundle over the usual Teichmuller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several coordinatizations of the total space of the bundle are developed. There is furthermore a natural cell-decomposition of the bundle. Finally, we compute the coordinate action of the mapping class group on the total space; the total space is found to have a rich (equivariant) geometric structure. We sketch some connections with arithmetic groups, diophantine approximations, and certain problems in plane euclidean geometry. Furthermore, these investigations lead to an explicit scheme of integration over the moduli spaces, and to the construction of a “universal Teichmuller space,” which we hope will provide a formalism for understanding some connections between the Teichmuller theory, the KP hierarchy and the Virasoro algebra. These latter applications are pursued elsewhere.

/pdf/the-decorated-teichmuller-space-of-punctured-surfaces-4na6yunrqs.pdf

The decorated Teichmüller space of punctured surfaces

Dichotomie dynamique de Fatou et Julia. Points periodiques. Consequences du theoreme de Montel. L'ensemble de Julia est la fermeture de l'ensemble des points periodiques repulseurs. Resultats classiques sur l'ensemble de Fatou. Classification de Sullivan de l'ensemble de Fatou. Une condition pour le developpement sur l'ensemble de Julia. La dynamique des polynomes. L'ensemble de Mandelbrot et le travail de Douady et Hubbard. Le theoreme d'application de Riemann mesurable et la dynamique analytique

/pdf/complex-analytic-dynamics-on-the-riemann-sphere-17he9zht2s.pdf

The Real Analytic Theory of Teichmüller Space

Citations

On iterated maps of the interval

The Euler characteristic of the moduli space of curves

The decorated Teichmüller space of punctured surfaces

Complex analytic dynamics on the Riemann sphere

Theory of Strings with Boundaries: Fluctuations, Topology, and Quantum Geometry

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