Non communiqué / Rauzy-type dynamics are group (or monoid) actions on a collection of combinatorial objects. The first and best known example concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects inducedby the group action are related to components of the moduli spaces of Abelian differentials with prescribed singularities, and, in two variants of the problem, have been classified by Kontsevich and Zorich, and by Boissy, through methods involving both combinatorics, algebraic geometry, topology and dynamical systems. In the first half of this thesis, we provide a purely combinatorial proof of both classification theorems. Our proof can be interpreted geometrically and the over archingidea is close to that of Kontsevich and Zorich, although the techniques arerather different. Not all Rauzy-type dynamics have a geometrical correspondence however, and some parts of this first proof do not seem to generalize well.In the second half of the thesis we develop a new method, that we call the labelling method. This second method is not completely disjoint from the first one, but it the new crucial ingredient of considering a sort of ‘monodromy’ for the dynamics, in away that we now sketch. Many statements in this thesis are proven by induction. It is conceivable to prove, by induction, a classification theorem for unlabelled objects. However, as the labelling method will show, it is easier to prove two statements in parallel within the same induction, the one on the unlabelled objects, and an apparently much harder one, on the monodromy of the labelled objects. Although the final result is stronger than the initial aim, by virtue of the stronger inductive hypothesis, the method may work more easily.This second approach extends to several other Rauzy-type dynamics. Our firststep is to apply the labelling method to derive a second proof of the classificationtheorem for the Rauzy dynamics. Then we apply it to the study of two other Rauzy-type dynamics (one of which is strictly related to the Rauzy dynamics on non-orientable surfaces), and finally we inventory a surprisingly high number of Rauzy-type dynamics for which the labelling.

本論文介紹Teichmuller 空間的參數化方法。我們會就此題目會作一歷史回顧，然後介紹Fenchel-Nielsen的參數化方法，最後集中討論在緊密且有界之曲面上的Teichmuller 空間以六角形分割之參數化方法。 / This thesis is an exposition of different parametrizations of the Teichmuller space. We will give a historical review on this subject, and in particular introduce the Fenchel-Nielsen coordinate. Our main focus would be the cellular decomposition method to parametrize the Teichmuller spaces of compact surface with boundary. / Detailed summary in vernacular field only. / Wong, Yun Shun Matthias. / "October 2012." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 42-43). / Abstracts also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Teichmüller Space --- p.4 / Chapter 2.1 --- Definition of Teichmüller Space --- p.4 / Chapter 2.2 --- Historical review --- p.6 / Chapter 3 --- Fenchel-Nielsen Coordinate --- p.10 / Chapter 4 --- Cellular Decomposition Method --- p.18 / Chapter 4.1 --- Ideal Triangulation --- p.19 / Chapter 4.2 --- Edge Invariant --- p.22 / Chapter 4.2.1 --- E-Invariant of Colored Hexagon --- p.23 / Chapter 4.3 --- E-coordinate and its Generalization --- p.24 / Chapter 4.4 --- Edge Path and Edge Cycle --- p.26 / Chapter 4.5 --- Parametrization Theorems --- p.26 / Chapter 4.6 --- Discussion on the Results --- p.30 / Chapter 4.7 --- The Variational Proof of Theorem 4.8 --- p.32 / Chapter 4.7.1 --- Overview of the Proof --- p.32 / Chapter 4.7.2 --- The Energy Function on Hexagon --- p.35 / Chapter 4.7.3 --- The Energy Function on Length Structure and the Proof of Theorem 4.8 --- p.37 / Bibliography --- p.42

Given a surface of genus g with fundamental group π, a representation of π into PSL(2,R) is a homomorphism that assigns to each generator of π an element of P SL(2, R). The group P SL(2, R) acts on Hom(π, P SL(2, R)) by conjugation. Define therepresentationspaceRg tobethequotientspaceHom(π,PSL(2,R))\PSL(2,R). Associated to each representation ρ is a number e(ρ) called its Euler class. Goldman showed that the space Rg has components that can be indexed by Euler classes of rep- resentations, and that there is one component for each integer e satisfying |e| ≤ 2g−2. The two maximal components correspond to Teichmu ̈ller space, the space of isotopy classes of hyperbolic structures on a surface. Teichmu ̈ller space is known to be homeomorphic to a ball of dimension 6g − 6. The other components of Rg are not as well understood. The theory of harmonic maps between non-positively curved manifolds has been used to study Teichmu ̈ller space. Given a harmonic map between hyperbolic surfaces, there is an associated quadratic differential on the domain surface called the Hopf differential. Wolf, following Sampson, proved that via the Hopf differential, harmonic maps parametrize Teichmu ̈ller space. This thesis extends his work to the case of branched hyperbolic structures, which correspond to certain elements in non- maximal components of representation space. More precisely, a branched hyperbolic structure is a pair (M, σ|dz|2) where M is a compact surface of genus g and σ|dz|2 is a hyperbolic metric with integral order cone singularities at a finite number of points expressed in terms of a conformal parameter. Fix a base surface (M, σ|dz|2). For each target surface (M, ρ|dw|2) with the same number and orders of cone points as (M,σ|dz|2), there is a unique harmonic map w : (M,σ|dz|2) → (M,ρ|dw|2) homotopic to the identity that fixes the cone points of M pointwise. Thus we may define another map from the space of branched hyperbolic structures with the same number and orders of cone points to the space of meromorphic quadratic differentials on the base surface M. This map, Φ, takes the harmonic map w associated with a metric ρ|dw|2 to the Hopf differential of w. This thesis shows that the map Φ is injective.

Lam, Ka Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 64-68). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Chan, Kin Wai. / "September 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 103-106). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.8 / Chapter 2 --- Background Knowledge --- p.13 / Chapter 2.1 --- Results from Riemann Surface Theory and Quasicon- formal Mappings --- p.13 / Chapter 2.1.1 --- Riemann Surfaces and the Uniformization The- orem --- p.13 / Chapter 2.1.2 --- Fuchsian Groups --- p.15 / Chapter 2.1.3 --- Quasiconformal Mappings and the Beltrami Equation --- p.17 / Chapter 2.1.4 --- Holomorphic Quadratic Differentials --- p.20 / Chapter 2.1.5 --- Nodal Riemann Surfaces --- p.21 / Chapter 2.2 --- Teichmuller Theory --- p.24 / Chapter 2.2.1 --- Teichmiiller Spaces --- p.24 / Chapter 2.2.2 --- Teichmuller's Distance --- p.26 / Chapter 2.2.3 --- The Bers Embedding --- p.26 / Chapter 2.2.4 --- Teichmuller Modular Groups and Moduli Spaces of Riemann Surfaces --- p.27 / Chapter 2.2.5 --- Infinitesimal Theory of Teichmiiller Spaces --- p.28 / Chapter 2.2.6 --- Boundary of Moduli Spaces of Riemann Sur- faces --- p.29 / Chapter 2.3 --- Schwarz-Yau Lemma --- p.30 / Chapter 3 --- Classical Canonical Metrics on the Teichnmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.31 / Chapter 3.1 --- Finsler Metrics and Bergman Metric --- p.31 / Chapter 3.1.1 --- Definitions and Properties of the Metrics --- p.32 / Chapter 3.1.2 --- Equivalences of the Metrics --- p.33 / Chapter 3.2 --- Weil-Petersson Metric --- p.36 / Chapter 3.2.1 --- Definition and Properties of the Weil-Petersson Metric --- p.36 / Chapter 3.2.2 --- Results about Harmonic Lifts --- p.37 / Chapter 3.2.3 --- Curvature Formula for the Weil-Petersson Met- ric --- p.41 / Chapter 4 --- Kahler Metrics on the Teichmiiller Spaces and the Moduli Spaces of Riemann Surfaces --- p.42 / Chapter 4.1 --- McMullen Metric --- p.42 / Chapter 4.1.1 --- Definition of the McMullen Metric --- p.42 / Chapter 4.1.2 --- Properties of the McMullen Metric --- p.43 / Chapter 4.1.3 --- Equivalence of the McMullen Metric and the Teichmuller Metric --- p.45 / Chapter 4.2 --- Kahler-Einstein Metric --- p.50 / Chapter 4.2.1 --- Existence of the Kahler-Einstein Metric --- p.50 / Chapter 4.2.2 --- A Conjecture of Yau --- p.50 / Chapter 4.3 --- Ricci Metric --- p.51 / Chapter 4.3.1 --- Definition of the Ricci Metric --- p.51 / Chapter 4.3.2 --- Curvature Formula of the Ricci Metric --- p.53 / Chapter 4.4 --- The Asymptotic Behavior of the Ricci Metric --- p.61 / Chapter 4.4.1 --- Estimates on the Asymptotics of the Ricci Metric --- p.61 / Chapter 4.4.2 --- Estimates on the Curvature of the Ricci Metric --- p.83 / Chapter 4.5 --- Perturbed Ricci Metric --- p.92 / Chapter 4.5.1 --- Definition and the Curvature Formula of the Perturbed Ricci Metric --- p.92 / Chapter 4.5.2 --- Estimates on the Curvature of the Perturbed Ricci Metric --- p.93 / Chapter 4.5.3 --- Equivalence of the Perturbed Ricci Metric and the Ricci Metric --- p.96 / Chapter 5 --- Equivalence of the Kahler Metrics on the Teichmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.98 / Chapter 5.1 --- Equivalence of the Ricci Metric and the Kahler-Einstein Metric --- p.98 / Chapter 5.2 --- Equivalence of the Ricci Metric and the McMullen Metric --- p.99 / Bibliography --- p.103

Ce travail est consacré à la recherche de surfaces de Riemann (\it compactes) extrê\-mes (i.e. maxima locaux) pour la systole, ou tout au moins parfaites. En genre 4, on donne une nouvelle surface extrême et deux surfaces parfaites non extrêmes (ce sont les premiers exemples de telles surfaces en genre $\leq 10$). La méthode consiste à réaliser géométriquement les groupes d'automorphismes à 4 points de branchements. En effet, le lieu des points fixes dans l'espace de Teichmüller $T_g$ d'un tel groupe, dépend d'un paramètre complexe qu'on peut alors ajuster pour maximiser la systole. On étudie ensuite les propriétés variationnelles dans $T_g$ des surfaces obtenues. Par extension de cette méthode, on trouve également une nouvelle surface extrême en genre 6, ainsi qu'une suite infinie de surfaces parfaites non extrêmes de genre $g>3$. En outre, on retrouve, de manière unifiée, les surfaces déjà connues en genre $\leq 5$. La méthode employée pour la recherche de surfaces parfaites, permet de trouver parallèlement un certain nombre de surfaces eutactiques, qui sont intéressantes à classifier en elles-mêmes puisque ce sont les points critiques de la fonction systole. Enfin, le dernier chapitre, développant une toute autre approche, concerne une méthode purement algébrique qui permet de redémontrer l'extrémalité des surfaces respectivement de Bolza et de Klein.

by Wong Chun-fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 81-82). / Chapter CHAPTER0 --- Introduction --- p.1 / Chapter CHAPTER1 --- Teichmuller Space of genus g --- p.5 / Chapter 1.1. --- Teichmiiller Space of genus g / Chapter 1.2. --- Fuchsian Model and Discrete subgroup of Aut(H) / Chapter 1.3. --- Fricke Space / Chapter CHAPTER2 --- Hyperbolic Geometry and Fenchel-Nielsen Coordinates --- p.14 / Chapter 2.1. --- Poincare Metric and Hyperbolic Geometry / Chapter 2.2. --- Fenchel-Nielsen Coordinates / Chapter 2.3. --- Fricke-Klein Embedding / Chapter CHAPTER3 --- Quasiconformal Mappings --- p.23 / Chapter 3.1. --- Definitions / Chapter 3.2. --- Existence Theorems on Quasiconformal Mappings / Chapter 3.3. --- Dependence on Beltrami Coefficients / Chapter CHAPTER4 --- Teichmuller Spaces --- p.37 / Chapter 4.1. --- Analytic Construction of Teichmiiller Spaces / Chapter 4.2. --- Teichmiiller mapping and Teichmiiller Theorem / Chapter 4.3. --- Teichmiiller Uniqueness Theorem / Chapter CHAPTER5 --- Complex Analytic Theory of Teichmiiller Spaces --- p.50 / Chapter 5.1. --- Bers' Embedding and the complex structure of Teichmiiller Space / Chapter 5.2. --- Invariance of Complex Structure of Teichmiiller Space / Chapter 5.3. --- Teichmiiller Modular Groups / Chapter 5.4. --- Classification of Teichmiiller Modular Transformations / Chapter CHAPTER6 --- Weil-Petersson Metric --- p.68 / Chapter 6.1. --- Petersson Scalar Product and Reproducing formula / Chapter 6.2. --- Infinitesimal Theory of Teichmuller Spaces / Chapter 6.3. --- Weil-Petersson Metric / BIBLIOGRAPHY --- p.81

We study isometric maps between Teichmüller spaces and bounded symmetric domains in their Kobayashi metric. We prove that every totally geodesic isometry from a disk to Teichmüller space is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. However, we prove that in dimensions two or more there are no holomorphic isometric immersions between Teichmüller spaces and bounded symmetric domains and also prove a similar result for isometric submersions. / Mathematics

Soit f une fraction rationnelle de degré d au moins 2. McMullen et Sullivan ont introduit l'espace de Teichmüller dynamique Teich(f), qui est une variété complexe de dimension au plus 2d-2 et qui paramétrise la classe de conjugaison quasiconforme de f dans l'espace des modules ratd via une application holomorphe F allant de Teich(f) dans ratd.Nous donnons une nouvelle construction élémentaire de Teich(f), et nous prouvonsque F est une immersion, ce qui répond à une question posée par McMullen et Sullivan.Ce dernier résultat nous permet d'obtenir des preuves simplifiées de résultats dus à Makienko et Levin sur la rigidité de f sous une hypothèse d'expansivité le long de l'orbite critique. Dans une seconde partie, nous construisons une famille d'exemples d'endomorphismes polynômiaux de P^2(C) ayant un domaine errant. Nos exemples sont des produits fibrés, de la forme (z,w) -> ( f(z) + aw, g(w)). De plus, on construira des exemples à coefficients réels où le domaine errant intersectera R^2. / Let f be a rational map of degree d at last 2. McMullen and Sullivan introduced the dynamical Teichmüller space Teich(f), which is a complex manifold of dimension at most 2d-2. It paramtrizes the quasiconformal conjugacy class of f in the moduli space ratdvia a holomorphic map F from Teich(f) to ratd. We give a new and elementary construction of Teich(f), and we prove that the parametrization F is an immersion, answering a question of McMullen and Sullivan. This last result enables us to give simplified proofs of rigidity results of Makienko and Levin under the assumption of expansion along the critical orbit. In a second part, we construct a family of examples of polynomial endomorphisms of¨P^2(C) with a wandering domain. Our examples are skew-products, of the form (z,w) -> (f(z)+aw, g(w)). Moreover, we will construct examples with real coefficients where the wandering domain intersects R^2.

Cette thèse porte sur certaines familles à un paramètre de surfaces de Riemann compactes de genre 2 définies par des surfaces de translation. Les familles que nous considérons constituent des géodésiques de Teichmüller dans l'espace des modules.Nous nous attachons en particulier à décrire ces surfaces par leurs matrices des périodes et par les équations des courbes algébriques associées.Nous étudions notamment les automorphismes admissibles par les surfaces de certaines de ces familles.Le principal résultat consiste en une caractérisation explicite des matrices des périodes des courbes réelles à trois composantes réelles appartenant à la famille obtenue par projection dans l'espace des modules de la SL(2,R)-orbite de la surface de translation en «L» pavée par trois carreaux.Nous montrons enfin, grâce à une interprétation en termes de transformations de Schwarz-Christoffel, comment calculer numériquement une équation de la courbe algébrique définie par une surface de translation en «L». / In this thesis we study some one parameter families of compact Riemann surfaces of genus 2 defined by translation surfaces.The families we consider are Teichmüller geodesics in the moduli space.We mainly describe these surfaces by means of period matrices and equations of the associated algebraic curves.We study admissible automorphisms for surfaces in some of those families.The main result is an explicit characterisation of period matrices of real curves with three real components belonging to the family obtained by projecting the SL(2,R)-orbit of the «L»-shaped translation surface tiled by three squares into the moduli space.We finally show, using an interpretation in terms of Schwarz-Christoffel transformations, how to numerically compute an equation of the algebraic curve defined by a «L»-shaped translation surface.