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This textbook offers an accessible introduction to translation surfaces. Building on modest prerequisites, the authors focus on the fundamentals behind big ideas in the field: ergodic properties of translation flows, counting problems for saddle connections, and associated renormalization techniques. Proofs that go beyond the introductory nature of the book are deftly omitted, allowing readers to develop essential tools and motivation before delving into the literature. Beginning with the fundamental example of the flat torus, the book goes on to establish the three equivalent definitions of translation surface. An introduction to the moduli space of translation surfaces follows, leading into a study of the dynamics and ergodic theory associated to a translation surface. Counting problems and group actions come to the fore in the latter chapters, giving a broad overview of progress in the 40 years since the ergodicity of the Teichmüller geodesic flow was proven. Exercises are included throughout, inviting readers to actively explore and extend the theory along the way. Translation Surfaces invites readers into this exciting area, providing an accessible entry point from the perspectives of dynamics, ergodicity, and measure theory. Suitable for a one- or two-semester graduate course, it assumes a background in complex analysis, measure theory, and manifolds, while some familiarity with Riemann surfaces and ergodic theory would be beneficial.
A translation surface is a two-dimensional manifold, equipped with a translation structure. It can be obtained by considering Euclidean polygons and identifying their edges via translations. The vertices of the polygons form singularities if the translation structure can not be extended to them. We study translation surfaces with wild singularities, regarding the topology (genus and space of ends), the geometry (behavior of the singularities), and how the topology and the geometry are related.
This encyclopedia presents an all-embracing collection of analytical surface classes. It provides concise definitions and description for more than 500 surfaces and categorizes them in 38 classes of analytical surfaces. All classes are cross references to the original literature in an excellent bibliography. The encyclopedia is of particular interest to structural and civil engineers and serves as valuable reference for mathematicians.
This volume constitutes the thoroughly refereed post-conference proceedings of the 9th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2016, held in Tønsberg, Norway, in June 2016. The 17 revised full papers presented were carefully reviewed and selected from 115 submissions. The topics range from mathematical theory to industrial applications.
This volume contains the proceedings of the conference on Riemann and Klein Surfaces, Symmetries and Moduli Spaces, in honor of Emilio Bujalance, held from June 24-28, 2013, at Linköping University. The conference and this volume are devoted to the mathematics that Emilio Bujalance has worked with in the following areas, all with a computational flavor: Riemann and Klein surfaces, automorphisms of real and complex surfaces, group actions on surfaces and topological properties of moduli spaces of complex curves and Abelian varieties.
This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey "Principal Structures of Volume 1A.The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations).. Written by experts in the field.. The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.
The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fme subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.
This book focuses on the interactions between discrete and geometric dynamical systems, and between dynamical systems and theoretical physics and computer science. Accordingly, the contributions revolve around two main topics: (1) interaction between geometric and symbolic systems, with emphasis on tiling problems for quasicrystals, substitutions and their multidimensional generalizations, geodesic and horocycle flow, adic systems; (2) dynamical systems: geometry and chaos, with special interest in smooth ergodic theory, statistical and multifractal properties of chaotic systems, stability and turbulence in extended complex systems.
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