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Non-Integrable Flat Dynamical Systems
Author(s): József Beck (Author), William W L Chen (Author), Yuxuan Yang (Author)
- Publisher finelybook 出版社: World Scientific Publishing
- Publication Date 出版日期: September 15, 2025
- Language 语言: English
- Print length 页数: 448 pages
- ISBN-10: 9819803152
- ISBN-13: 9789819803156
Book Description
This book provides a systematic introduction and treatment of non-integrable flat systems. More precisely, it develops analogues of the well-known Kronecker–Weyl equidistribution theorem for various non-integrable flat systems. The distribution properties of half-infinite geodesics in non-integrable flat dynamical systems are investigated, with particular emphasis on density and uniformity on various flat surfaces, as well as generalizations to higher dimensions and related problems.
The approach to addressing some of these uniformity questions combines traditional ergodic theory methods, such as the Birkhoff ergodic theorem, with tools from number theory and new ideas outside the field of ergodic theory. Several non-trivial results on 3-dimensional flat systems are included, expanding upon the extensive literature on 2-dimensional non-integrable flat dynamical systems, which often focuses on plane-specific methods that do not readily adapt to 3-dimensional systems. New approaches are developed to address these challenges.
A systematic study of a class of dissipative systems is also presented, where the flow is neither time-reversible nor measure-preserving. The overwhelming majority of the material in this book represents previously unpublished research.
About the Author
József Beck and William Chen were both heavily influenced early in their careers by Klaus Roth and Wolfgang Schmidt, pioneers in the broad area of uniformity and irregularity, specializing in irregularities of distribution, known nowadays as discrepancy theory. They carried on their investigations in discrepancy theory throughout the 1980s and beyond after Roth and Schmidt had stopped work in the area, and began a long and fruitful collaboration that included writing a monograph on the subject entitled Irregularities of Distribution, published in 1987. This work played a crucial role in the current prominence of the subject by having attracted a number of strong researchers into the area since the early 1990s.
József Beck spent a number of years at the Mathematical Institute of the Hungarian Academy of Sciences in Budapest before his move in the late 1980s to Rutgers University in the United States, eventually becoming Harold H. Martin Professor of Mathematics. He is currently Distinguished Professor Emeritus of Mathematics. He received the Alfréd Rényi Prize of the Hungarian Academy of Sciences in 1984 as well as the Delbert Ray Fulkerson Prize of the American Mathematical Society in 1985, and was elected an External Member of the Hungarian Academy of Sciences in 2004. Further to the book Irregularities of Distribution in 1987, he has written a number of other monographs, Combinatorial Games in 2008, Inevitable Randomness in Discrete Mathematics in 2009, Probabilistic Diophantine Approximation in 2014, Strong Uniformity and Large Dynamical Systems in 2017 and Equidistribution of Dynamical Systems in 2020, the last two published by World Scientific.
William Chen spent a number of years in the Department of Mathematics at the Imperial College of Science, Technology and Medicine in London before his move at the end of 1990 to Macquarie University in Australia, eventually becoming Professor of Mathematics. He also served twice as the Head of Department, and is currently Emeritus Professor of Mathematics. Further to the book Irregularities of Distribution in 1987, he has co-edited two volumes, Analytic Number Theory: Essays in Honour of Klaus Roth (with Tim Gowers, Heini Halberstam, Wolfgang Schmidt and Robert Vaughan) in 2009 and A Panorama of Discrepancy Theory (with Anand Srivastav and Giancarlo Travaglini) in 2014. Currently he is Managing Editor of Mathematika, a journal published by the London Mathematical Society on behalf of University College London.
Yuxuan Yang obtained his PhD in mathematics at Rutgers University in 2022 under the guidance of József Beck. He is currently an Assistant Professor at the Beijing University of Posts and Telecommunications.
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