Non-Integrable Dynamics: Time-Quantitative Results

The subject of this monograph is to describe orbits of slowly chaotic motion. The study of geodesic flow on the unit torus is motivated by the irrational rotation sequence, where the most outstanding result is the Kronecker–Weyl equidistribution theorem and its time-quantitative enhancements, including superuniformity. Another important result is the Khinchin density theorem on superdensity, a best possible form of time-quantitative density. The purpose of this monograph is to extend these classical time-quantitative results to some non-integrable flat dynamical systems.

The theory of dynamical systems is on the most part about the qualitative behavior of typical orbits and not about individual orbits. Thus, our study deviates from, and indeed is in complete contrast to, what is considered the mainstream research in dynamical systems. We establish non-trivial results concerning explicit individual orbits and describe their long-term behavior in a precise time-quantitative way. Our non-ergodic approach gives rise to a few new methods. These are based on a combination of ideas in combinatorics, number theory, geometry and linear algebra.

Approximately half of this monograph is devoted to a time-quantitative study of two concrete simple non-integrable flat dynamical systems. The first concerns billiard in the L-shape region which is equivalent to geodesic flow on the L-surface. The second concerns geodesic flow on the surface of the unit cube. In each, we give a complete description of time-quantitative equidistribution for every geodesic with a quadratic irrational slope.

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.