This book concentrates on the modern theory of dynamical systems and its interactions with number theory and combinatorics. The greater part begins with a course in analytic number theory and focuses on its links with ergodic theory, presenting an exhaustive account of recent research on Sarnak's conjecture on Moebius disjointness. Selected topics involving more traditional connections between number theory and dynamics are also presented, including equidistribution, homogenous dynamics, and Lagrange and Markov spectra. In addition, some dynamical and number theoretical aspects of aperiodic order, some algebraic systems, and a recent development concerning tame systems are described.

These volumes collect most of the papers of Anatole Katok, one of the founders of the modern theory of dynamical systems. Katok's work reflects half a century of research in mathematics and includes ergodic theory, hyperbolic, elliptic, and parabolic smooth dynamics, as well as higher-rank actions. Katok's papers cover an extremely broad range of topics in dynamics, and they contain many seminal contributions that had great impact on later developments and are now widely recognized as classical.Katok also authored numerous historical and biographical papers, and these contain accounts of crucial developments from the point of view of one of the main protagonists.Besides papers which have already appeared in academic journals, this collection includes several previously unpublished papers as well as some whose English translation appears here for the first time.These collected works are organized by topic into six chapters, each featuring an introduction written by respective leading specialists. Volume I focuses on the following topics: Hyperbolicity, Entropy, Geodesic Flows, Interval Exchange Transformations, Billiards, Twist Maps, Spectral Theory, Approximations, Combinatorial Constructions, and History of Dynamics. Volume II focuses on these topics: Cohomology and Geometric Rigidity, and Measure Rigidity.

These volumes collect most of the papers of Anatole Katok, one of the founders of the modern theory of dynamical systems. Katok's work reflects half a century of research in mathematics and includes ergodic theory, hyperbolic, elliptic, and parabolic smooth dynamics, as well as higher-rank actions. Katok's papers cover an extremely broad range of topics in dynamics, and they contain many seminal contributions that had great impact on later developments and are now widely recognized as classical.Katok also authored numerous historical and biographical papers, and these contain accounts of crucial developments from the point of view of one of the main protagonists.Besides papers which have already appeared in academic journals, this collection includes several previously unpublished papers as well as some whose English translation appears here for the first time.These collected works are organized by topic into six chapters, each featuring an introduction written by respective leading specialists. Volume I focuses on the following topics: Hyperbolicity, Entropy, Geodesic Flows, Interval Exchange Transformations, Billiards, Twist Maps, Spectral Theory, Approximations, Combinatorial Constructions, and History of Dynamics. Volume II focuses on these topics: Cohomology and Geometric Rigidity, and Measure Rigidity.