Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics
Résumé
This volume consists of minicourses notes, survey, research/survey, and research articles that have arisen as an outcome of workshops, research in pairs, and other scientific work held under the auspices of the Jean Morlet Chair at CIRM between August 1, 2016 and January 31, 2017. The semester had a substantial core support and funding by CIRM, Aix-Marseille University, and the city of Marseille. Additionally, it was supported by the LABEX Archimède, and the ANR grants of Christian Mauduit (Aix-Marseille University) and Joël Rivat (Aix-Marseille University).
The minicourses were those given in the framework of the doctoral school Applications of Ergodic Theory in Number Theory organized by Sébastien Fer- enczi (Aix-Marseille University), Joanna Kułaga-Przymus (Nicolaus Copernicus University Torun ́ and Aix-Marseille University), Mariusz Leman ́czyk (Nicolaus Copernicus University Torun ́), and Serge Troubetzkoy (Aix-Marseille University). The main aim of this school was, on one hand, to provide participants with modern methods of ergodic theory and topological dynamics oriented toward applications in number theory and combinatorics, and, on the other hand, to present them with a broad spectrum of number theory problems that can be treated with the use of such tools. These tasks were realized in four minicourses by Vitaly Bergelson (Ohio State University), “Mutually enriching connections between ergodic theory and combinatorics,” Manfred Einsiedler (ETH Zürich), “Equidistribution on homoge- neous spaces, a bridge between dynamics and number theory,” Carlos Matheus Silva Santos (CNRS - Université Paris 13), “The Lagrange and Markov spectra from the dynamical point of view,” and Joël Rivat “Introduction to analytic number theory."
The main conference Ergodic Theory and its Connections with Arithmetic and Combinatorics was organized by Julien Cassaigne (Aix-Marseille University), Sébastien Ferenczi, Pascal Hubert (Aix-Marseille University), Joanna Kułaga- Przymus, Mariusz Leman ́czyk with the scientific committee consisting of Artur Avila (University Paris Diderot and IMPA, Rio de Janeiro), Vitaly Bergelson, Mandred Einsiedler, Hillel Furstenberg (The Hebrew University of Jerusalem), Anatole Katok (Penn State University), Christian Mauduit, Imre Ruzsa (Alfred Rényi Institute Budapest), and Peter Sarnak (IAS Princeton). The conference was aimed at interactions between ergodic theory and dynamical systems and number theory. Its main subjects were disjointness in ergodic theory and randomness in number theory, ergodic theory and combinatorial number theory, and homogenous dynamics and its applications.
Important events of the semester were two smaller specialized workshops. The first one Ergodic Theory and Möbius Disjointness was organized by Sébastien Fer- enczi, Joanna Kułaga-Przymus, Mariusz Leman ́czyk, Christian Mauduit, and Joël Rivat. The meeting focused on the recent progress on Sarnak’s conjecture on Möbius disjointness: methods, results, and the feedback in ergodic theory. The second one Spectral Theory of Dynamical Systems and Related Topics was organized by Alexander Bufetov (Aix-Marseille University), Sébastien Ferenczi, Joanna Kułaga- Przymus, Mariusz Leman ́czyk, and Arnaldo Nogueira (Aix-Marseille University). The meeting was aimed at the recent progress in the spectral theory and joinings of dynamical systems, especially, in the recent spectacular progress toward the solutions of some open classical problems of ergodic theory: Rokhlin problem on mixing of all orders, stability of spectral properties under smooth changes for the parabolic systems, the Banach problem on the existence of dynamical systems with simple Lebesgue spectrum, and the problem of spectral multiplicity.
The scientific part of the semester was completed by two research in pairs: Dynamical Properties of Systems Determined by Free Points in Lattices and On the Stability of Möbius Disjointness in Topological Models and a special program of invitations with participation of Michael Baake (University of Bielefeld), Jean- Pierre Conze (University of Rennes 1), Alexandre Danilenko (Institute of Low Temperature, Kharkov), Christian Huck (University of Bielefeld), Joanna Kułaga- Przymus, El Houcein El Abdalaoui (University of Rouen), Mariusz Leman ́ czyk and Thierry de la Rue (University of Rouen).
The contents of this volume are as follows. It begins with Part I which is entirely the course.
• Joël Rivat, Bases of Analytic Number Theory. Among other aspects, the course contains a presentation of the main properties of the Riemann ζ function with a generous introduction to the theory of Dirichlet series. Large sieve method together with a beautiful application to Twin Prime conjecture and deep relations with the theory of multiplicative functions are dealt with. We find also a detailed presentation of Vinogradov’s method of major and minor arcs, together with a deep analysis of sums of type I and II which are of great use in current research. The final chapter is devoted to the van der Corput method of computing and estimating trigonometric sums.
Part II of the volume consists of articles devoted to interactions between arithmetic and dynamics. They are all of research/survey/course type:
• M. Baake, A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems is a survey which presents the basic notions and reviews facts concerning the reversing symmetry of dynamical systems, focusing on systems (subshifts) of algebraic and number-theoretic origin.
• M. Einsiedler, M. Luethi, Kloosterman Sums, Disjointness, and Equidistribution summarizes the aforementioned minicourse of M. Einsiedler. Various appli- cations of Kloosterman sums are shown: equidistribution properties of sparse subsets of horocycle orbits in the modular case, disjointness results on the torus, mixing properties.
• S. Ferenczi, J. Kułaga-Przymus, M. Leman ́czyk, Sarnak’s Conjecture: What’s New? is a survey presenting an exhaustive list of methods and results concerning the problem of Möbius disjointness. Some new results are also included.
• A. Gomilko, D. Kwietniak, M. Leman ́czyk, Sarnak’s Conjecture Implies the Chowla Conjecture Along a Subsequence proves this elementary but new result.
• C. Huck, On the Logarithmic Probability That a Random Integral Ideal Is A-free
is an article which extends a theorem of Davenport and Erdös on sets of multiples with integers to the existence of logarithmic density for unions of integral ideals in number fields.
• C. Matheus, The Lagrange and Markov Spectra from the Dynamical Point of View summarizes the aforementioned minicourse of C. Matheus. The notes introduce the world of Lagrange and Markov spectra with a special focus on the proof of Moreira’s theorem on the intricate structure of such spectra.
• O. Ramaré, On the Missing Log Factor is a “journey” around the Axer-Landau Equivalence Theorem of the Prime Number Theorem and properties of the Möbius and von Mangoldt functions.
• O. Ramaré, Chowla’s Conjecture: From the Liouville Function to the Möbius Function is a note focusing on proofs of implications between various versions of the Chowla conjecture in which we use either Liouville or Möbius function.
Part III of the volume consists of three articles of survey or research/survey type from selected topics in dynamics:
• T. Adams, C. Silva, Weak Mixing for Infinite Measure Invertible Transformations surveys and studies mixing properties of transformations preserving infinite measure.
• E. Glasner, M. Megrelishvili, More on Tame Dynamical Systems surveys and amplifies old results in (topological) tame dynamical systems, proves some new results, and provides new examples of tame systems.
• K. Inoue, H. Nakada, A Piecewise Rotation of the Circle, IPR Maps and Their Connection with Translation Surfaces reviews a construction of translation surfaces in terms of a continuous version of the cutting-and-stacking systems and proves a new result of realization of Rauzy classes.