Rigidity in dynamical systems
Project Goals and Description:
Mixing properties of dynamical systems are of interest in a broad class of applications. Many systems exhibit a combination of mixing and non-mixing behavior, and the study of weak mixing rigid measure preserving dynamical systems reveals the extent to which these two extremes can exist in combination. These systems are studied from a variety of perspectives and contribute to the fundamental theory in several overlapping fields, including ergodic theory, harmonic analysis, and combinatorics. Many new results about such systems have appeared in the past decade, as well as a plethora of new open problems. This project will undertake one or more of the following paths to advance the field:
- Solve, or make substantial progress on the following major open question: is every lacunary sequence of integers a rigidity sequence for a weak mixing measure preserving transformation?
- Introduce refinements of the concept of rigidity for measure preserving systems and study their relationships with known rigidity properties.
- Produce novel examples of rigidity sequences with prescribed properties (such as being dense in the Bohr topology).
Grand Challenge: Not applicable.
Textbooks for background: Walters, Peter, An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. ix+250 pp. ISBN: 0-387-90599-5 Petersen, Karl, Ergodic theory. Corrected reprint of the 1983 original. Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1989. xii+329 pp. ISBN: 0-521-38997-6 Articles (for recent developments and open problems): Bergelson, Del Junco, Lemanczyk, Rosenblatt: Rigidity and non-recurrence along sequences Badea, Grivaux, and Matheron: Rigidity sequences, Kazhdan sets and group topologies on the integers
John Griesmer, firstname.lastname@example.org
Knowledge of measure theory (Math 501) or very strong background in real analysis (Math 301) and probability.
TIME COMMITMENT (HRS/WK)
Facility with constructing abstract dynamical systems, understanding asymptotics of measure preserving dynamical systems through harmonic analysis.
We will meet at least once per week. The first 3 - 5 weeks will be spent developing background and understanding recent results. I will present some background in ergodic theory and harmonic analysis. The student(s) will be asked to present more background material, as warm up for the main project. The remainder of the semester will be spent attacking open problems and developing the theory outlined in the project description. The student(s) will work on their own throughout the week, and we will discuss their ideas, and work together, during our meeting time. I will offer constructive feedback and suggest new avenues for investigation. Once new results are obtained, I will guide the student in writing an article, with the aim of submitting to a research journal.
PREFERRED STUDENT STATUS