Polynomial ergodic theorems and Ramsey theory

点击翻译按钮获取中文摘要

ABSTRACT:The project concentrates on multiple recurrence and convergence indynamical systems with a focus on polynomial actions of abelian andnilpotent groups. The problems considered lead to diverse applications ofergodic theory to combinatorics, number theory and algebra which areinaccessible so far by conventional methods. The polynomial Szemeredi andHales-Jewett theorems proved by the proposers in the recent years havesince been extended by the proposers and their colleagues in differentdirections, each providing a better understanding of the phenomenon ofmultiple recurrence along polynomials and offering new vistas of research.The directions of study touched upon in this proposal include(but are not limited to) the deeper study of multiple recurrence fornilpotent group actions and the investigation of polynomial multiplerecurrence in the framework of IP-systems. Another interesting areaconsidered in this proposal has to do with convergence of ergodic averagesnaturally appearing in the theory of multiple recurrence and itsapplications. The results recently obtained by the proposers indicate adichotomy between the behavior of ergodic averages depending on whetherthe acting groups have polynomial or exponential growth. Therelated conjectures formulated in the last section of the proposal areshedding new light on these questions. The theory of multiple recurrence that we focus on in this project usesthe ideas from several diverse areas of mathematics and aims to advanceour knowledge about the intrinsic properties of dynamical systems whichare related to their long range behavior. An example of a highlynontrivial fact stemming from our investigations is the regularity of thebehavior of dynamical systems along polynomial time measurements. Thisfact, in its turn, has strong applications to seemingly distant areas ofnumber theory and combinatorics.