Extension of structural results in ergodic theory and applications
Final Report Abstract
The aim of this project was to explore the structure theory of measure-preserving dynamical systems for the action of uncountable groups on inseparable spaces and to study applications and connections of such a general theory to multiple recurrence, ergodic Ramsey theory and higher-order Fourier analysis in additive combinatorics. The generality of uncountable groups and inseparable spaces causes several foundational measure-theoretic and functional analytic problems. We overcame these problems by developing a suitable pointfree framework in which uncountable ergodic theory can be studied systematically. Within this framework we further developed several basic tools such as a canonical model and a canonical disintegration of measures which help to address basic needs in structural ergodic theory. This framework generalizes the usual frameworks in countable ergodic theory and explains many of its shortcomings if extended to an uncountable setting. Moreover, the suggested framework naturally connects to operatortheoretic ergodic theory for commutative von Neumann algebras. To solve technical problems in the relative analysis of systems conditioned on a factor, we introduced topos-theoretic methods in the form of conditional analysis (or, conditional set theory) to ergodic theory. Combining the pointfree framework and the topos-theoretic techniques for the analysis in relative settings, we established uncountable versions of several fundamental results in structural ergodic theory such as an uncountable Moore-Schmidt theorem, an uncountable Mackey-Zimmer theorem, an uncountable Furstenberg-Zimmer structure theory, and an uncountable version of Austin’s machinery of sated extensions. As a first application, we used these uncountable structural results to establish an extension of the double recurrence theorem for amenable groups by Bergelson, McCutcheon and Zhang and the multiple recurrence theorem for amenable groups of Austin to arbitrary not necessarily countable amenable groups acting on arbitrary not necessarily separable spaces respectively. We derived from these uncountable recurrence theorems novel combinatorial applications and uniformity results. We plan to continue this line of research by establishing a general Host-Kra-Ziegler structure theorem for arbitrary uncountable abelian groups and unifying such structure theorem with an inverse theorem for the Gowers uniformity norms for arbitrary finite abelian groups.
