Compactness and decompositions in ergodic and operator theory

• Applicant: Professor Dr. Balint Farkas
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 451698284

• Final Report Year: 2024

Final Report Abstract

Via Furstenberg’s famous correspondence principle statements of number theory can be translated to assertions about measure-preserving transformations on probability spaces, the central objects of mathematical ergodic theory. To prove such statements, a deep structure theoretic understanding, related to compactness and decompositions, of these measure-preserving systems is necessary. In our project we used operator-theoretic and topological-algebraic tools to obtain a better comprehension of this theory. The major step from classical structural results (e.g., the famous Halmos–von Neumann theorem from the 1940s) to the modern Furstenberg–Zimmer and Host–Kra–Ziegler theories for measurepreserving systems is to consider the “relative behavior” of dynamical systems. In order to gain insight into the structure of a dynamical system, one tries to find interesting factors of it, and then studies the properties of the original system relative to this factor. For example, instead of considering compact (i.e., “structured” ) systems, investigating compact extensions can help to understand a much bigger class of dynamical systems. In turn, one can derive a relative decomposition of a system with respect to one of its factors. While the original results rely on measure-theoretic instruments like measure disintegration, we propose an approach to the Furstenberg–Zimmer theory based on “relative topology” and “relative functional analysis”. In this framework we use concepts like Kaplansky–Hilbert modules (generalizations of Hilbert spaces) and topological bundles of compact groups to derive structural results in ergodic theory in a very general form, relying on relativized notions of compactness and decompositions. This unveils the topological-algebraic core behind the measure-theoretic results, and also removes “countability assumptions”, making the theory more flexible, for example, when using powerful constructions such as ultraproducts. We extend this approach to the fundamental Host– Kra machinery in follow-up research.

Publications

• A dynamical proof of the van der Corput inequality. Dynamical Systems, 37(4), 648-665.
  Edeko, Nikolai; Kreidler, Henrik & Nagel, Rainer
  
• Distal systems in topological dynamics and ergodic theory. Ergodic Theory and Dynamical Systems, 43(8), 2651-2672.
  EDEKO, NIKOLAI & KREIDLER, HENRIK
  
• A Halmos–von Neumann Theorem for Actions of General Groups. Applied Categorical Structures, 31(5).
  Hermle, Patrick & Kreidler, Henrik
  
• A Peter–Weyl theorem for compact group bundles and the geometric representation of relatively ergodic compact extensions.
  Nikolai Edeko, Asgar Jamneshan & Henrik Kreidler
  
• A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem. Analysis Mathematica, 50(2), 377-454.
  Edeko, N.; Haase, M. & Kreidler, H.