Combinatorial constructions in Smooth Ergodic Theory
Final Report Abstract
Historically motivated by problems in statistical mechanics Ergodic Theory examines statistical properties of dynamical systems. In particular, one is interested in the long-term behaviour of the system as well as the relationship between its time and space averages. In a pioneering Annals-paper from 1932, John von Neumann introduced Ergodic Theory to the mathematical community. In this research project we addressed two major problems dating back to this foundational paper. The so-called isomorphism problem asks to classify invertible measure-preserving transformations (MPT's) up to isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown in a rigorous way that such a classication is impossible. In joint work with Shilpak Banerjee we show that von Neumann's isomorphism problem is impossible even when restricting to real-analytic area-preserving diffeomorphisms of the 2-torus. Besides isomorphism, Kakutani equivalence is the best known and most natural equivalence relation on ergodic MPT's for which the classication problem can be considered. In joint work with Marlies Gerber we prove that the Kakutani equivalence relation of ergodic MPT's is not a Borel set. This shows in a precise way that the problem of classifying such transformations up to Kakutani equivalence is also intractable. Another important question in ergodic theory is the smooth realization problem, which asks if there are smooth versions to the objects and concepts of abstract ergodic theory. While only very few restrictions are known, there is a scarcity of general results on the smooth realization problem. One of the most powerful tools of constructing smooth diffeomorphisms with prescribed ergodic, spectral or topological properties is the so-called approximation by conjugation-method developed by Dmitri Anosov and Anatole Katok. In this research project we could provide the smooth realization of further spectral properties. Moreover, we continued to extend the approximation by conjugation-method from C∞ diffeomorphisms to the real-analytic category.
Publications
- Loosely Bernoulli Odometer-Based Systems Whose Corresponding Circular Systems Are Not Loosely Bernoulli, Preprint
M. Gerber & P. Kunde
(See online at https://doi.org/10.48550/arXiv.1912.12684) - Slow entropy of higher rank abelian unipotent actions, Preprint
A. Kanigowski, P. Kunde, K. Vinhage, D. Wei
(See online at https://doi.org/10.48550/arXiv.2005.02212) - Spectral disjointness of powers of diffeomorphisms with arbitrary Liouvillean rotation behavior, Studia Mathematica 259: 271-304, 2021
P. Kunde
(See online at https://doi.org/10.4064/sm191202-31-8)
