Searcharxiv⌕ Search

arXiv subjects

Nikolai Edeko

Publications and source records attributed to Nikolai Edeko.

9 recordsLinked to original sources
arXiv2024

A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem

In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof rests heavily on the operator theory on Kaplansky-Hilbert modules, in particular the spectral theorem for Hilbert-Schmidt homomorphisms on such modules. As an application, a generalization of the celebrated Furstenberg-Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.

math.DS↗
arXiv2023

A Peter-Weyl theorem for compact group bundles and the geometric representation of relatively ergodic compact extensions

We show that a relatively ergodic extension of measure-preserving dynamical systems has relative discrete spectrum if and only if it can be represented as a skew-product by a bundle of compact homogeneous spaces. Our result holds without restrictions on the acting group or the underlying probability spaces. This generalizes previous work by Mackey, Zimmer, Ellis, Austin, and the second author and Tao, and is inspired by the Furstenberg-Zimmer and Host-Kra structure theories for actions of uncountable groups. Our approach translates the ergodic-theoretic question into topological dynamics, where we establish a corresponding classification: an extension in topological dynamics has relative discrete spectrum precisely when it admits a skew-product representation by bundles of compact homogeneous spaces. A key step in our argument is establishing a Peter-Weyl-type theorem for bundles of compact groups which might be of independent interest.

math.DS↗
arXiv2022

A dynamical proof of the van der Corput inequality

We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on the Furstenberg correspondence principle. This is done by reducing the inequality to the mean ergodic theorem for contractions on Hilbert spaces. The key difficulty therein is that the Furstenberg correspondence principle is, a priori, limited to scalar-valued sequences. We therefore discuss how interpreting the Furstenberg correspondence principle via the Gelfand--Naimark--Segal construction for C*-algebras allows to study not just scalar but general Hilbert space-valued sequences in terms of unitary operators. This yields a proof of the van der Corput inequality in the spirit of the Furstenberg correspondence principle and the flexibility of this method is discussed via new proofs for different variants of the inequality.

math.DS↗
arXiv2022

Distal systems in topological dynamics and ergodic theory

We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg--Zimmer tower of a measurably distal system and showing that at each step, there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.

math.DS↗
arXiv2020

Uniform enveloping semigroupoids for groupoid actions

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems. We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniform enveloping semigroupoid is actually a compact groupoid. We also provide an operator theoretic characterization based on an abstract Peter-Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest.

math.DS↗
arXiv2019

On equicontinuous factors of flows on locally path-connected compact spaces

We consider a locally path-connected compact metric space $K$ with finite first Betti number $b_1(K)$ and a flow $(K, G)$ on $K$ such that $G$ is abelian and all $G$-invariant functions $f\in\mathrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than $b_1(K)$. For this purpose, we use and provide a new proof for [HJop, Theorem 2.12] which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces $K$ and $L$ that we obtain by characterizing the local connectedness of $K$ in terms of the Banach lattice $\mathrm{C}(K)$.

math.DS↗
arXiv2018

On the isomorphism problem for non-ergodic systems with discrete spectrum

The article presents a new perspective on the isomorphism problem for non-ergodic measure-preserving dynamical systems with discrete spectrum which is based on the connection between ergodic theory and topological dynamics constituted by topological models. By first solving the isomorphism problem for a certain class of topological dynamical systems, it is shown that the measure-preserving case can in fact be deduced from the topological one via the construction of topological models. As a byproduct, a new characterization of mean ergodicity for topological dynamical systems is obtained.

math.DS↗