Workshop on
Operator Theoretic Aspects of Ergodic Theory
May 5-6, 2017, Feldkirch
Pädagogische Hochschule Vorarlberg, Liechtensteinerstraße 33 – 37, 6800 Feldkirch, Austria, Hörsaal B.
Friday, May 5th:
• 9:20–
Opening
• 9:30–10:30 Eli Glasner (Tel-Aviv): Tame minimal dynamical systems for general groups
A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a
dynamical system is either very large and contains a topological copy of βN, or it is a “tame” topological space
whose topology is determined by the convergence of sequences. In the latter case the dynamical system is called
tame. Minimal tame dynamical systems with an abelian acting group were studied by several authors. Their
structure is rather simple and by now well understood. What happens when the acting group is assumed to be
amenable? Is there a simplified structure theorem for minimal tame actions of an arbitrary acting group? I will
address these questions and present some answers, as well as several interesting new examples.
• 10:35–11:05
Coffee break
• 11:05–12:05 Philipp Kunde (Hamburg): Smooth diffeomorphisms with homogeneous spectrum
One of the main problems in the spectral theory of dynamical systems at the interface of unitary operator theory
and ergodic theory is the question about possible spectral properties for a Koopman operator associated with
a measure-preserving transformation. More specifically, one can search for transformations T possessing specific
essential values MUT of the spectral multiplicities. Over the past years, impressive progress concerning this
question has been made.
Another important question in ergodic theory asks if there are smooth versions to the objects and concepts of
abstract ergodic theory. With regard to the spectral multiplicity problem there are only few results in this direction. In this talk we present the recent construction of smooth diffeomorphisms T with homogeneous spectrum
MUT ×T = {2} on arbitrary smooth compact connected manifold of dimension at least 2 admitting a non-trivial
circle action. This construction is based on the Approximation by Conjugation-method developed by D.V. Anosov
and A. Katok.
• 12:10–12:35 Tanja Eisner (Leipzig): Orbits along primes
We discuss some results and questions concerning behaviour of orbits along primes in both ergodic theory and
operator theory.
• 12:35–14:30
Lunch break
• 14:30–15:30 Manfred Einsiedler (ETH Zürich): Joinings and Orthogonal lattices
We will discuss how measure classification theorems and in particular joining classification theorems can help
to prove the joint equidistribution of integer points on spheres and the shape of the lattice in the orthogonal
complement. This is joint work with Menny Aka and Uri Shapira, respectively Rene Rühr and Philipp Wirth. The
dynamical inputs are depending on the dimension unipotent dynamics or dynamics of higher rank diagonalisable
groups.
• 15:35–16:00 Nikolai Edeko (Paris): The isomorphism problem for systems with discrete spectrum.
Motivated by the classical Halmos-von Neumann theorem, we consider the question whether it is possible to give
a complete isomorphism invariant for not necessarily minimal/ergodic systems with discrete spectrum.
• 16:05–16:35
Coffee break
• 16:35–17:00 Christian Budde (Wuppertal): Mean ergodic theorems for bi-continuous semigroups
In this talk we will introduce the class of bi-continuous semigroups and will discuss some mean ergodic properties
of these semigroups. The basis of this talk will be the article from Albanese, Lorenzi and Manco which has the
same title as the talk.
• 17:05–17:30 Martin Grothaus (Kaiserslautern): Weak Poincaré inequalities for convergence rate of
degenerate diffusion processes
We investigate the ergodicity of operator semigroups generated by degenerate non-sectorial operators.
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• 17:35–18:00 Jürgen Voigt (Dresden): Bands in Lp -spaces
Let M be a band in a space Lp (µ), with 1 ≤ p < ∞ and a measure space (Ω, A, µ). If the measure space is
σ-finite, then M is of the form
M = {f ∈ Lp (µ); f = 0 µ-a.e. on Ω \ Ω0 },
with a suitable set Ω0 ∈ A. We show that, for a general measure space (Ω, A, µ), there exists a decomposition
µ = µ0 + µ00 such that
M = Lp (µ0 ) = {f ∈ Lp (µ) | f = 0 µ00 -a.e.}.
In fact, the possibility of describing bands with subsets Ω0 as above is equivalent to the weak localisability of
the measure space. The theory will be illustrated by an example, with an application to absorption semigroups.
The talk is a report on joint work with H. Vogt.
• 19:00–
Conference dinner, Schattenburg
Saturday, May 6th.
• 9:30–10:30 Markus Haase (Kiel): Dynamical Systems with quasi-discrete spectrum
In this talk we re-examine the theory of systems with quasi-discrete spectrum initiated in the 1960’s by Abramov,
Hahn, and Parry. In the first part, we give a simpler proof of the Hahn–Parry theorem stating that each minimal
topological system with quasi-discrete spectrum is isomorphic to a certain affine automorphism system on some
compact Abelian group. Next, we show that a suitable application of Gelfand’s theorem renders Abramov’s theorem — the analogue of the Hahn-Parry theorem for measure-preserving systems — a straightforward corollary of
the Hahn-Parry result. In the second part, independent of the first, we present a shortened proof of the fact that
each factor of a totally ergodic system with quasi-discrete spectrum (a ”QDS-system”) has again quasi-discrete
spectrum and that such systems have zero entropy. Moreover, we obtain a complete algebraic classification of
the factors of a QDS-system. In the third part, we apply the results of the second to the (still open) question
whether a Markov quasi-factor of a QDS-system is already a factor of it. We show that this is true when the
system satisfies some algebraic constraint on the group of quasi-eigenvalues, which is satisfied, e.g., in the case
of the skew shift.
• 10:35–11:00 Henrik Kreidler (Tübingen): The primitive spectrum of a dynamical system
The primitive spectrum, i.e., the set of all kernels of irreducible representations, has proven to be a useful
tool in the theory of C ∗ -algebras. Based on this concept we introduce the primitive spectrum of a topological
dynamical system and examine its properties.
• 11:05–11:35
Coffee break
• 11:35–12:00 Kari Küster (Tübingen): Koopman operators with one-dimensional fixed space
For measure-theoretic dynamical systems a one-dimensional fixed space of the Koopman operator is equivalent
to ergodicity of the underlying system. In the topological case the situation is more complex. Transitivity
implies a one-dimensional fixed space but the converse does not hold true in general. In my talk I introduce supertransitivity as a sufficient and potentially necessary condition.
• 12:05–12:30 Jochen Glück (Ulm): Dilation theorems on general Banach spaces
The celebrated dilation theorem of Akcoglu and Sucheston is a deep and powerful tool in operator theory. It
asserts that a positive contraction on an Lp -space possesses a so-called dilation to a bijective isometry on a larger
Lp -space. From this, one can deduce a maximal inequality needed to prove the pointwise ergodic theorem on Lp .
Despite the usefulness of dilation theory, until today no systematic approach is known to prove dilation theorems
on general Banach spaces. In this talk we present such an approach. We show that, on many classes of reflexive
Banach spaces, dilation theory is in a sense compatible with taking convex combinations of linear operators.
Therefore, proving a dilation theorem on a given space X comes down to studying the convex hull of all bijective
isometries on X. This also yields a new and structure theoretic proof for the theorem of Akcoglu–Sucheston on
Lp . This talk is based on joint work with Stephan Fackler (Institut für Angewandte Analysis, Universität Ulm,
Germany).
• 12:35–13:00 Mike Schnurr (Leipzig): Generic properties of extensions
Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing
transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions
on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not
have an intermediate nilfactor.
• 13:00
Closing
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