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Aspects of aperiodic order
Speakers:
Maria Isabel Cortez (Pontifical Catholic University of Chile)
Johannes Kellendonk (Claude Bernard University Lyon 1)
David Damanik (Rice University)
Participants:
Francisco Araújo (Paderborn University)
Michael Baake (Bielefeld University)
Ruiwen Dong (Saarland University)
Franz Gähler (Bielefeld University)
Markus Kirschmer (Paderborn University)
Anna Klick (Bielefeld University)
Simon Machado (ETH Zurich)
Neil Manibo (Bielefeld University)
Jan Mazac (Bielefeld University)
Eden Delight P. Miro (Ateneo de Manila University)
Timo Spindeler (Bielefeld University)
Schedule:
Organisers: Michael Baake,
Franz Gähler
Abstracts:
Maria Isabel Cortez: Toeplitz subhifts, equicontinuous systems and residually finite groups
In 1969, Jacobs and Keane introduced Toeplitz subshifts in
the context of $\mathbb{Z}$-actions. Since then, different works have shown
that there are dynamical systems within this family of subshifts
with varied behavior. For example, Downarowicz has shown in 1991
that any possible set of invariant probability measures is
realizable by some Toeplitz subshift in $\{0,1\}^{\mathbb{Z}}$. The flexibility of
Toeplitz subshifts motivated further generalizations beyond
$\mathbb{Z}$-actions, providing examples of group actions on the Cantor set
with exciting properties. Another consequence of the generalization
of the Toeplitz subshift concept is the characterization of infinite
residually finite groups as those admitting actions corresponding to
non-periodic Toeplitz subshifts.
In this talk, we gather some results about Toeplitz subshifts and
their relationship with minimal equicontinuous systems. For example,
applying results concerning Furstenberg-Weiss type almost 1-1
extensions, we will show that Toeplitz subshifts are a test family for
the amenability of residually finite groups.
Johannes Kellendonk: Semigroup methods in aperiodic order theory
One of the basic tools to study aperiodic tilings is the theory of ergodic
dynamical systems. Properties of tiling dynamical systems reflect properties
of the tiling, a point of view which has been very fruitful in the
diffraction theory of aperiodic media and also in the description of their
topological phases.
We consider a tiling to be highly ordered if its associated dynamical system
is “nearly equicontinuous”. By this we mean that the tiling dynamical system
factors onto a group rotation so that the fibres of the factor map are
finite.
The perhaps simplest class of such tilings is defined by constant shape
substitutions on a finite alphabet $A$. In this case the substitution is given
by a shape with a collection of maps from $A$ to $A$. These maps define a
semigroup whose algebraic structure can be employed to study factors of the
dynamical system of various types. This, in turn, is also useful to study
another semigroup associated to the dynamical system, namely the enveloping
(or Ellis) semigroup.
David Damanik: Deterministic Delocalization
We present joint work with Artur Avila on delocalizing Schrödinger
operators in arbitrary dimensions via arbitrarily small perturbations
of the potential.