134,315 research outputs found
Almost maximally almost-periodic group topologies determined by T-sequences
A sequence in a group is a {\em -sequence} if there is a
Hausdorff group topology on such that
. In this paper, we provide several
sufficient conditions for a sequence in an abelian group to be a -sequence,
and investigate special sequences in the Pr\"ufer groups
. We show that for , there is a Hausdorff group
topology on that is determined by a -sequence,
which is close to being maximally almost-periodic--in other words, the von
Neumann radical is a non-trivial finite
subgroup. In particular, . We also prove that the
direct sum of any infinite family of finite abelian groups admits a group
topology determined by a -sequence with non-trivial finite von Neumann
radical.Comment: v2 - accepted (discussion on non-abelian case is removed, replaced by
new results on direct sums of finite abelian groups
Rotation sets and almost periodic sequences
We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segment as well as non-convex and even plane-separating continua. This shows that the restriction which hold for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences
Hybrid Quasicrystals, Transport and Localization in Products of Minimal Sets
We consider convex combinations of finite-valued almost periodic sequences
(mainly substitution sequences) and put them as potentials of one-dimensional
tight-binding models. We prove that these sequences are almost periodic. We
call such combinations {\em hybrid quasicrystals} and these studies are related
to the minimality, under the shift on both coordinates, of the product space of
the respective (minimal) hulls. We observe a rich variety of behaviors on the
quantum dynamical transport ranging from localization to transport.Comment: 3 figures. To appear in Journal of Stat. Physic
Analysis of two step nilsequences
Nilsequences arose in the study of the multiple ergodic averages associated
to Furstenberg's proof of Szemer\'edi's Theorem and have since played a role in
problems in additive combinatorics. Nilsequences are a generalization of almost
periodic sequences and we study which portions of the classical theory for
almost periodic sequences can be generalized for two step nilsequences. We
state and prove basic properties for 2-step nilsequences and give a
classification scheme for them
- …