## Journals

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### Open Access Journals

ERA-MS

We consider the spectrum of the Fibonacci Hamiltonian for small
values of the coupling constant. It is known that this set is a
Cantor set of zero Lebesgue measure. Here we study the limit, as the
value of the coupling constant approaches zero, of its thickness and
its Hausdorff dimension. We announce the following results and
explain some key ideas that go into their proofs. The thickness
tends to infinity and, consequently, the Hausdorff dimension of the
spectrum tends to one. Moreover, the length of every gap tends to
zero linearly. Finally, for sufficiently small coupling, the sum of
the spectrum with itself is an interval. This last result provides a
rigorous explanation of a phenomenon for the Fibonacci square
lattice discovered numerically by Even-Dar Mandel and Lifshitz.

keywords:
Fibonacci Hamiltonian
,
Hausdorff dimension
,
Spectrum
,
trace map
,
thickness.
,
dynamically
defined Cantor set

DCDS-S

We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of GL$(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the set of points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.

PROC

We consider symbolic flows over finite alphabets and study certain
kinds of repetitions in these sequences. Positive and negative
results for the existence of such repetitions are given for
codings of interval exchange transformations and codings of
quadratic polynomials.

DCDS

We discuss the long-time behavior of solutions to the Schrödinger equation in some separable Hilbert space, with particular emphasis on the spreading over some orthonormal basis. Various ways of studying wavepacket spreading from this perspective are described and their inter-relations investigated. We also state and discuss known results for concrete quantum systems relative to this general framework.

CPAA

We investigate the spectral properties of Schrödinger operators
in $l^2(Z)$ with limit-periodic potentials. The perspective we
take was recently proposed by Avila and is based on regarding such
potentials as generated by continuous sampling along the orbits of
a minimal translation of a Cantor group. This point of view allows
one to separate the base dynamics and the sampling function. We
show that for any such base dynamics, the spectrum is of positive
Lebesgue measure and purely absolutely continuous for a dense set
of sampling functions, and it is of zero Lebesgue measure and
purely singular continuous for a dense $G_\delta$ set of sampling
functions.

JMD

We discuss discrete one-dimensional Schrödinger operators whose
potentials are generated by an invertible ergodic transformation
of a compact metric space and a continuous real-valued sampling
function. We pay particular attention to the case where the
transformation is a minimal interval-exchange transformation.
Results about the spectral type of these operators are
established. In particular, we provide the first examples of
transformations for which the associated Schrödinger operators
have purely singular spectrum for every nonconstant continuous
sampling function.

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