We show that under natural technical conditions, the sum of a $C^2$
dynamically defined Cantor set with a compact set in most cases (for almost all
parameters) has positive Lebesgue measure, provided that the sum of the
Hausdorff dimensions of these sets exceeds one. As an application, we show that
for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive
Lebesgue measure, while at the same time the density of states measure is
purely singular.

The spectra of tridiagonal square Fibonacci Hamiltonians, which are two-dimensional quasicrystal models, are given by sums of two Cantor sets, and the spectrum of the Labyrinth model, which is another two-dimensional quasicrystal model, is given by products of two Cantor sets. We consider spectral properties of those models and also obtain the optimal estimates in terms of thickness that guarantee that products of two Cantor sets is an interval.

We consider Jacobi matrices with zero diagonal and off-diagonals given by
elements of the hull of the Fibonacci sequence and show that the spectrum has
zero Lebesgue measure and all spectral measures are purely singular continuous.
In addition, if the two hopping parameters are distinct but sufficiently close
to each other, we show that the spectrum is a dynamically defined Cantor set,
which has a variety of consequences for its local and global fractal dimension.

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.

We consider random products of $SL(2, \mathbb{R})$ matrices that depend on a
parameter in a non-uniformly hyperbolic regime. We show that if the dependence
on the parameter is monotone then almost surely the random product has upper
(limsup) Lyapunov exponent that is equal to the value prescribed by the
Furstenberg Theorem (and hence positive) for all parameters, but the lower
(liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set of
parameters of zero Hausdorff dimension. As a byproduct of our methods, we
provide a purely geometrical proof of Spectral Anderson Localization for
discrete Schr\"odinger operators with random potentials (including the
Anderson-Bernoulli model) on a one dimensional lattice.

For a compact set K ⊂ R1 and a family {Cλ}λ∈J of dynamically defined Cantor sets sufficiently close to affine with dimH K +dimH Cλ > 1 for all λ ∈ J, we prove that the sum K+Cλ has positive Lebesgue measure for almost all values of the parameter λ. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.

We construct multidimensional Schr\"odinger operators with a spectrum that
has no gaps at high energies and that is nowhere dense at low energies. This
gives the first example for which this widely expected topological structure of
the spectrum in the class of uniformly recurrent Schr\"odinger operators,
namely the coexistence of a half-line and a Cantor-type structure, can be
confirmed. Our construction uses Schr\"odinger operators with separable
potentials that decompose into one-dimensional potentials generated by the
Fibonacci sequence and relies on the study of such operators via the trace map
and the Fricke-Vogt invariant. To show that the spectrum contains a half-line,
we prove an abstract Bethe--Sommerfeld criterion for sums of Cantor sets which
may be of independent interest.

We consider a minimal action of a finitely generated semigroup by
homeomorphisms of a circle, and show that the collection of translation numbers
of individual elements completely determines the set of generators (up to a
common continuous change of coordinates). One of the main tools used in the
proof is the synchronization properties of random dynamics of circle
homeomorphisms: Antonov's theorem and its corollaries.