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Substitution Delone Sets
This paper addresses the problem of describing aperiodic discrete structures
that have a self-similar or self-affine structure. Substitution Delone set
families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an
inflation functional equation under the action of an expanding integer matrix
in R^d. This paper studies such functional equation in which each X_i is a
discrete multiset (a set whose elements are counted with a finite
multiplicity). It gives necessary conditions on the coefficients of the
functional equation for discrete solutions to exist. It treats the case where
the equation has Delone set solutions. Finally, it gives a large set of
examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems
reformulate
On stochastic sea of the standard map
Consider a generic one-parameter unfolding of a homoclinic tangency of an
area preserving surface diffeomorphism. We show that for many parameters
(residual subset in an open set approaching the critical value) the
corresponding diffeomorphism has a transitive invariant set of full
Hausdorff dimension. The set is a topological limit of hyperbolic sets
and is accumulated by elliptic islands.
As an application we prove that stochastic sea of the standard map has full
Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure
Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian
We show that under natural technical conditions, the sum of a
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.Comment: 13 page
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