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.

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# Your search: "author:"Gorodetski, A""

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## Scholarly Works (26 results)

We obtain the asymptotics of the optimal global Hölder exponent of the integrated density of states of the Fibonacci Hamiltonian for large and small couplings. © 2013 Springer-Verlag Berlin Heidelberg.

We prove estimates for the transport exponents associated with the weakly coupled Fibonacci Hamiltonian. It follows in particular that the upper transport exponent (Formula presented.) approaches the value one as the coupling goes to zero. Moreover, for sufficiently small coupling, (Formula presented.) strictly exceeds the fractal dimension of the spectrum. © 2014 Hebrew University of Jerusalem.

© 2016, Springer-Verlag Berlin Heidelberg. We study the synchronization properties of the random double rotations on tori. We give a criterion that show when synchronization is present in the case of random double rotations on the circle and prove that it is always absent in dimensions two and higher.

© 2016 Elsevier Inc. We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented. On the one hand, we show that the Schrödinger operators on ℓ2(Z) with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on ℓ2(Z). More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on ℓ2(Z). On the other hand, we show that Schrödinger operators on ℓ2(Z) with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasi-periodic potentials with ‖·‖∞ norm exceeding 2, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.

© Instytut Matematyczny PAN, 2018. For a compact set K⊂ℝ1 and a family {Cλ}λϵJ of dynamically defined Cantor sets sufficiently close to affine with dimH K + dimH Cλ > 1 for all λ ϵ J, under natural technical conditions we prove that the sum K+C\ 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 1.

© 2015, Springer-Verlag Berlin Heidelberg. We consider the spectrum of discrete Schrödinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.

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.

© 2017 American Mathematical Society. We show that the space of hyperbolic ergodic measures of a given index supported on an isolated homoclinic class is path connected and entropy dense provided that any two hyperbolic periodic points in this class are ho-moclinically related. As a corollary we obtain that the closure of this space is also path connected.

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.