Anton Gorodetski | University of California, Irvine (original) (raw)

Papers by Anton Gorodetski

Journal of spectral theory, Oct 22, 2018

We study the spectrum and the density of states measure of the square Fibonacci Hamiltonian. We d... more We study the spectrum and the density of states measure of the square Fibonacci Hamiltonian. We describe where the transitions from positive-measure to zero-measure spectrum and from absolutely continuous to singular density of states measure occur. This shows in particular that for almost every parameter from some open set, a positive-measure spectrum and a singular density of states measure coexist. This provides the first physically relevant example exhibiting this phenomenon.

Funkcionalʹnyj analiz i ego priloženiâ, 1999

Nonlinearity, Dec 5, 2008

We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the in... more We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of the Fibonacci Hamiltonian all coincide and are smooth functions of the coupling constant.

Communications in Mathematical Physics, Oct 29, 2011

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface... more 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 Ω of full Hausdorff dimension. The set Ω 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.

Communications in Mathematical Physics, Mar 12, 2011

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. ... more 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 prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for a 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. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents. Contents

Contemporary mathematics, 2017

We show that the space of hyperbolic ergodic measures of a given index supported on an isolated h... more 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 homoclinically related. As a corollary we obtain that the closure of this space is also path connected.

Duke Mathematical Journal, Jun 1, 2015

We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely co... more We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.

Annales Henri Poincaré, Jun 21, 2013

We study Schrödinger operators on the real line whose potentials are generated by an underlying e... more We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik-Lenz and Klassert-Lenz-Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible. We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke-Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.

Ergodic Theory and Dynamical Systems, Feb 3, 2009

We prove that there is a residual subset S in Diff 1 (M) such that, for every f ∈ S, any homoclin... more We prove that there is a residual subset S in Diff 1 (M) such that, for every f ∈ S, any homoclinic class of f containing saddles of different indices (dimension of the unstable bundle) contains also an uncountable support of an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of f .

arXiv (Cornell University), May 8, 2023

We prove a version of pointwise Ergodic Theorem for nonstationary random dynamical systems. Also,... more We prove a version of pointwise Ergodic Theorem for nonstationary random dynamical systems. Also, we discuss two specific examples where the result is applicable: non-stationary iterated function systems and non-stationary random matrix products.

arXiv (Cornell University), Jun 24, 2012

We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small v... more We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V , this measure is exact-dimensional and the almost everywhere value d V of the local scaling exponent is a smooth function of V , is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as V tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V-dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.

arXiv (Cornell University), Jul 3, 2013

We prove estimates for the transport exponents associated with the weakly coupled Fibonacci Hamil... more We prove estimates for the transport exponents associated with the weakly coupled Fibonacci Hamiltonian. It follows in particular that the upper transport exponentα ± u approaches the value one as the coupling goes to zero. Moreover, for sufficiently small coupling,α ± u strictly exceeds the fractal dimension of the spectrum.

Communications in Mathematical Physics, Feb 24, 2015

We consider the spectrum of discrete Schrödinger operators with Sturmian potentials and show that... more 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.

Annales Henri Poincaré, Jan 25, 2019

We construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower... more We construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower box counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.

Advances in Mathematics, Feb 1, 2021

Abstract We consider random products of S L ( 2 , R ) matrices that depend on a parameter in a no... more Abstract We consider random products of S L ( 2 , 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 δ 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 Schrodinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

Communications in Mathematical Physics, Mar 4, 2008

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fract... more We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as λ → ∞, dim(σ(H λ)) • log λ converges to an explicit constant (≈ 0.88137). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.

arXiv (Cornell University), May 8, 2015

arXiv (Cornell University), Jul 18, 2008

We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of... more 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.

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arXiv (Cornell University), Sep 3, 2018

We consider random products of SL(2, R) matrices that depend on a parameter in a non-uniformly hy... more We consider random products of SL(2, 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 δ 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 Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

arXiv (Cornell University), Jan 7, 2016

We show that under natural technical conditions, the sum of a C 2 dynamically defined Cantor set ... more 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.