Wednesday, February 12, 2014

Fibonacci Hamiltonian fractual dim The nonlinear Schrödinger equation is given by replacing the on-site Fibonacci potential with a random potential

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Nov 8, 2012 - The nonlinear Schrödinger equation is given by replacing the on-site Fibonacci potential with a random potential in (1). As is well known, all the ...
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    ANTON GORODETSKI

    www.math.uci.edu/.../Gorodetski%20rese...
    University of California, Ir...
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    Schrödinger operator with Fibonacci potential, the so-called Fibonacci Hamiltonian, obtained by methods of the modern theory of dynamical systems (uniformly ...

  • |❡s❡❛r❝* st❛t❡♠❡♥t

    *✇✇✳♠❛t❤✳✉❝✐✳❡*✉✴ *s❣♦r

    Here I mention only results that were published or obtained during the period July 1, 2007 –

    September 30, 2010. A brief description of results completed before 2007 can be found on my

    webpage ( *✇✇✳♠❛t❤✳✉❝✐✳❡*✉✴ *s❣♦r ).

    *♣*❝tr❛❧ *r♦♣*rt✐❡s *❢ t❤❡ **s❝r❡t❡ *❝*rö*✐♥❣❡r *♣*r*t♦r *✐t❤ *✐❜*♥❛❝❝✐ **t❡♥t✐*❧

    It is always exciting to obtain a new connection between two different branches of mathematics.

    Here I describe a series of new results concerning the spectral properties of the discrete

    Schrödinger operator with Fibonacci potential, the so-called Fibonacci Hamiltonian, obtained by

    methods of the modern theory of dynamical systems (uniformly hyperbolic and normally hyperbolic

    dynamics).

    The Fibonacci Hamiltonian is a central model in the study of electronic properties of onedimensional

    quasicrystals. It is given by the following bounded self-adjoint operator in `

    where V > 0,
    =

    [H

    V;!

    ](n) = (n + 1) + (n 1) + V

    p

    51

    2

    [1
    ;1)

    (n
    + ! mod 1) (n);

    , and ! 2 T = R=Z. The spectrum is easily seen to be independent of !

    and may therefore be denoted by

    V

    . It is known that

    is a zero-measure Cantor set for every

    V 6 = 0.

    V

    Naturally, one is interested in quantitative fractal properties of

    , such as its dimension,

    thickness, and denseness. While such a study is well-motivated from a purely mathematical perspective,

    there is also significant additional interest in these quantities. In particular, in a paper

    [1] joint with Damanik, Embree and Tcheremchantsev we show that t❤❡ *r*❝t❛❧ *✐♠❡♥s✐♦♥ *❢ t❤❡

    s♣**tr✉♠ *s *♥t✐♠❛t❡❧② r*❧❛t❡* *✐t❤ t❤❡ *♦♥❣✲t✐♠❡ *s②♠♣t♦t✐❝s *❢ t*❡ s♦❧✉t✐♦♥ t♦ t❤❡ *ss♦*✐❛t❡* t✐♠❡✲

    *❡♣*♥❞❡♥t *❝❤rö*✐♥❣❡r *q✉❛t✐♦♥✱ t❤❛t *s✱ i@

    t

    - = H

    -*

    Results by Casdagli and Süt*o show that for V 16,

    V;!

    V

    V

    is a dynamically defined Cantor

    set. This implies that the Hausdorff dimension and the upper and lower box counting dimension

    of

    V

    all coincide; let us denote this common value by dim

    . Using this result, in [1] we have

    shown upper and lower bounds for the dimension. A particular consequence of these bounds is

    the identification of the asymptotic behavior of the dimension as V tends to infinity:

    lim

    V !1

    dim

    V

    log V = log(1 +

    p

    2):

    V

    This represents the *nal word in a topic with a long and reach history. One of the ingredients

    of the proof is application of results well known in hyperbolic dynamics to the trace map of the

    Fibonacci Hamiltonian.

    In a joint paper with Damanik [2] we also prove that t❤❡ s♣**tr✉♠

    *❢ t❤❡ *✐❜*♥❛❝*✐ *❛♠✐❧✲

    t♦♥✐❛♥ H

    V

    V

    *s * *②♥❛♠✐❝*❧*② *❡✜♥❡* *❛♥t♦r s❡t *♦r s♠❛❧* *♥♦✉❣❤ *❛❧✉❡s *❢ t❤❡ **✉♣❧✐♥❣ **♥st❛♥t,

    and obtain some new properties of the spectrum (in particular, smooth dependence of the Hausdorff

    dimension of the spectrum on coupling constant) for these values of V . In [3] we study the

    spectrum and the spectral type of the off-diagonal Fibonacci operator and obtain similar results

    *

    *❤❡ *♦♥t❡♥t *❢ *✸❪ *s *✉❜❧✐s❤❡❞ *s *♥ *♣♣*♥❞✐① t♦ *✺❪✱ *♦t *s * s❡♣❛r❛t❡ *❛♣*r✳

    2

    (Z),

    *

    .

    *✐❣✉r❡ *✿ *❤❡ s❡t f(E; V ) : E 2

    Clearly, as V approaches zero, H

    approaches the free Schrödinger operator

    [H

    0

    V;!

    ](n) = (n + 1) + (n 1);

    which is a well-studied object whose spectral properties are completely understood. In particular,

    is given by the interval [2; 2]. Lebesgue measure of the spectrum does not

    extend continuously to the case V = 0. Given this situation, one would at least hope that the

    dimension of the spectrum is continuous at V = 0. In [4, 5] we prove that t❤❡ t❤✐❝❦♥❡ss t❡♥❞s t♦

    *♥✜♥✐t② *♥❞✱ *❡♥❝*✱ t❤❡ *❛✉s❞♦r✛ *✐♠❡♥s✐♦♥ *❢ t❤❡ s♣**tr✉♠ t❡♥❞s t♦ *♥❡✱ lim

    the spectrum of H

    0

    = 1*

    Also, it is natural to ask about the size of the gaps in the spectrum

    V !0

    , which can in fact

    be parametrized by a canonical countable set of gap labels. These gap openings were studied by

    Bellissard for a Thue-Morse potential and by Bellissard-Bovier-Ghez for period doubling potential.

    However, for the important Fibonacci case, the problem remained open. In fact, Bovier and Ghez

    remarked: It is a quite perplexing feature that even in the simplest case of all, the golden Fibonacci sequence,

    the opening of the gaps at small coupling is not known! In [5] we prove that *t s♠❛❧* **✉♣❧✐♥❣✱ *❧* *❛*s

    *❧*♦✇❡* *② t❤❡ *❛♣ *❛❜*❧✐♥❣ t❤❡*r*♠ *r* *♣*♥ *♥❞ t❤❡ *❡♥❣t❤ *❢ *✈❡r② *❛♣ t❡♥❞s t♦ *❡r* *✐♥❡*r❧②, see

    Figure 1.

    ; 0 V 2g*

    Next result concerns the sum set

    V

    +

    V

    = fE

    1

    V

    + E

    2

    : E

    1

    ; E

    2

    2

    V

    V

    g: This set is equal to

    the spectrum of the so-called square Fibonacci Hamiltonian. Here, one considers the Schrödinger

    operator

    in `

    2

    (Z

    2

    [H

    (2)

    V

    ](m; n) = (m + 1; n) + (m 1; n) + (m; n + 1) + (m; n 1)+

    + V



    [1
    ;1)

    (m
    mod 1) +

    [1
    ;1)

    (n
    mod 1)



    (m; n)

    ). This operator and its spectrum have been studied numerically and heuristically by

    Even-Dar Mandel and Lifshitz (a similar model was studied by Sire). Their study suggested that

    at small coupling, the spectrum

    V

    +

    is not a Cantor set; quite on the contrary, it has no gaps

    at all. In [5] we confirm this observation and prove rigourously that *♦r s✉✣❝✐❡♥t❧② s♠❛❧* **✉♣❧✐♥❣✱

    t❤❡ s✉♠

    V

    +

    V

    V

    *s *♥ *♥t❡r✈❛❧✳ Certainly, the same statement holds for the cubic Fibonacci

    Hamiltonian (i.e., the analogously defined Schrödinger operator in `

    2

    (Z

    3

    *✐♠

    H



    V

    ) with spectrum

    +



    V

    +

    ).

    Besides, in [5] we prove a version of the Dry Ten Martini Problem for Fibonacci Hamilto-

    V

    nian (i.e. we prove that all labels given by the gap labeling theorem correspond to gaps in the

    spectrum), describe the rate of the linear gap opening in terms of labels provided by the gap labeling

    theorem, give explicit upper and lower bounds for the solutions to the associated difference

    equation, and use them to study the spectral measures and the transport exponents.

    V

    In our current work in progress with Damanik we intend to show that as coupling constant

    increases, the spectrum of the square Fibonacci Hamiltonian bifurcates from an interval to the so

    called cantorval (compact set with dense interior and without isolated connected components).

    That will give a new type of spectrum for "natural" potentials.

    *♦♥s❡r✈*t*** *♦♠♦*❧✐♥✐❝ *✐❢✉r❝❛t✐♦♥s *♥❞ **♣*r❜*❧✐❝ s*ts *❢ *❛r❣❡ *❛✉s❞♦r✛ *✐♠❡♥s✐♦♥

    In the case of dissipative dynamical systems on surfaces homoclinic bifurcations were in-

    tensively investigated; some of the dynamical phenomena that appear after a bifurcation in this

    case are persistent tangencies and infinite number of sinks (Newhose phenomena), strange attractors

    (Mora, Viana), arbitrarily degenerate periodic points of arbitrary high periods (Gonchenko,

    Shilnikov, Turaev), and superexponential growth of periodic orbits (Kaloshin).

    The conservative (area preserving) case is known to be more complicated. For example,

    it took over two decades to prove an analog of Newhouse results for area preserving surface

    diffeomorphisms (Duarte). I proved that *♦**❧*② *❛①✐♠❛❧ *②♣*r❜*❧✐❝ s*ts *❢ *❛✉s❞♦r✛ *✐♠❡♥s✐♦♥

    *r❜✐tr*r② *❧♦s❡ t♦ t✇♦ *♣♣**r *❢t❡r * *❡♥❡r✐❝ *♥❡✲♣*r*♠❡t❡r *♥❢♦❧❞✐♥❣ *❢ * *♦♠♦*❧✐♥*❝ t❛♥❣❡♥❝②.

    This fact has numerous applications, two of them are presented below.

    The results described in this section and in two sections after that will be published in a

    series of papers. The *rst paper of the series [6] is published, and the second one [7] is currently

    submitted for a publication.

    *♥ t❤❡ s✐③❡ *❢ t❤❡ st♦**❛st✐❝ *❛**r *❢ t❤❡ st❛♥❞❛r❞ *❛♣

    The KAM theorem on the conservation of quasiperiodic motions in near-integrable Hamil-

    tonian systems gave rise to the question on dynamical behavior in the regions where invariant

    tori are destroyed. In a more general form this (open) question can be stated in the following

    way: "Can an analytic symplectic map have a chaotic component of positive measure and the

    Kolmogorov-Arnold-Moser (KAM) tori coexist? "

    The simplest and most famous system where one would expect *✐①❡* **❤❛✈✐♦r (KAM tori

    and orbits with non-zero Lyapunov exponents (stochastic sea) both have positive measure) is the

    Taylor-Chirikov standard map of the two–dimensional torus T

    , given by

    f

    k

    2

    (x; y) = (x + y + k sin(2 x); y + k sin(2 x)) *♦* Z

    2

    : *✶✮

    I proved that st♦*❤❛st✐❝ *❛②❡r *❢ t❤❡ st❛♥❞❛r* *❛♣ *❛s *✉❧* *❛✉s❞♦r✛ *✐♠❡♥s✐♦♥ *♦r *❛r*❡

    **r*♠❡t❡rs *r*♠ * r*s✐❞✉❛❧ s❡t *♥ t❤❡ s♣*❝* *❢ **r*♠❡t❡rs✱ see [6, 7] for the precise statement.

    Notice that this result gives a partial explanation of the dif*culties that we encounter study-

    ing the standard family. Indeed, one of the possible approaches is to consider an invariant hyperbolic

    set in the stochastic layer and to try to extend the hyperbolic behavior to a larger part of the

    phase space through homoclinic bifurcations. Unavoidably Newhouse domains associated with

    absence of hyperbolicity appear after small change of the parameter. If the Hausdorff dimension

    of the initial hyperbolic set is less than one, then the measure of the set of parameters that correspond

    to Newhouse domains is small and has zero density at the critical value, as was shown by

    Newhose-Palis and Palis-Takens. For the case when the Hausdorff dimension of the hyperbolic

    set is slightly bigger than one, similar result was recently obtained by Palis and Yoccoz, and the

    proof is astonishingly involved. They also conjectured that analogous property holds for an initial

    hyperbolic set of any Hausdorff dimension, but the proof would require even more technical and

    complicated considerations. Here is what Palis and Yoccoz wrote:

    *❖❢ **✉rs❡✱ *❡ *①♣**t t*❡ s❛♠❡ t♦ ** tr✉❡ *♦r *❧* **s❡s 0 < *✐♠

    ( ) < 2* **r t❤❛t✱ *t s❡*♠s

    t♦ *s t❤❛t *✉r *❡t❤♦*s *❡** t♦ ** **♥s✐❞❡r*❜❧② s❤❛r♣*♥❡*✿ *❡ *❛✈❡ t♦ st✉❞② ***♣*r t❤❡ *②♥❛♠✐❝*❧

    r**✉rr*♥❝* *❢ **✐♥ts *❡*r t❛♥❣❡♥❝✐❡s *❢ *✐❣❤❡r *r*❡r *❝✉❜✐❝✱ q✉❛rt✐❝✱ *✳✳✮ **t*❡*♥ st❛❜❧❡ *♥* *♥st❛❜❧❡

    *✉r✈❡s✳ ** *❧s♦ *♦♣* t❤❛t t❤❡ *❞❡*s *♥tr**✉❝** *♥ t❤❡ *r*s❡♥t **♣*r *✐❣❤t ** *s❡❢✉❧ *♥ *r**❞❡r

    H

    **♥t❡①ts✳ ¦♥ t❤❡ *♦r✐③♦♥ *✐❡s t** *❛♠♦✉s q✉❡st✐♦♥ *❤❡t❤❡r *♦r t❤❡ st❛♥❞❛r* *❛♠✐❧② *❢ *r** *r*s❡r✈✐♥❣

    *❛♣s *♥❡ **♥ *♥❞ s❡ts *❢ **s✐t✐✈❡ **❜*s❣✉❡ *r*❜*❜✐❧✐t② *♥ **r*♠❡t❡r s♣*❝* s✉❝❤ t❤❛t t❤❡ **rr*s♣*♥❞✐♥❣

    *❛♣s *✐s♣❧❛② *♦♥✲③❡r* *②❛♣✉♥♦✈ *①♣*♥❡♥ts *♥ s❡ts *❢ **s✐t✐✈❡ **❜*s❣✉❡ *r*❜*❜✐❧✐t② *♥ *❤*s* s♣*❝*✳✑

    The result stated above shows that in order to understand the dynamics of the stochastic

    layer of the standard map one has to face these dif*culties.

    *♦♠♦*❧✐♥✐❝ *✐❢✉r❝❛t✐♦♥s *♥ s♦♠❡ r❡str✐❝t❡❞ **rs✐♦♥s *❢ t❤❡ t❤r❡❡ ***② *r♦❜❧❡♠s

    Initially my interest in the conservative bifurcations was motivated by the fact that it appears

    in the three body problem. The classical three–body problem consists in studying the dynamics of

    3 point masses in the plane or in the three-dimensional space mutually attracted under Newton

    gravitation. The three–body problem is called r*str✐❝t❡* if one of the bodies has mass zero and the

    other two are strictly positive. In his pioneering work Alexeev found important use of hyperbolic

    dynamics for the three–body problem. He proved existence of the so called *s❝✐❧*❛t♦r② *♦t✐♦♥s. A

    motion of the three–body problem is called *s❝✐❧*❛t♦r② if the limsup of the mutual distances is infinite

    and the liminf is *nite. Existence of such motions was a long standing open problem. The *rst

    rigorous example of existence of such motions is due to Sitnikov for the restricted spacial three–

    body problem. Alexeev extended the Sitnikov example to the spatial three–body problem. Later

    Moser gave a conceptually transparent proof of existence of oscillatory motions for the Sitnikov

    example interpreting homoclinic intersections. This paved a road to a variety of applications of

    hyperbolic dynamics to the three–body problem.

    A famous open conjecture (that goes back to Kolmogorov or even Chazy) claims that the set

    of initial conditions that correspond to oscillatory motions has zero measure. In our joint work

    with Kaloshin we show that *♥ *✐t**❦♦✈ *r*❜❧❡♠ *♥❞ *♥ t❤❡ r*str✐❝t❡* *❧❛♥❛r *✐r*✉❧❛r t❤r** ***②

    *r*❜❧❡♠ *♥✈❛r✐❛♥t *②♣*r❜*❧✐❝ s❡ts *❢ *❛r*❡ **❧♦s❡ t♦ t❤❡ *✐♠❡♥s✐♦♥ *❢ t❤❡ *❤❛s❡ s♣*❝*✮ *❛✉s❞♦r✛

    *✐♠❡♥s✐♦♥ *♣♣**r *♦r *❛♥② *❛❧✉❡s *❢ t❤❡ **r*♠❡t❡r. Then we apply this result to show that in the

    considered cases t❤❡ s❡t *❢ *s❝✐❧*❛t♦r② *♦t✐♦♥s *❛s *✉❧* *❛✉s❞♦r✛ *✐♠❡♥s✐♦♥ *♦r *❛♥② *❛❧✉❡s *❢ t❤❡

    **r*♠❡t❡r, see [6, 8] for precise statements.

    *♦♥✲❤*♣*r❜*❧✐❝ *♥**r✐❛♥t *❡❛s✉r❡s *♦r *♦♥✲❤*♣*r❜*❧✐❝ *♦♠♦*❧✐♥✐❝ *❧❛ss❡s

    To what extent is the behavior of a generic dynamical system hyperbolic? A number of

    problems in modern theory of smooth dynamical systems can be viewed as some forms of this

    question. It was shown by Abraham and Smale in the 1960s that uniformly hyperbolic systems

    (Anosov diffeomorphisms, Axiom A) are not dense in the space of dynamical systems. This forced

    weakening the notion of hyperbolicity and gave rise to notions of partial and nonuniform hyperbolicity.

    Some systems of these types are studied by famous Pesin theory, where hyperbolic behavior

    is characterized by nonzero Lyapunov exponents for some invariant measure. The most

    natural case is that of a system with a smooth invariant measure. However, the question about

    Lyapunov exponents can also be considered for maps that do not a priori have a natural invariant

    measure. In particular, it is reasonable to conjecture that *♥✐❢♦r♠❧② *②♣*r❜*❧✐❝ *❛♣s t♦*❡t*❡r

    *✐t❤ *✐✛❡*♠♦r♣❤✐s♠s *①**❜✐t**❣ * *♦♥✲*②**r❜*❧✐❝ *r*♦*✐❝ *♥✈❛r✐❛♥t *❡*s✉r* *♦r♠ *♥ *♣*♥ *♥* *❡♥s❡

    s✉❜s❡t *♥ t❤❡ s♣*❝* *❢ s♠♦*t❤ *②♥❛♠✐❝*❧ s②st❡♠s✳ In [9] we prove that t❤✐s *s tr✉❡ *♦r * *❡♥❡r✐❝

    *✐✛❡*♠♦r♣❤✐s♠ *①❤✐❜✐t✐♥❣ * *♦♠♦*❧✐♥✐❝ *❧❛ss *✐t❤ s❛❞❞*❡s *❢ *✐✛*r*♥t *♥❞✐❝*s✳

    The conjecture above provides a nice description of the space of smooth dynamical systems

    on ergodic level in a spirit of the famous Palis’ Conjectures. In our joint paper with Diaz and

    Bonatti [10] we prove that *♥ * **rt✐❛❧*② *②♣*r❜*❧✐❝ **s❡ *✇✐t❤ *♥❡✲❞*♠*♥s*♦*❛❧ **♥tr*❧ *✉♥❞*❡✮ *♥❡

    **♥ *✉❛r*♥t❡* t❤❛t t❤❡ s✉♣♣*rt *❢ t❤❡ **♥str✉❝t❡* *♦♥✲❤②♣*r❜*❧✐❝ *❡*s✉r* *s t❤❡ *❤♦❧❡ *♦♠♦*❧✐♥✐❝

    *❧❛ss✳ In particular, this explains the dif*culties that appear when one is trying to investigate such

    systems numerically.

    *②♥❛♠✐❝❛❧ *r♦♣*rt✐❡s *❢ *✐❡❝❡✇✐s❡ *s♦♠❡tr✐❡s

    Consider a ball in R

    . Cut the ball into several pieces, and shift each piece in such a way

    that the image is still inside of the initial ball. Certainly, images of some pieces may overlap. This

    simple construction defines a (discontinuous) dynamical system with highly non-trivial behavior.

    Interestingly enough, very little is known about these piecewise isometries. A class of systems

    of this kind appeared in a natural way in some problems related to machine learning, due to

    the approach suggested by Max Welling. Currently this promising direction of research is in the

    beginning stage. Recently we received an NSF grant

    n

    (joint with Max Welling) to study these

    systems in details and to apply the results to machine learning.

    |❡❢❡r❡♥❝❡s

    *

    [1] D. Damanik, M. Embree, A. Gorodetski, S. Tcheremchantsev, The Fractal Dimension of the

    Spectrum of the Fibonacci Hamiltonian, *♦♠♠✉♥✐❝*t✐♦♥s *♥ *❛t❤❡♠❛t✐❝*❧ P❤②s✐❝s, vol. 280

    (2008), no. 2, pp. 499-516.

    [2] D. Damanik, A. Gorodetski, Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci

    Hamiltonian, *♦♥❧✐♥❡*r✐t②✱ vol. 22 (2009), pp. 123–143.

    [3] D. Damanik, A. Gorodetski, The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci

    Operator, 10 pages, arXiv:0807.3024v2.

    [4] D. Damanik, A. Gorodetski, The Spectrum of the Weakly Coupled Fibonacci Hamiltonian,

    *❧❡*tr*♥✐❝ |*s❡*r*❤ **♥♦✉♥❝*♠❡♥ts *♥ *❛t❤❡♠❛t✐❝*❧ *❝✐❡♥❝*s, vol. 16 (2009), pp. 23–29.

    [5] D. Damanik, A. Gorodetski, Spectral and Quantum Dynamical Properties of the Weakly Coupled

    Fibonacci Hamiltonian, 53 pages, arXiv:1001.2552v1, to appear in Communications in

    Mathematical Physics.

    [6] A. Gorodetski, V. Kaloshin, Conservative homoclinic bifurcations and some applications, Pr*✲

    ****✐♥❣s *❢ t❤❡ *t❡❦❧♦✈ ¦♥st✐t✉t❡ ** *❛t❤❡♠❛t✐❝s, vol. 267 (2009) (volume dedicated to the 70th

    anniversary of Vladimir Arnold), pp. 76–90.

    [7] A. Gorodetski, On stochastic sea of the standard map, 36 pages, arXiv:1007.1474v1, submitted.

    [8] A. Gorodetski, V. Kaloshin, On Hausdorff dimension of oscillatory motions in Sitnikov problem,

    preprint, http : ==math:uci:edu= asgor=Sitnikov:pdf

    [9] L. Diaz, A. Gorodetski, Non-hyperbolic ergodic measures for non-hyperbolic homoclinic

    classes, *r*♦*✐❝ *❤❡*r② *♥❞ *②♥❛♠✐❝*❧ *②st❡♠s, vol. 29 (2009), pp. 1479-1513.

    [10] Ch. Bonatti, L. Diaz, A. Gorodetski, Non-hyperbolic ergodic measures with large support,

    *♦♥❧✐♥❡*r✐t②, vol. 23 (2010), no. 3, pp. 687–705


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