Asymptotic Analysis of the Spectrum of the Discrete...

Period Doubling HamiltonianThe Spectrum

Methods and Results

Asymptotic Analysis of the Spectrum of theDiscrete Hamiltonian with Period Doubling

Potential

Meg Fields, Tara Hudson, Maria Markovich

Cornell Summer Math Institute 2013

August 2, 2013

Within ε of Awesome Spectrum of the Period Doubling Hamiltonian

Methods and Results

MotivationQuasi-Crystals

The discovery of quasi-crystals in the 1980’s has provoked manynew questions in physics and mathematics [1].

Quasi-crystals are materials that share some properties withcrystals, but have aperiodic lattices.

Mathematically, we can model quasi-crystals using one-dimensionalHamiltonian operators with aperiodic potentials [1], [2], [5].

Fractal properties of the spectra of these operators affectquantum diffusion patterns of electrons in quasi-crystallinematerials [7], [4], [8].

Methods and Results

Period Doubling SequenceDefining the Operator

The Period Doubling Sequence

The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA

S1(A) = A BS2(A) =

A B A A

S3(A) = A B A A A B A B

To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter

...B A B A A A B A A B A A A B A B...

For computational purposes, we define A = 1 and B = −1.

Methods and Results

S1(A) = A B

S2(A) =

A B A A

Methods and Results

S1(A) = A BS2(A) = A B

A AS3(A) = A B A A A B A B

Methods and Results

S1(A) = A BS2(A) = A B A A

Methods and Results

S1(A) = A BS2(A) = A B A AS3(A) = A B A A A B A B

Methods and Results

...B A B A A A B A

A B A A A B A B...

Methods and Results

Defining the Operator

The Hamiltonian operator (HV ) with period doubling potential is:

HV = −∆ + V

Where,

∆ is the discrete Laplacian of Z,

0 −1 0 0 . . .−1 0 −1 0 . . .0 −1 0 −1 . . ....

......

.... . .

V is a multiplication operator which inputs the perioddoubling sequence into the main diagonal.

This operator acts on a sequence in `2(Z) [2].

Methods and Results

Defining the Operator

The Hamiltonian operator (HV ) with period doubling potential is:

HV = −∆ + V

Where,

∆ is the discrete Laplacian of Z,

0 −1 0 0 . . .−1 0 −1 0 . . .0 −1 0 −1 . . ....

......

.... . .

V is a multiplication operator which inputs the perioddoubling sequence into the main diagonal.

This operator acts on a sequence in `2(Z) [2].Within ε of Awesome Spectrum of the Period Doubling Hamiltonian

Methods and Results

The Trace MapVisualizing the Spectrum

Methods of Approximating the Spectrum

The spectrum of an operator is a generalization of eigenvalues fora finite dimensional matrix.

To numerically approximate the spectrum of the Hamiltonian withperiod doubling potential, two methods will be employed:

1 Truncate the matrix representation of the Hamiltonian andfind the eigenvalues

2 Iterate a trace map and identify initial values which are notunstable

Methods and Results

Defining the Trace Map

The trace map, is a recursively defined function developed fromthe eigenvalue equation [2].

The map describes solutions to the equation in the following way:

Initial values will be defined in terms of E, V ∈ R.

For initial points that are not unstable, E will be a value inthe spectrum.

Methods and Results

The initial values for the map are:

x0 := E − V and y0 := E + V ,

for some E, V ∈ R.

The trace map is,xn+1 := xnyn − 2 and yn+1 := (xn)2 − 2.

Methods and Results

The initial values for the map are:

x0 := E − V and y0 := E + V ,

for some E, V ∈ R.

The trace map is,xn+1 := xnyn − 2 and yn+1 := (xn)2 − 2.

Methods and Results

Approximating the Spectrum

First iterate the trace map and find the unstable points.

Definition (Unstable from [2])

A point (x, y) ∈ R2 is unstable ifx = x0 = E − V, y = y0 = E + V, and there is some N such thatfor all n > N , |xn| > 2.

Define a region

A := {(x, y) | y > 2 , |x| > 2}.

Theorem (Bellisard, Bovier, Ghez)

A point (x, y) is unstable if and only if there exists an n such that(xn, yn) ∈ A.

Methods and Results

Define a region

A := {(x, y) | y > 2 , |x| > 2}.

Methods and Results

Define a region

A := {(x, y) | y > 2 , |x| > 2}.

Methods and Results

Let U be the set of unstable points.

E is in the spectrum of HV if and only if x = E − V andy = E + V are such that (x, y) ∈ UC .

We call V the coupling constant, a real valued parameter.

For a fixed coupling constant greater than zero, the spectrum ofthe Hamiltonian is a Cantor set of Lebesgue measure zero.

So it is natural to ask questions about the Hausdorff dimension,box-counting dimension, and thickness of the spectrum.

Methods and Results

−3 −2 −1 0 1 2 3−3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

Student Version of MATLAB

Top: A numerical approximation of UC .

Bottom: An approximation of the spectrum for a 0 ≤ V ≤ 1 [2].

Methods and Results

The Spectrum for a Fixed Coupling Constant

We are interested in the fractal properties of the spectrum as thecoupling constant approaches 0 [5].

For V = 0, the spectrum is the closed interval [−2, 2] ∈ R, and forV > 0, recall that the spectrum is a Cantor set.

To numerically approximate fractal measures for these sets, we usea MATLAB function crossSection to discretize the spectrum fora fixed V .

Methods and Results

Cross Sections of the Spectrum

The function crossSection iterates the trace map for initialpoints, x0 := E − V , y0 := E + V and outputs a representation ofthe spectrum.

We fix the value of V ∈ (0, 1) while varying E ∈ [−2, 2] to definedifferent initial points. Recall that if an initial point is notunstable, the corresponding E is in the spectrum.

After iterating the trace map, crossSection outputs a vector of0’s and 1’s, representing a cross section of the spectrum.

A value in the spectrum is represented by 1, while 0 denotes avalue not in the spectrum.

Methods and Results

Box-Counting DimensionThicknessHausdorff Dimension

Definition of Box-Counting Dimension

Definition (Box-Counting Dimension from [6])

The box-counting dimension of a Cantor set, K, is given by,

dimBK = limδ→0

logNδ(K)

− log δ,

if the limit exists, where Nδ(K) is the number of covers of lengthδ needed to cover the set.

Methods and Results

Using MATLAB to Approximate Box-Counting Dimension

Using the function crossSection we obtain a vectorrepresentation of the spectrum:

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

We then define a diameter δ and determine the minimum numberof δ-covers needed to cover the set.

In this discrete calculation, δ will correspond to a number ofentries from ~E to include in a cover.

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

Methods and Results

We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).

A δ-cover will include 2k × 2−i entries from ~E.

For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.

~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

The minimum number of δ-covers is 4.

Methods and Results

~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

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~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

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~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

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~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]

Methods and Results

For each δi, we determine Nδi(~E), the minimum number of

δi-covers needed to cover ~E.

We then plot the points (− log δi , logNδi(~E)) for each i and find

the slope of the regression line for these points.

This estimates the box-counting dimension, i.e.

limδ→0

logNδ(K)

− log δ

for the spectrum at the particular coupling constant we used togenerate ~E.

Methods and Results

limδ→0

logNδ(K)

− log δ

Methods and Results

limδ→0

logNδ(K)

− log δ

Methods and Results

Results for Box-Counting Dimension

We find that as coupling constant approaches zero, box-countingdimension approaches one.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

Coupling Constant, V

Box Counting Dimension as Coupling Constant Approaches Zero

Student Version of MATLAB

Methods and Results

ThicknessPreliminary Definitions

Let K ⊂ R be a Cantor set.

Definition (Bounded Gap from [9])

A bounded gap of K is a bounded connected component of R\K.

Definition (Bridge from [9])

Let U be any bounded gap of K and u be a boundary point of U ,that is u ∈ K. Then a bridge of K at u, denoted C, is themaximal interval in R such that

u is a boundary point of C, and

C contains no point of another gap U ′, whose length, `(U ′),is at least the length U (where length is defined in the usualway for R).

Methods and Results

ThicknessPreliminary Definitions

Let K ⊂ R be a Cantor set.

Definition (Bounded Gap from [9])

A bounded gap of K is a bounded connected component of R\K.

Definition (Bridge from [9])

Let U be any bounded gap of K and u be a boundary point of U ,that is u ∈ K. Then a bridge of K at u, denoted C, is themaximal interval in R such that

u is a boundary point of C, and

C contains no point of another gap U ′, whose length, `(U ′),is at least the length U (where length is defined in the usualway for R).

Methods and Results

Definition of Thickness

Define thickness of K at a particular u to be

τ(K,u) =`(C)

where `(C) is the length of the bridge and `(U) the length of itsrespective gap [9].

Definition (Thickness from [9])

The thickness of K, denoted τ(K) is defined as:

inf τ(K,u)

for all boundary points u of bounded gaps.

Methods and Results

Definition of Thickness

Define thickness of K at a particular u to be

τ(K,u) =`(C)

where `(C) is the length of the bridge and `(U) the length of itsrespective gap [9].

Definition (Thickness from [9])

The thickness of K, denoted τ(K) is defined as:

inf τ(K,u)

for all boundary points u of bounded gaps.

Methods and Results

Using MATLAB to Approximate Thickness

To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.

Using the function crossSection we obtain a representation ofthe spectrum,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [

2 2 1 3 3

interval = [ 1 2 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2

2 1 3 3

interval = [ 1 2 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2

interval = [ 1 2 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2 1

interval = [ 1 2 1 1 ]

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~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2 1 3 3 ]

interval = [ 1 2 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

To simplify the process, first consider the thickness of rightboundary points.

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

For Example,

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

where we calculate,

gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

We obtain the following:

gap = [ 2 2 1 3 3 ]bridge = [ 4 1 1 ]

To calculate the thickness, we divide the ith bridge length by the(i+ 1)th gap and take the minimum of these values.

For left boundary points, we reflect the sequence and repeat thesame process.

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2 1 3 3 ]bridge = [ 4 1 1 ]

Methods and Results

~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]

gap = [ 2 2 1 3 3 ]bridge = [ 4 1 1 ]

Methods and Results

Resulting Trends

We find that as coupling constant approaches zero, thicknessapproaches infinity,

Thickness

V n = 1000 n = 10000

0.03 0.0106 0.0116

0.003 0.0833 0.0125

0.0003 3.0 0.0909

0.00003 383 1.0

Issue: Theoretically, the calculation of thickness should improvewith a better approximation of the Cantor Set.

Methods and Results

Resulting Trends

We find that as coupling constant approaches zero, thicknessapproaches infinity,

Thickness

V n = 1000 n = 10000

0.03 0.0106 0.0116

0.003 0.0833 0.0125

0.0003 3.0 0.0909

0.00003 383 1.0

Issue: Theoretically, the calculation of thickness should improvewith a better approximation of the Cantor Set.

Methods and Results

Introduction to Hausdorff Dimension

Consider covers of a set U 6= ∅ ⊂ R

Definition (Diameter from [6])

The diameter of U , denoted |U |, is defined as:

|U | := sup{|x− y| : x, y ∈ U}

Take diameters 0 ≤ |Ui| ≤ δ.

Definition (Hausdorff Measure from [6])

Let K ⊂ R be a set, and let s > 0. We define Hausdorff Measure,Hs(K), in the following way:

Hs(K) := limδ→0

{ ∞∑i=1

|Ui|s : {Ui} is a δ cover

Methods and Results

Introduction to Hausdorff Dimension

Consider covers of a set U 6= ∅ ⊂ R

Definition (Diameter from [6])

The diameter of U , denoted |U |, is defined as:

|U | := sup{|x− y| : x, y ∈ U}

Take diameters 0 ≤ |Ui| ≤ δ.

Definition (Hausdorff Measure from [6])

Let K ⊂ R be a set, and let s > 0. We define Hausdorff Measure,Hs(K), in the following way:

Hs(K) := limδ→0

{ ∞∑i=1

|Ui|s : {Ui} is a δ cover

Methods and Results

Hausdorff Dimension

Definition (Hausdorff Dimension from [6])

Let K ⊂ R, s > 0. Hausdorff dimension is given by

dimH(K) := inf {s ≥ 0 : Hs(K) = 0} = sup {s : Hs(K) =∞} .

It is known that the following relationship between Hausdorffmeasure and Hausdorff dimension holds:

Hs(K) =

{∞ if 0 ≤ s < dimH K0 if s > dimH K

Methods and Results

Hausdorff Dimension

Definition (Hausdorff Dimension from [6])

Let K ⊂ R, s > 0. Hausdorff dimension is given by

dimH(K) := inf {s ≥ 0 : Hs(K) = 0} = sup {s : Hs(K) =∞} .

It is known that the following relationship between Hausdorffmeasure and Hausdorff dimension holds:

Hs(K) =

{∞ if 0 ≤ s < dimH K0 if s > dimH K

Methods and Results

Graph of Hausdorff Dimension

dimH(K)s

Methods and Results

Estimate of Hausdorff DimensionUsing Box-Counting Dimension

In similar models (e.g. the Fibonacci Hamiltonian) thebox-counting and Hausdorff dimensions were found to coincide [5].

Recall, we found that as coupling constant approaches zero,box-counting dimension approaches one.

Therefore, we hypothesize that the Hausdorff dimension of ourspectrum will be approximately one.

Methods and Results

Estimate of Hausdorff DimensionUsing Thickness

Theorem (Palis, Takens)

Thickness provides a lower bound of the Hausdorff dimension(dimH K) in the following way:

dimH K ≥log(2)

log(2 + 1τ(K))

According to our results,

limV→0

τ(K) =∞

it follows that when the coupling constant is zero,

limV→0

dimH(K) ≥ log(2)

log(2)= 1.

Methods and Results

dimH K ≥log(2)

log(2 + 1τ(K))

limV→0

τ(K) =∞

limV→0

dimH(K) ≥ log(2)

log(2)= 1.

Methods and Results

dimH K ≥log(2)

log(2 + 1τ(K))

limV→0

τ(K) =∞

limV→0

dimH(K) ≥ log(2)

log(2)= 1.

Methods and Results

Methods of Approximating Hausdorff Dimension

We numerically estimate the Hausdorff dimension of the spectrumusing methods employed by Chorin [3].

Consider the quantity nδs, where n is the number of covers neededfor diameter δ, and s is a nonnegative number.

We have the following bounds on the Hausdorff dimension

Lower bound: the value of s1, where nδs1 is strictly increasing

Upper bound: the value of s2 where nδs2 is strictly decreasing.

Methods and Results

Using MATLAB to Approximate Hausdorff Dimension

Recall that HV = −∆ + V .

We approximate the spectrum by first truncating the Hamiltonianoperator:

Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:

S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

Call the function truncatedHamiltonian which

Creates a matrix with S3(1) on the main diagonal

Multiples the matrix by V

Inputs 1’s along the super and sub-diagonals

Computes the eigenvalues of the resulting matrix.

Methods and Results

S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

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S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

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S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

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S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

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S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

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S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

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S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]

Methods and Results

At this point, we employ a numerical approximation method [3].

We will be computing the quantity: nδs

We call the function hausdorffDimension which:

Defines δ = 2−i for i ∈ {1, 2, ..., 10}Calculates the number of covers, n, needed for each δ

Computes nδs, for s ∈ [0, 1].

Methods and Results

Defines δ = 2−i for i ∈ {1, 2, ..., 10}

Calculates the number of covers, n, needed for each δ

Methods and Results

Results for Hausdorff Dimension

The value of the quantity: nδs where δ = 2−i, is displayed in thefollowing table (V = 0.001).

si 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

1 4.757 4.724 4.691 4.659 4.627 4.595 4.563 4.531 4.5002 4.748 4.683 4.619 4.555 4.492 4.430 4.369 4.309 4.2503 4.872 4.771 4.673 4.577 4.483 4.391 4.300 4.212 4.1254 5.071 4.933 4.798 4.667 4.539 4.415 4.294 4.177 4.0635 5.319 5.138 4.963 4.794 4.631 4.473 4.321 4.173 4.0316 5.601 5.373 5.154 4.944 4.742 4.549 4.364 4.186 4.0167 5.909 5.629 5.362 5.108 4.866 4.636 4.416 4.207 4.008

Therefore, the Hausdorff dimension is bounded below by 0.92 andabove by 1.00.

As V approaches 0, Hausdorff dimension approaches 1.

Methods and Results

Results for Hausdorff Dimension

The value of the quantity: nδs where δ = 2−i, is displayed in thefollowing table (V = 0.001).

si 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

1 4.757 4.724 4.691 4.659 4.627 4.595 4.563 4.531 4.5002 4.748 4.683 4.619 4.555 4.492 4.430 4.369 4.309 4.2503 4.872 4.771 4.673 4.577 4.483 4.391 4.300 4.212 4.1254 5.071 4.933 4.798 4.667 4.539 4.415 4.294 4.177 4.0635 5.319 5.138 4.963 4.794 4.631 4.473 4.321 4.173 4.0316 5.601 5.373 5.154 4.944 4.742 4.549 4.364 4.186 4.0167 5.909 5.629 5.362 5.108 4.866 4.636 4.416 4.207 4.008

Therefore, the Hausdorff dimension is bounded below by 0.92 andabove by 1.00.

As V approaches 0, Hausdorff dimension approaches 1.Within ε of Awesome Spectrum of the Period Doubling Hamiltonian

Methods and Results

Future Work

Further work for this problem includes refining the approximationsof the two fractal dimensions, refining the approach to thethickness computations, and analyzing the physical implications ofthe results.

Methods and Results

Acknowledgements

We would like to thank our advisor, May Mei, and our project TA,Drew Zemke. Thanks also to Ravi Ramakrishna, Summer MathInstitute program director, and the math department at CornellUniversity. This work was supported by NSF grant DMS-0739338.

Methods and Results

Michael Baake, Uwe Grimm, and Robert V. Moody.What is aperiodic order?Spektrum der Wissenschaft, pages 64–74, 2002.The translated version was used for our research.

Jean Bellissard, Anton Bovier, and Jean-Michel Ghez.Spectral properties of a tight binding hamiltonian with perioddoubling potential.Communications in Mathematical Physics, 135(2):379–399,1991.

Alexandre Joel Chorin.Numerical estimates of hausdorff dimension.Journal of Computational Physics, 46(3):390 – 396, 1982.

Jean-Michel Combes.Connections between quantum dynamics and spectralproperties of time-evolution operators.

Methods and Results

In E.M. Harrell W.F. Ames and J.V. Herod, editors,Differential Equations with Applications to MathematicalPhysics, volume 192 of Mathematics in Science andEngineering, pages 59 – 68. Elsevier, 1993.

Daminik, Embree, and Gorodetski.Spectral properties of schrodinger operators arising in thestudy of quasicrystalsl.1210.5753, October 2012.

Kenneth Falconer.Hausdorff Measure and Dimension, pages 27–38.John Wiley & Sons, Ltd, 2005.

I. Guarneri.Spectral properties of quantum diffusion on discrete lattices.EPL, 10(2):95–100, 1989.

Yoram Last.

Methods and Results

Quantum dynamics and decompositions of singular continuousspectra.Journal of Functional Analysis, 142(2):406 – 445, 1996.

Jacob Palis and Floris Takens.Hyperbolicity and sensitive chaotic dynamics at homoclinicbifurcations, volume 35 of Cambridge Studies in AdvancedMathematics.Cambridge University Press, Cambridge, 1993.Fractal dimensions and infinitely many attractors.

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