Period Doubling Hamiltonian The Spectrum Methods and Results Asymptotic Analysis of the Spectrum of the Discrete 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
115
Embed
Asymptotic Analysis of the Spectrum of the Discrete ...personal.denison.edu/~meim/SMI/MegTaraMaria_Presentation.pdfPeriod Doubling Hamiltonian The Spectrum Methods and Results Motivation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
Period Doubling HamiltonianThe Spectrum
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].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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 B
S2(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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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 AS3(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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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 AS3(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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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 AS3(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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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 AS3(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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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 AS3(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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
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
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
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
Period Doubling HamiltonianThe Spectrum
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
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
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
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
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.
[2]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
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.
[2]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
Let U be the set of unstable points.
Theorem (Bellisard, Bovier, Ghez)
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.
Theorem (Bellisard, Bovier, Ghez)
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
Let U be the set of unstable points.
Theorem (Bellisard, Bovier, Ghez)
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.
Theorem (Bellisard, Bovier, Ghez)
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
Let U be the set of unstable points.
Theorem (Bellisard, Bovier, Ghez)
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.
Theorem (Bellisard, Bovier, Ghez)
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
y
x
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
V
E
Student Version of MATLAB
Top: A numerical approximation of UC .
Bottom: An approximation of the spectrum for a 0 ≤ V ≤ 1 [2].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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 .
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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 .
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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 .
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
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.
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.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian