Spectral Properties of an Operator-Fractal Keri Kornelson University of Oklahoma - Norman NSA Texas A&M University July 18, 2012 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 1 / 25
Acknowledgements This is joint work with Palle Jorgensen (University of Iowa) Karen Shuman (Grinnell College). K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 2 / 25
Outline 1 Bernoulli-Cantor Measures 2 Fourier bases Families of ONBs 3 4 Operator-fractal K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 3 / 25
Bernoulli-Cantor Measures Iterated Function Systems (IFSs) We will construct an L 2 space via an iterated function system. Definition An Iterated Function System (IFS) is a finite collection { τ i } k i = 1 of contractive maps on a complete metric space. The map on the compact subsets given by k � A �→ τ i ( A ) i = 1 is a contraction in the Hausdorff metric. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 4 / 25
Bernoulli-Cantor Measures IFS Attractor Set By the Banach Contraction Mapping Theorem, there exists a unique “fixed point” of the map. In other words, there is a compact set X satisfying the invariance relation: k � τ i ( X ) = X . (1) i = 1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 5 / 25
Bernoulli-Cantor Measures IFS Attractor Set By the Banach Contraction Mapping Theorem, there exists a unique “fixed point” of the map. In other words, there is a compact set X satisfying the invariance relation: k � τ i ( X ) = X . (1) i = 1 The set X is called the attractor of the IFS. We say ( 1 ) is an invariance held by X . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 5 / 25
Bernoulli-Cantor Measures IFS Attractor Set By the Banach Contraction Mapping Theorem, there exists a unique “fixed point” of the map. In other words, there is a compact set X satisfying the invariance relation: k � τ i ( X ) = X . (1) i = 1 The set X is called the attractor of the IFS. We say ( 1 ) is an invariance held by X . Given any compact set A 0 , successive iterations of our contraction k � A n + 1 = τ i ( A n ) i = 1 converge (in Hausdorff metric) to the attractor X . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 5 / 25
Bernoulli-Cantor Measures Examples The Cantor ternary set in R : τ 0 ( x ) = 1 τ 1 ( x ) = 1 3 x + 2 3 x 3 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 6 / 25
Bernoulli-Cantor Measures Examples The Cantor ternary set in R : τ 0 ( x ) = 1 τ 1 ( x ) = 1 3 x + 2 3 x 3 The Sierpinski gasket in R 2 . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 6 / 25
Bernoulli-Cantor Measures Examples The Cantor ternary set in R : τ 0 ( x ) = 1 τ 1 ( x ) = 1 3 x + 2 3 x 3 The Sierpinski gasket in R 2 . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 6 / 25
Bernoulli-Cantor Measures Examples The Cantor ternary set in R : τ 0 ( x ) = 1 τ 1 ( x ) = 1 3 x + 2 3 x 3 The Sierpinski gasket in R 2 . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 6 / 25
Bernoulli-Cantor Measures Examples The Cantor ternary set in R : τ 0 ( x ) = 1 τ 1 ( x ) = 1 3 x + 2 3 x 3 The Sierpinski gasket in R 2 . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 6 / 25
Bernoulli-Cantor Measures Examples The Cantor ternary set in R : τ 0 ( x ) = 1 τ 1 ( x ) = 1 3 x + 2 3 x 3 The Sierpinski gasket in R 2 . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 6 / 25
Bernoulli-Cantor Measures IFS Measure Hutchinson, 1981 { τ i } k i = 1 an IFS i = 1 probability weights, i.e. p i ≥ 0, � k { p i } k i = 1 p i = 1 Define a map on measures: k � p i ( ν ◦ τ − 1 ν �→ ) (2) i i = 1 The Banach Theorem yields again a unique “fixed point”, in this case a probability measure supported on X . This measure µ is called an equilibrium or IFS measure for the IFS. As a fixed point, µ satisfies the invariance property: k � p i ( µ ◦ τ − 1 µ = ) . (3) i i = 1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 7 / 25
Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ − on R of the form τ + ( x ) = λ ( x + 1 ) τ − ( x ) = λ ( x − 1 ) for 0 < λ < 1. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 8 / 25
Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ − on R of the form τ + ( x ) = λ ( x + 1 ) τ − ( x ) = λ ( x − 1 ) for 0 < λ < 1. Bernoulli attractor set — X λ K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 8 / 25
Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ − on R of the form τ + ( x ) = λ ( x + 1 ) τ − ( x ) = λ ( x − 1 ) for 0 < λ < 1. Bernoulli attractor set — X λ Bernoulli convolution measure ( p + = p − = 1 2 ) — µ λ — supported on X λ . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 8 / 25
Bernoulli-Cantor Measures Bernoulli IFS Definition A Bernoulli IFS consists of two affine maps τ + and τ − on R of the form τ + ( x ) = λ ( x + 1 ) τ − ( x ) = λ ( x − 1 ) for 0 < λ < 1. Bernoulli attractor set — X λ Bernoulli convolution measure ( p + = p − = 1 2 ) — µ λ — supported on X λ . Historical note: The Bernoulli measures date back to work of Erdös and others, long before this IFS approach came along. µ λ is the distribution of the random variable � k ± λ k where + and − have equal probability. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 8 / 25
Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 ( µ λ ) . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 9 / 25
Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 ( µ λ ) . Is it ever possible for L 2 ( µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 9 / 25
Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 ( µ λ ) . Is it ever possible for L 2 ( µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? If so, does the existence of a Fourier basis depend on λ ? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 9 / 25
Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 ( µ λ ) . Is it ever possible for L 2 ( µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? If so, does the existence of a Fourier basis depend on λ ? Given a spectral measure, what are the possible spectra? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 9 / 25
Fourier bases Fourier bases for Bernoulli measures? Consider the Hilbert space L 2 ( µ λ ) . Is it ever possible for L 2 ( µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? If so, does the existence of a Fourier basis depend on λ ? Given a spectral measure, what are the possible spectra? Could a non-spectral measure have a frame of exponential functions? K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 9 / 25
Fourier bases Getting started: � µ λ Recall that, given λ ∈ ( 0 , 1 ) , the Bernoulli IFS is: τ + ( x ) = λ ( x + 1 ) and τ − ( x ) = λ ( x − 1 ) . The Bernoulli measure µ λ satisfies the invariance µ λ = 1 2 ( µ λ ◦ τ − 1 + µ λ ◦ τ − 1 − ) . + Then the Fourier transform of µ λ is: � e 2 π ixt d µ λ ( x ) µ λ ( t ) = � � � 1 e 2 π i ( λ x + λ ) t d µ λ ( x ) + 1 e 2 π i ( λ x − λ ) t d µ λ ( x ) = 2 2 = cos ( 2 πλ t ) � µ λ ( λ t ) cos ( 2 πλ t ) cos ( 2 πλ 2 t ) � µ λ ( λ 2 t ) = . . . . . . � ∞ cos ( 2 πλ k t ) = k = 1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 10 / 25
Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2 π i γ · in L 2 ( µ λ ) . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 11 / 25
Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2 π i γ · in L 2 ( µ λ ) . � γ d µ λ � e γ , e ˜ γ � L 2 = e γ − ˜ = � µ λ ( γ − ˜ γ ) � � � � k � ∞ = cos 2 π λ ( γ − ˜ γ ) k = 1 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 11 / 25
Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2 π i γ · in L 2 ( µ λ ) . � γ d µ λ � e γ , e ˜ γ � L 2 = e γ − ˜ = � µ λ ( γ − ˜ γ ) � � � � k � ∞ = cos 2 π λ ( γ − ˜ γ ) k = 1 Lemma The two exponentials e γ , e ˜ γ are orthogonal if and only if one of the factors in the infinite product above is zero. This is equivalent to � 1 � 4 λ − k ( 2 m + 1 ) : k ∈ N , m ∈ Z γ − ˜ γ ∈ =: Z λ . K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 11 / 25
Fourier bases Surprising first results Theorem (Jorgensen, Pedersen 1998) L 2 ( µ 1 4 ) has an ONB of exponential functions. K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 12 / 25
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