Polynomial eigenfunctions associated to affine IFS Helena Pe na, - PowerPoint PPT Presentation

Polynomial eigenfunctions associated to affine IFS Helena Pe˜ na, Uni Greifswald 3rd Bremen Winter School and Symposium Universit¨ at Bremen March 28, 2015

Polynomial eigenfunctions associated to affine IFS 1. Bernoulli IFS 2. Transfer and Hutchinson operators 3. Numerical approach for the Bernoulli IFS 4. Analytical approach for affine IFS

1. Bernoulli IFS

1.1 Bernoulli IFS IFS on R with mappings f 1 ( x ) = tx f 2 ( x ) = tx + 1 − t each with probability 1 2 , parameter 0 . 5 ≤ t < 1 The invariant set is [0 , 1].

1.2 Invariant measure ν = 1 2 ν ◦ f − 1 + 1 2 ν ◦ f − 1 depends highly on t 1 2 Does ν have a density?

1.3 Bernoulli convolution problem ν is normalized distribution of the random sum � � � n ≥ 0 ± t n with P (+) = P ( − ) = 1 2 . S = { 0 . 5 ≤ t < 1 : ν singular } Jessen, Winter 1935: ν is either Lebesgue absolutely continuous or purely singular. Erd¨ os 1939: countably many examples in S . Garsia 1962: countably many examples in [ 1 2 , 1 ) \ S . Solomyak 1995: S has Lebesgue measure 0. Shmerkin 2013: S has Hausdorff dimension 0.

2. Transfer and Hutchinson operators

2.1 General IFS setting IFS on X = R or C with f 1 , . . . , f M mappings X → X and a corresponding ( p 1 , . . . , p M ) probability vector Assume there is a non-empty compact set K ⊂ X such that f i ( K ) ⊂ K for all i .

2.2 The transfer operator T C ( K ) space of continuous functions K → K T : C ( K ) → C ( K ) � M � � Th ( x ) = p i h ( f i ( x )) i =1 For a random trajectory starting at x 0 ∈ K : E h ( x 1 ) = Th ( x 0 ) E h ( x n ) = T n h ( x 0 )

2.3 Dual operator of T Let M ( K ) be the dual space of C ( K ) i.e. the space of Borel regular measures on K . The Hutchinson operator H : M ( K ) → M ( K ) M � � � p i µ ◦ f − 1 H µ = i i =1 is dual to the transfer operator , i.e.: ( H µ, h ) = ( µ, Th ) duality

2.4 Hutchinson operator H : M ( K ) → M ( K ) M � � � p i µ ( f − 1 H µ ( A ) = ( A )) i i =1 Start with points in K distributed by µ 0 . Expected distribution after n steps of chaos game: µ n = H n µ 0

2.4 Hutchinson operator – Bernoulli IFS t = 0 . 9

2.4 Hutchinson operator – Bernoulli IFS t = 0 . 6

2.5 Implications of duality Spectra: The transfer operator T and the Hutchinson operator H have the same spectra . Invariant measure of the IFS: ν = H ν is orthogonal to all eigenfunctions of T with eigenvalue � = 1 . T might help understand H and the invariant measure.

3. Numerical approach for the Bernoulli IFS

3.1 Discrete Transfer operator T N Instead of Th ( x ) = 1 2 h ( tx ) + 1 2 h ( tx + 1 − t ) we have a step function h = ( h 1 , . . . , h N ) T N ( h 1 , . . . , h N ) ≈ 1 2 ( h 1 , . . . , h [ tN ] ) + 1 2 ( h [ tN + t − 1] , . . . , h N )

3.1 Discrete Transfer operator T N T N as a Markov chain on intervals I i | f − 1 | f − 1 ( T N ) ij = 1 ( I j ) ∩ I i | + 1 ( I j ) ∩ I i | 1 2 2 | I i | 2 | I i |

3.1 Discrete Transfer operator T N Proposition The matrix T N defines an irreducible and aperiodic Markov chain.

3.2 Spectra of T N By Perron-Frobenius, 1 is a simple eigenvalue and remaining eigenvalues lie in the unit circle.

3.3 Eigenfunctions of T N T N keeps the centrosymmetry of T . Left and right eigenspaces contain centrosymmetric vectors. First eigenfunctions for T 100 with t = 0 . 9.

3.4 Discrete Hutchinson operator H N M N prob. measures on the grid { 0 , 1 N , 2 N , . . . , 1 } Continuous: H δ x = 1 2 δ f 1 ( x ) + 1 2 δ f 2 ( x ) Discrete: H N : M N → M N is linear operator with H N δ x = 1 δ f 1 ( x ) + 1 � � δ f 2 ( x ) 2 2 where � δ x ∈ M N is the measure which minimizes the Wasserstein-distance to δ x conditioned on having first moment x .

3.4 Wasserstein metric The Wasserstein distance W ( µ, ν ) is minimum cost (distance × mass) of turning one pile into the other � � � � � � W ( µ, ν ) = inf E | X − Y | = inf | x − y | ds ( x , y ) where X ∼ µ, Y ∼ ν and the infimum is taken over all joint distributions s with marginal µ and ν

3.4 Discrete Hutchinson operator H N � δ x = p δ x + ( 1 − p ) δ x with p = x − x 1 N

3.4 Properties of H N Remark For any measure m ∈ M N and k ∈ N , the measures ( H k ) m and ( H k N ) m have the same first moment. Proposition Let ν be the Bernoulli IFS invariant measure and k ∈ N . Then, the Wasserstein distance 1 W (( H N ) k m , ν ) ≤ N (1 − t ) + t n

3.5 Iterations of H N Iterations of H 100 with t = 0 . 7 .

4. Analytical approach for affine IFS

4.1 The operator T on polynomials Let f i ( x ) = t i x + v i be contractions on the invariant set K ⊂ K . Transfer operator: T : C ( K ) → C ( K ) M � � � Th ( x ) = p i h ( t i x + v i ) i =1

4.1 The operator T on polynomials The space P n ( K ) of polynomials p : K → K of degree ≤ n is invariant under T , i.e. T n : P n ( K ) → P n ( K ) is well-defined.

4.2 Matrix for T n : P n ( K ) ← ֓   1 ∗ ∗ ∗ . . . ∗ 0 � � �     i p i t i ∗ ∗ ∗ . . .   � � �   i p i t 2 0 ∗ . . . ∗   i    .  . T n = .       . ... ... .  .        � � � i p i t n 0 0 . . . i with respect to the basis 1 , x , x 2 , . . . , x n .

4.3 Eigenvalues of T n λ 0 = 1 � � � λ 1 = p i t i i � � � p i t 2 λ 2 = i i . . . � � � p i t n λ n = i i i = 1 , . . . , N

4.4 Basis of eigenfunctions Theorem. The transfer operator T : P n ( K ) → P n ( K ) � M � � Th ( x ) = p i h ( tx + v i ) i =1 has eigenvalues t k for 0 ≤ k ≤ n and the eigenfunctions build a basis of P n ( K ) .

4.5 Bernoulli IFS We get the eigenpolynomials of Th ( x ) = 1 2 h ( tx − 1 + t ) + 1 2 h ( tx + 1 − t ) of degree ≤ 3 from the matrix   0 (1 − t ) 2 1 0   3 t (1 − t ) 2 0 t 0   T 3 =   t 2 0 0 0 t 3 0 0 0 q 0 ( x ) = 1 , q 1 ( x ) = x , q 2 ( x ) = x 2 + t − 1 t + 1 , q 3 ( x ) = x 3 + 3 t − 1 t + 1

4.5 Bernoulli IFS Polynomial eigenfunctions q 0 , . . . , q 5 for t = 0 . 7

4.5 Bernoulli IFS Theorem. The transfer operator T has the eigenfunctions ⌊ n 2 ⌋ � � � a n , k x n − 2 k q n ( x ) = n ∈ N 0 k =0 with eigenvalues λ n = t n . The coefficients are given recursively by � n − 2 j � k − 1 � � � 1 ( 1 − t ) 2 k − 2 j a n , j a n , k = t 2 k − 1 n − 2 k j =0 for k ≥ 1 and a n , 0 = 1 else.

4.6 Approximation of the invariant measure Consider an IFS on K ⊂ K with invariant measure H ν = ν . Assumption: the eigenfunctions of the transfer operator T q k ∈ P k ( K ) , k ∈ N 0 build a basis of P ( K ) (this is the case for the Bernoulli IFS)

4.7 Approximating densities v n Duality implies: ν ⊥ q k for k = 1 , 2 , . . . We get a sequence of polynomial probability densities v n ∈ P n ( K ) by solving v n ⊥ q k for 1 ≤ k ≤ n or equivalently, � v n , x k � = m k for 1 ≤ k ≤ n m k is the k th moment of the invariant measure ν

4.8 Linear system of equations for v n Theorem. The approximation v n ( x ) = � � � n k =0 u k x k satisfies G ( u 0 , u 1 , . . . , u n ) ′ = ( m 0 , m 1 , . . . , m n ) ′ with the Hilbert matrix G ∈ K ( n +1) × ( n +1) � x i + j dx . G ij = K � � � K x k d ν are the moments of ν . m k =

4.9 Approximating measures ν n Theorem. The approximating measures ν n � � � ν n ( A ) = v n ( x ) dx A converge ν n → ν weakly to the invariant measure of the IFS.

4.10 Bernoulli IFS – Densities The approximation v n ( x ) = � � � n k =0 u k x k satisfies G ( u 0 , u 1 , . . . , u n ) ′ = ( m 0 , m 1 , . . . , m n ) ′ with the Hilbert matrix G ∈ K ( n +1) × ( n +1) .   1 1 1 0 3 0 5 . . .   1 1 0 3 0 5 0 . . .     1 1 1 3 0 5 0 7 . . . G = 2     1 1 0 5 0 7 0 . . .   . . . · · · · . . . m k are the moments of ν

4.11 Bernoulli IFS – Densities

4.11 Bernoulli IFS – Densities

References • Shmerkin. On the exceptional set for absolute continuity of Bernoulli convolutions . Geometric and Functional Analysis, 2014 • Peres, Schlag, Solomyak. Sixty years of Bernoulli convolutions . Springer 2000 • Solomyak. On the random series � ± λ n . Ann. of Math., 1995 • Lasota, Mackey. Chaos, fractals and noise . Springer 1994 • Barnsley, Demko. Iterated function systems and the global construction of fractals . Proc. Roy. Soc. London, 1985

References • Hutchinson. Fractals and self-similarity . Indiana Universitiy Math. Journal, 1981 • Kato. Perturbation theory for linear operators . Springer 1980 • Hilbert. Ein Beitrag zur Theorie des Legendre’schen Polynoms . 1894 • Rua Murray. Invariant measures and stochastic discretisations of dynamical systems. Proceedings of 15th IMACS World Congress, 1997.