Approximation of the invariant measure of an IFS Helena Pe na, Uni - PowerPoint PPT Presentation

Approximation of the invariant measure of an IFS Helena Pe˜ na, Uni Greifswald Berlin - Padova Young researchers Meeting in Probability WIAS, TU Berlin and Uni Potsdam October 23 - 25, 2014

Approximation of the invariant measure of an IFS 1. Iterated Function System (IFS) 2. The transfer operator T for an IFS 3. Eigenfunctions of T for affine IFS 4. Approximation for the invariant measure

1. Iterated Function System (IFS)

IFS Let X = R d or C d . An IFS on X consists of f 1 , . . . , f N mappings X → X and a corresponding ( p 1 , . . . , p N ) probability vector Assume there is a non-empty compact set K ⊂ X such that f i ( K ) ⊂ K for all i .

IFS as a stochastic dynamical system f 1 , . . . , f N mappings X → X ( p 1 , . . . , p N ) probability vector Start at a point x 0 ∈ X . Given a point x n , choose a function f according to P ( f = f i ) = p i and set x n+1 = f ( x n ) random trajectory in X x 0 , x 1 , x 2 , . . .

IFS – right-angled triangle D right-angled triangle mappings D 1 = f 1 ( D ) D 2 = f 2 ( D ) with prob. p 1 = p 2 = 1 2

IFS – random trajectory Points x 0 , x 1 , . . . , x 100000

IFS – right-angled triangle D right-angled triangle mappings D 1 = f 1 ( D ) and D 2 = f 2 ( D ) with prob. p 1 = | D 1 | | D | = 1 4 and p 2 = | D 2 | | D | = 3 4

IFS – random trajectory Points x 0 , x 1 , . . . , x 100000 of a random trajectory

Bernoulli IFS IFS on R with mappings f 1 ( x ) = λ x − 1 with prob. 1 2 f 2 ( x ) = λ x + 1 with prob. 1 2 parameter λ ∈ [ 1 2 , 1 ) 1 1 Interval I λ = [ − 1 − λ , 1 − λ ] with I λ = f 1 ( I λ ) ∪ f 2 ( I λ )

Bernoulli IFS – trajectories f 1 ( x ) = λ x − 1 f 2 ( x ) = λ x + 1 n th iteration: ± 1 ± λ ± λ 2 ± λ 3 ± . . . ± λ n x n

Bernoulli IFS - random trajectory First 10 5 points (histogram)

2. The transfer operator T for an IFS

The transfer operator T C ( K ) space of continuous functions K → K T : C ( K ) → C ( K ) N � � � Th ( x ) = p i h ( f i ( x )) i=1 For a trajectory with starting point x 0 ∈ K we have E h ( x 1 ) = Th ( x 0 ) E h ( x n ) = T n h ( x 0 )

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 ) N � � � p i µ ◦ f − 1 H µ = i i=1 is dual to the transfer operator , i.e.: ( H µ, h ) = ( µ, Th ) duality

Hutchinson operator H : M ( K ) → M ( K ) N � � � p i µ ( f − 1 H µ ( A ) = ( A )) i i=1 Start with a distribution µ 0 auf K . The mass in A ⊂ K after one step comes with prob. p i from the set f − 1 ( A ) , so that i µ 1 ( A ) = H µ 0 ( A ) Distribution after n steps: µ n = H n µ 0

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 .

Bernoulli IFS - Hutchinson Operator λ = 0 . 9

Bernoulli IFS - Hutchinson Operator λ = 0 . 8

Bernoulli IFS - Hutchinson Operator λ = 0 . 7

Bernoulli IFS - Hutchinson Operator λ = 0 . 6

Bernoulli IFS - Hutchinson Operator λ = 0 . 55

Bernoulli convolution problem ν distribution of the random sum � � � ∞ n=0 ± λ n Does ν have a density?

Results on Bernoulli convolutions � ν distribution of the random sum � � ∞ n=0 ± λ n Jessen, Winter 1935: ν is either absolutely continuous or purely singular with respect to the Lebesgue measure.

Results on Bernoulli convolutions S = { λ ∈ [ 1 2 , 1 ) : ν singular } Erd¨ os 1939: countably many examples in S . Garsia 1962: countably many examples in S ∩ [ 1 2 , 1 ) . Solomyak 1995: S has Lebesgue measure 0. Shmerkin 2013: S has Hausdorff dimension 0.

3. Eigenfunctions of T for affine IFS

T for affine IFS Transfer operator for an IFS on K ⊂ K d with affine contractions f i ( x ) = A i x + v i T : C ( K ) → C ( K ) N � � � Th ( x ) = p i h ( A i x + v i ) i=1 with matrix A i and translation vector v i .

Invariant eigenspaces for affine IFS Let P n ( K ) be the space of polynomials p : K → K of degree ≤ n . Then, T : P n ( K ) → P n ( K ) is well-defined, i.e. P n ( K ) is an invariant eigenspace of T.

T for affine IFS on K Consider an IFS on K ⊂ K with mappings f i ( x ) = λ i x + v i , i = 1 , . . . , N λ i , v i ∈ K The operator T : P n ( K ) → P n ( K ) is represented by a matrix T n ∈ K (n+1) × (n+1) .

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

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

Eigenvectors of T n The eigenvectors of T n correspond to the polynomial eigenfunctions of T : P n ( K ) → P n ( K ) . The constant function 1 is always an eigenfunction with eigenvalue 1, since N � � � T1 = p i = 1 i=1

Basis of eigenfunctions Theorem. The transfer operator for an IFS with equal contraction factors λ T : P n ( K ) → P n ( K ) N � � � Th ( x ) = p i h ( λ x + v i ) i=1 has eigenvalues λ k for 0 ≤ k ≤ n and the eigenfunctions build a basis of P n ( K ) .

Basis of eigenfunctions Theorem. The transfer operator T : C ( K ) → C ( K ) N � � � Th ( x ) = p i h ( λ x + v i ) i=1 has eigenvalues { λ k : k ∈ N 0 } and their eigenfunctions build a basis of P ( K ) .

Bernoulli IFS – Transfer operator T : C ([ − 1 , 1 ]) → C ([ − 1 , 1 ]) Th ( x ) = 1 + 1 2h ( λ x − 1 + λ ) 2h ( λ x + 1 − λ ) � �� � � �� � f 2 (x) f 1 (x) with λ ∈ [ 1 2 , 1 ) .

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

Bernoulli IFS – eigenfunctions of T Polynomial eigenfunctions q 0 , . . . , q 5 for λ = 0 . 7

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

4. Approximation of the invariant measure

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

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

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 =

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.

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 G = 2 7 . . .   1 1   0 5 0 7 0 . . .   . . . · · · · . . . m k are the moments of ν

Bernoulli IFS – Densities

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