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 f1(x) = tx f2(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

2ν ◦ f −1 1

+ 1

2ν ◦ f −1 2

depends highly on t Does ν have a density?

1.3 Bernoulli convolution problem

ν is normalized distribution of the random sum

• n≥0 ±tn 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¨

• s 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
• perators

2.1 General IFS setting

IFS on X = R or C with f1, . . . , fM mappings X → X and a corresponding (p1, . . . , pM) probability vector Assume there is a non-empty compact set K ⊂ X such that fi(K) ⊂ K for all i.

2.2 The transfer operator T

C(K) space of continuous functions K → K T : C(K) → C(K) Th(x) =

M

• i=1

pi h(fi(x)) For a random trajectory starting at x0 ∈ K: E h(x1) = Th(x0) E h(xn) = T nh(x0)

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) Hµ =

M

• i=1

pi µ ◦ f −1

i

is dual to the transfer operator, i.e.: (Hµ, h) = (µ, Th) duality

2.4 Hutchinson operator

H : M(K) → M(K) Hµ(A) =

M

• i=1

pi µ(f −1

i

(A)) Start with points in K distributed by µ0. Expected distribution after n steps of chaos game: µn = Hnµ0

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

• perator H have the same spectra.

Invariant measure of the IFS: ν = Hν 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 TN

Instead of Th(x) = 1

2h(tx) + 1 2h(tx + 1 − t) we

have a step function h = (h1, . . . , hN) TN(h1, . . . , hN) ≈ 1

2(h1, . . . , h[tN]) + 1 2(h[tN+t−1], . . . , hN)

3.1 Discrete Transfer operator TN

TN as a Markov chain on intervals Ii (TN)ij = 1 2 |f −1

1

(Ij) ∩ Ii| |Ii| + 1 2 |f −1

2

(Ij) ∩ Ii| |Ii|

3.1 Discrete Transfer operator TN

Proposition

The matrix TN defines an irreducible and aperiodic Markov chain.

3.2 Spectra of TN

By Perron-Frobenius, 1 is a simple eigenvalue and remaining eigenvalues lie in the unit circle.

3.3 Eigenfunctions of TN

TN keeps the centrosymmetry of T. Left and right eigenspaces contain centrosymmetric vectors. First eigenfunctions for T100 with t = 0.9.

3.4 Discrete Hutchinson operator HN

MN prob. measures on the grid {0, 1

N, 2 N, . . . , 1}

Continuous: Hδx = 1 2δf1(x) + 1 2δf2(x) Discrete: HN : MN → MN is linear operator with HNδx = 1 2

• δf1(x) + 1

2

• δf2(x)

where δx ∈ MN is the measure which minimizes the Wasserstein-distance to δx 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

• ther

W (µ, ν) = inf E|X−Y | = inf

• |x−y|ds(x, y)

where X ∼ µ, Y ∼ ν and the infimum is taken

• ver all joint distributions s with marginal µ and ν

3.4 Discrete Hutchinson operator HN

• δx = pδx + (1 − p)δx

with p = x − x

1 N

3.4 Properties of HN

Remark

For any measure m ∈ MN and k ∈ N, the measures (Hk)m and (Hk

N)m have the same first

moment.

Proposition

Let ν be the Bernoulli IFS invariant measure and k ∈ N. Then, the Wasserstein distance W ((HN)km, ν) ≤ 1 N(1 − t) + tn

3.5 Iterations of HN

Iterations of H100 with t = 0.7.

• 4. Analytical approach for affine

IFS

4.1 The operator T on polynomials

Let fi(x) = tix + vi be contractions on the invariant set K ⊂ K. Transfer operator: T : C(K) → C(K) Th(x) =

M

• i=1

pi h(tix + vi)

4.1 The operator T on polynomials

The space Pn(K) of polynomials p : K → K

• f degree ≤ n is invariant under T, i.e.

Tn : Pn(K) → Pn(K) is well-defined.

4.2 Matrix for Tn : Pn(K) ← ֓

Tn =                1 ∗ ∗ ∗ . . . ∗

• i piti

∗ ∗ . . . ∗

• i pit2

i

∗ . . . ∗ . . . ... ... . . . . . .

• i pitn

i

               with respect to the basis 1, x, x2, . . . , xn.

4.3 Eigenvalues of Tn

λ0 = 1 λ1 =

• i

piti λ2 =

• i

pit2

i

. . . λn =

• i

pitn

i

i = 1, . . . , N

4.4 Basis of eigenfunctions

• Theorem. The transfer operator

T : Pn(K) → Pn(K) Th(x) =

M

• i=1

pi h(tx + vi) has eigenvalues tk for 0 ≤ k ≤ n and the eigenfunctions build a basis of Pn(K).

4.5 Bernoulli IFS

We get the eigenpolynomials of Th(x) = 1 2h(tx − 1 + t) + 1 2h(tx + 1 − t)

• f degree ≤ 3 from the matrix

T3 =     1 0 (1 − t)2 t 3t(1 − t)2 t2 t3     q0(x) = 1, q1(x) = x, q2(x) = x2 + t − 1 t + 1, q3(x) = x3 + 3 t − 1 t + 1

4.5 Bernoulli IFS

Polynomial eigenfunctions q0, . . . , q5 for t = 0.7

4.5 Bernoulli IFS

• Theorem. The transfer operator T has the

eigenfunctions qn(x) =

⌊ n

2⌋

• k=0

an,kxn−2k n ∈ N0 with eigenvalues λn = tn. The coefficients are given recursively by an,k = 1 t2k − 1

k−1

• j=0

n − 2j n − 2k

• (1 − t)2k−2jan,j

for k ≥ 1 and an,0 = 1 else.

4.6 Approximation of the invariant measure

Consider an IFS on K ⊂ K with invariant measure Hν = ν. Assumption: the eigenfunctions of the transfer

• perator T

qk ∈ Pk(K), k ∈ N0 build a basis of P(K) (this is the case for the Bernoulli IFS)

4.7 Approximating densities vn

Duality implies: ν ⊥ qk for k = 1, 2, . . . We get a sequence of polynomial probability densities vn ∈ Pn(K) by solving vn ⊥ qk for 1 ≤ k ≤ n

• r equivalently,

vn, xk = mk for 1 ≤ k ≤ n mk is the kth moment of the invariant measure ν

4.8 Linear system of equations for vn

• Theorem. The approximation

vn(x) =

• n

k=0 ukxk satisfies

G(u0, u1, . . . , un)′ = (m0, m1, . . . , mn)′ with the Hilbert matrix G ∈ K(n+1)×(n+1) Gij =

• K

xi+j dx. mk =

• K xk dν are the moments of ν.

4.9 Approximating measures νn

• Theorem. The approximating measures νn

νn(A) =

• A

vn(x) dx converge νn → ν weakly to the invariant measure of the IFS.

4.10 Bernoulli IFS – Densities

The approximation vn(x) =

• n

k=0 ukxk satisfies

G(u0, u1, . . . , un)′ = (m0, m1, . . . , mn)′ with the Hilbert matrix G ∈ K(n+1)×(n+1). G = 2        1 0

1 3 0 1 5 . . . 1 3 0 1 5 0 . . . 1 3 0 1 5 0 1 7 . . . 1 5 0 1 7 0 . . .

. . . · · · · . . .        mk 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 ±λn. Ann.
• f 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.

Polynomial eigenfunctions of T

Let λ = 0.5 − 0.5i. We get the eigenpolynomials

• f

Th(z) = 1 2h(λz) + 1 2h(λz + 1)

• f degree ≤ 3 from the matrix

T3 =     1 1 1 1 0 0.5 − 0.5 i 1 − i 1.5 − 1.5 i −0.5 i −1.5 i −0.25 − 0.25 i    

Polynomial eigenfunctions of T

degree 1 left: real part, right: imaginary part

Polynomial eigenfunctions of T

degree 2 left: real part, right: imaginary part

Polynomial eigenfunctions of T

degree 3 left: real part, right: imaginary part

Polynomial eigenfunctions of T

degree 4 left: real part, right: imaginary part

Polynomial eigenfunctions of T

degree 5 left: real part, right: imaginary part

Recursions

Moments of νλ m2k = −

k

• i=1

a2k,im2k−2i with m0 = 1 and an,i = 1 λ2k−1

i−1

• j=0

n − 2j n − 2i

• (1 − λ)2i−2jan,j

the coefficient of xi in the eigenpolynomial pn of A with an,0 = 1.

Recursions

Legendre moments of νλ (νλ, L2n) =

n

• k=0

(−1)n−k4−n 2n + 2k 2k 2n n + k

• m2k

where m2k = (νλ, x2k) are the moments of the convolution measure.

Recursions

Moments of νλ, another recursion: m2k = 1 1 − λ2k

k−1

• j=0

b2k,λ(2j)m2j where bn,λ(·) are the weights of the binomial distribution with parameters n and λ and m0 = 1