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 = Rd or Cd. An IFS on X consists of f1, . . . , fN mappings X → X and a corresponding (p1, . . . , pN) probability vector Assume there is a non-empty compact set K ⊂ X such that fi(K) ⊂ K for all i.

IFS as a stochastic dynamical system

f1, . . . , fN mappings X → X (p1, . . . , pN) probability vector Start at a point x0 ∈ X. Given a point xn, choose a function f according to P(f = fi) = pi and set xn+1 = f(xn) random trajectory in X x0, x1, x2, . . .

IFS – right-angled triangle

D right-angled triangle mappings D1 = f1(D) D2 = f2(D) with prob. p1 = p2 = 1 2

IFS – random trajectory

Points x0, x1, . . . , x100000

IFS – right-angled triangle

D right-angled triangle mappings D1 = f1(D) and D2 = f2(D) with prob. p1 = |D1| |D| = 1 4 and p2 = |D2| |D| = 3 4

IFS – random trajectory

Points x0, x1, . . . , x100000 of a random trajectory

Bernoulli IFS

IFS on R with mappings f1(x) = λx − 1 with prob. 1 2 f2(x) = λx + 1 with prob. 1 2 parameter λ ∈ [1

2, 1)

Interval Iλ = [−

1 1−λ, 1 1−λ] with Iλ = f1(Iλ)∪f2(Iλ)

Bernoulli IFS – trajectories

f1(x) = λx − 1 f2(x) = λx + 1 nth iteration: ±1 ± λ ± λ2 ± λ3 ± . . . ± λnxn

Bernoulli IFS - random trajectory

First 105 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) Th(x) =

N

• i=1

pi h(fi(x)) For a trajectory with starting point x0 ∈ K we have E h(x1) = Th(x0) E h(xn) = Tnh(x0)

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 =

N

• i=1

pi µ ◦ f−1

i

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

Hutchinson operator

H : M(K) → M(K) Hµ(A) =

N

• i=1

pi µ(f−1

i

(A)) Start with a distribution µ0 auf K. The mass in A ⊂ K after one step comes with prob. pi from the set f−1

i

(A), so that µ1(A) = H µ0(A) Distribution after n steps: µn = Hnµ0

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.

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 ±λn

Does ν have a density?

Results on Bernoulli convolutions

ν distribution of the random sum

• ∞

n=0 ±λn

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¨

• s 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 ⊂ Kd with affine contractions fi(x) = Aix + vi T : C(K) → C(K) Th(x) =

N

• i=1

pi h(Aix + vi) with matrix Ai and translation vector vi.

Invariant eigenspaces for affine IFS

Let Pn(K) be the space of polynomials p : K → K

• f degree ≤ n. Then,

T : Pn(K) → Pn(K) is well-defined, i.e. Pn(K) is an invariant eigenspace of T.

T for affine IFS on K

Consider an IFS on K ⊂ K with mappings fi(x) = λix + vi, i = 1, . . . , N λi, vi ∈ K The operator T : Pn(K) → Pn(K) is represented by a matrix Tn ∈ K(n+1)×(n+1).

Matrix for Tn : Pn(K) → Pn(K)

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

• i piλi

∗ ∗ . . . ∗

• i piλ2

i

∗ . . . ∗ . . . ... ... . . . . . .

• i piλn

i

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

Eigenvalues of Tn

ω0 = 1 ω1 =

• i

piλi ω2 =

• i

piλ2

i

. . . ωn =

• i

piλn

i

i = 1, . . . , N

Eigenvectors of Tn

The eigenvectors of Tn correspond to the polynomial eigenfunctions of T : Pn(K) → Pn(K). The constant function 1 is always an eigenfunction with eigenvalue 1, since T1 =

N

• i=1

pi = 1

Basis of eigenfunctions

• Theorem. The transfer operator for an IFS with

equal contraction factors λ T : Pn(K) → Pn(K) Th(x) =

N

• i=1

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

Basis of eigenfunctions

• Theorem. The transfer operator

T : C(K) → C(K) Th(x) =

N

• i=1

pi h(λx + vi) has eigenvalues {λk : k ∈ N0} and their eigenfunctions build a basis of P(K).

Bernoulli IFS – Transfer operator

T : C([−1, 1]) → C([−1, 1]) Th(x) = 1 2h(λx − 1 + λ)

• f1(x)

+1 2h(λx + 1 − λ

• f2(x)

) with λ ∈ [1

2, 1).

Bernoulli IFS – Transfer operator

We get the eigenpolynomials of Th(x) = 1 2h(λx − 1 + λ) + 1 2h(λx + 1 − λ)

• f degree ≤ 3 from the matrix

T3 =     1 0 (1 − λ)2 λ 3λ(1 − λ)2 λ2 λ3     q0(x) = 1, q1(x) = x, q2(x) = x2 + λ − 1 λ + 1, q3(x) = x3 + 3 λ − 1 λ + 1

Bernoulli IFS – eigenfunctions of T

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

Bernoulli IFS

• Theorem. The transfer operator T has the

eigenfunctions qn(x) =

⌊ n

2⌋

• k=0

an,kxn−2k n ∈ N0 with eigenvalues ωn = λn. The coefficients are given recursively by an,k = 1 λ2k − 1

k−1

• j=0

n − 2j n − 2k

• (1 − λ)2k−2jan,j

for k ≥ 1 and an,0 = 1 else.

• 4. Approximation of the

invariant measure

Setting

Consider an IFS on K ⊂ K with invariant measure Hν = ν. Assumption: the eigenfunctions of T qk ∈ Pk(K), k ∈ N0 build a basis of P(K) (this is the case for the Bernoulli IFS)

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

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.

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.

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

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 ±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

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

are orthogonal to a measure supported on the set dragon curve

Polynomial eigenfunctions of T

degree 0 left: real part, right: imaginary part

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