Multiplicative Ergodic Theorems Anthony Quas (with Gary Froyland, - - PowerPoint PPT Presentation

Multiplicative Ergodic Theorems

Anthony Quas (with Gary Froyland, Cecilia Gonz´ alez Tokman and Simon Lloyd) June 2013

Anthony Quas Multiplicative Ergodic Theorems

For two subspaces, U and V , of Rd of the same dimension, we define ∠(U, V ) = dH(U ∩ B, V ∩ B), where dH denotes Hausdorff distance and B is the unit ball. For two subspaces U and W of complementary dimensions, we define ⊥ (U, W ) = (1/ √ 2) inf{u∈U∩S, w∈W ∩S} u − w, where S denotes the unit sphere. Thus ⊥ (U, W ) is a measure of complementarity of subspaces, taking values between 0 and 1, with 0 indicating that the spaces intersect and 1 indicating that the spaces are orthogonal complements. Note that ⊥ (U, V ) ≥⊥ (U, W ) − ∠(W , V ). Let sj (A) denote the jth singular value of the matrix A and let Ξj (A) denote log s1(A) + . . . + log sj (A). Note that Ξj (A) = log Λj A, so that Ξj (AB) ≤ Ξj (A) + Ξj (B). The structure of the proof of the main theorem closely follows that of Ledrappier and Young, in which the orbit of ω is divided into blocks of length ≈ | log ǫ|. These are classified as good if a number of conditions hold (separation

• f Lyapunov spaces, closeness of averages to integrals etc.) and bad otherwise. The crucial modifications that we

make are in estimations for the bad blocks. In the case of [LY], the matrices (and hence their perturbations) have uniformly bounded inverses, so that for bad blocks one can give uniform lower bounds on the contribution to the singular value. By contrast, here, there is no uniform lower bound. Upper bounds are straightforward, so all of the work is concerned with establishing lower bounds for the exponents. Absent the invertibility, a similar argument would yield (random) bounds of order log ǫ, which turn out to be too weak to give the lower bounds that we need. Anthony Quas Multiplicative Ergodic Theorems

Ulam’s method

A method for computing (absolutely continuous) invariant measures for dynamical systems

Anthony Quas Multiplicative Ergodic Theorems

Ulam’s method

Anthony Quas Multiplicative Ergodic Theorems

Ulam’s method

• 1. Divide the space into a finite number of cells.

Anthony Quas Multiplicative Ergodic Theorems

Ulam’s method

j i

• 1. Divide the space into a finite number of cells
• 2. For each pair of cells, compute the ‘probability’ that an

element of i will map into cell j.

Anthony Quas Multiplicative Ergodic Theorems

Ulam’s method

j i

• 1. Divide the space into a finite number of cells
• 2. For each pair of cells, compute the ‘probability’ that an

element of i will map into cell j.

• 3. Pretend that the d.s. is just a Markov Chain

Anthony Quas Multiplicative Ergodic Theorems

Ulam’s method

Anthony Quas Multiplicative Ergodic Theorems

The top eigenvectors coming from Ulam’s method have been proved to give a convergent sequence (as #partition elements→ ∞) of approximations to the acim for 1D expanding maps

Anthony Quas Multiplicative Ergodic Theorems

The top eigenvectors coming from Ulam’s method have been proved to give a convergent sequence (as #partition elements→ ∞) of approximations to the acim for 1D expanding maps Ulam’s method seems to work much more generally.

Anthony Quas Multiplicative Ergodic Theorems

Second Eigenvalue

In the Markov chain, the second eigenvalue gives information about the rate of convergence to the invariant distribution.

Anthony Quas Multiplicative Ergodic Theorems

Second Eigenvalue

In the Markov chain, the second eigenvalue gives information about the rate of convergence to the invariant distribution. An analogous quantity in the dynamical system is the rate of decay

• f correlations.
• f ◦ T n · g −
• f
• g
• .

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius Operator

P-F is the pre-dual of the Koopman operator, defined by

• f ◦ T(x)g(x) dx =
• f (x)Lg(x) dx.

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius Operator

P-F is the pre-dual of the Koopman operator, defined by

• f ◦ T(x)g(x) dx =
• f (x)Lg(x) dx.
• f ◦ T n(x)g(x) dx =
• f (x)Lng(x) dx.

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius Operator

P-F is the pre-dual of the Koopman operator, defined by

• f ◦ T(x)g(x) dx =
• f (x)Lg(x) dx.
• f ◦ T n(x)g(x) dx =
• f (x)Lng(x) dx.

Describes the evolution of densities: if a random variable X has an absolutely continuous distribution with density g(x), then T(X) has density (Lg)(x).

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius Operator

P-F is the pre-dual of the Koopman operator, defined by

• f ◦ T(x)g(x) dx =
• f (x)Lg(x) dx.
• f ◦ T n(x)g(x) dx =
• f (x)Lng(x) dx.

Describes the evolution of densities: if a random variable X has an absolutely continuous distribution with density g(x), then T(X) has density (Lg)(x). Correspondence: fixed points of L ↔ acims.

Anthony Quas Multiplicative Ergodic Theorems

If

• g = 0, decay of correlations is governed by
• (f ◦ T n)g =
• f · Lng.

L is an averaging operator. We can hope Lng → 0 if

• g = 0.

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius operators and Decay of Correlation

For nice T (expanding maps), L is often quasi-compact as an

• perator on Banach spaces that measure the smoothness of

functions: BV , Sobolev spaces, C k(X).

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius operators and Decay of Correlation

For nice T (expanding maps), L is often quasi-compact as an

• perator on Banach spaces that measure the smoothness of

functions: BV , Sobolev spaces, C k(X). Never quasi-compact on Lp or C(X).

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius operators and Decay of Correlation

For nice T (expanding maps), L is often quasi-compact as an

• perator on Banach spaces that measure the smoothness of

functions: BV , Sobolev spaces, C k(X). Never quasi-compact on Lp or C(X). The decay of correlations is governed by the second eigenvalue.

Anthony Quas Multiplicative Ergodic Theorems

Interpretation of Peripheral Eigenvectors/Eigenvalues

Dellnitz and Froyland have interpreted peripheral spectrum as arising from global obstruction to mixing.

Anthony Quas Multiplicative Ergodic Theorems

Interpretation of Peripheral Eigenvectors/Eigenvalues

Dellnitz and Froyland have interpreted peripheral spectrum as arising from global obstruction to mixing. Essential spectrum corresponds to rates of mixing arising from local data (expansion rates etc.)

Anthony Quas Multiplicative Ergodic Theorems

Interpretation of Peripheral Eigenvectors/Eigenvalues

Dellnitz and Froyland have interpreted peripheral spectrum as arising from global obstruction to mixing. Essential spectrum corresponds to rates of mixing arising from local data (expansion rates etc.) Idea: You can recover almost invariant sets (‘coherent structures’) as level sets of eigenfunctions.

+1 −1

Anthony Quas Multiplicative Ergodic Theorems

Anthony Quas Multiplicative Ergodic Theorems

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius and Finite-dimensional Approximation

For a single transformation,

• 1. The leading eigenvector of the Perron-Frobenius operator

describes the invariant measure

• 2. Other peripheral eigenvectors describe global obstructions to

mixing (almost invariant sets).

• 3. The Ulam matrix approximates the spectrum of the

Perron-Frobenius operator [Keller-Liverani]

Anthony Quas Multiplicative Ergodic Theorems

Perron-Frobenius and Finite-dimensional Approximation

For a single transformation,

• 1. The leading eigenvector of the Perron-Frobenius operator

describes the invariant measure

• 2. Other peripheral eigenvectors describe global obstructions to

mixing (almost invariant sets).

• 3. The Ulam matrix approximates the spectrum of the

Perron-Frobenius operator [Keller-Liverani] Now we’d like to extend this to forced dynamical systems.

Anthony Quas Multiplicative Ergodic Theorems

Oseledets Multiplicative Ergodic Theorem

Setting: σ: Ω → Ω is an arbitrary ergodic measure-preserving transformation; A is a (d × d) matrix-valued function of Ω with integrable log-norm.

Anthony Quas Multiplicative Ergodic Theorems

Oseledets Multiplicative Ergodic Theorem

Setting: σ: Ω → Ω is an arbitrary ergodic measure-preserving transformation; A is a (d × d) matrix-valued function of Ω with integrable log-norm. We define A(n)

ω

= Aσn−1ω · · · Aω

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(invertible)

If σ is invertible and the matrices Aω are invertible, then there exist λ1 > λ2 > . . . > λk and V1(ω), . . . , Vk(ω) such that:

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(invertible)

If σ is invertible and the matrices Aω are invertible, then there exist λ1 > λ2 > . . . > λk and V1(ω), . . . , Vk(ω) such that:

• 1. Rd = V1(ω) ⊕ . . . ⊕ Vk(ω);

Splitting

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(invertible)

If σ is invertible and the matrices Aω are invertible, then there exist λ1 > λ2 > . . . > λk and V1(ω), . . . , Vk(ω) such that:

• 1. Rd = V1(ω) ⊕ . . . ⊕ Vk(ω);

Splitting

• 2. Aω(Vi(ω)) = Vi(σω)

Equivariance

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(invertible)

If σ is invertible and the matrices Aω are invertible, then there exist λ1 > λ2 > . . . > λk and V1(ω), . . . , Vk(ω) such that:

• 1. Rd = V1(ω) ⊕ . . . ⊕ Vk(ω);

Splitting

• 2. Aω(Vi(ω)) = Vi(σω)

Equivariance

• 3. x ∈ Vi(ω) \ {0} implies limn→±∞ 1

n log A(n) ω x = λi.

Growth Rate

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(non-invertible)

If no invertibility assumption is made on σ or the matrices, then there exist λ1 > λ2 > . . . > λk and U1(ω), . . . , Uk(ω) such that:

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(non-invertible)

If no invertibility assumption is made on σ or the matrices, then there exist λ1 > λ2 > . . . > λk and U1(ω), . . . , Uk(ω) such that:

• 1. Rd = U1(ω) ⊇ U2(ω) . . . ⊇ Uk(ω);

Filtration

• 2. Aω(Ui(ω)) = Ui(σω)
• 3. x ∈ Ui(ω) \ Ui+1(ω) implies limn→∞ 1

n log A(n) ω x = λi.

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(semi-invertible)

If σ is invertible but no invertibility assumption is made on the matrices, then there exist λ1 > λ2 > . . . > λk and V1(ω), . . . , Vk(ω) such that:

Anthony Quas Multiplicative Ergodic Theorems

Oseledets(semi-invertible)

If σ is invertible but no invertibility assumption is made on the matrices, then there exist λ1 > λ2 > . . . > λk and V1(ω), . . . , Vk(ω) such that:

• 1. Rd = V1(ω) ⊕ . . . ⊕ Vk(ω);

Splitting

• 2. Aω(Vi(ω)) = Vi(σω)
• 3. x ∈ Vi(ω) \ {0} implies limn→∞ 1

n log A(n) ω x = λi.

Anthony Quas Multiplicative Ergodic Theorems

Proof of semi-invertible case

Anthony Quas Multiplicative Ergodic Theorems

Proof of semi-invertible case

Uses both parts of the standard Oseledets theorem as a black box

Anthony Quas Multiplicative Ergodic Theorems

Operator Oseledets theorems

A variety of these exist due to Ma˜ n´ e, Ruelle, Thieullen, Lian and Lu and others. We make use of Thieullen’s result and that of Lian and Lu.

Anthony Quas Multiplicative Ergodic Theorems

Operator Oseledets theorems

Standard template: σ invertible. Quasi-compactness assumption: κ = limn→∞ 1

n log

• ic(Lσn−1ω ◦ · · · ◦ Lω)
• is smaller than

λ = limn→∞ 1

n log

• Lσn−1ω ◦ · · · ◦ Lω
• .

Anthony Quas Multiplicative Ergodic Theorems

Operator Oseledets theorems

Standard template: σ invertible. Quasi-compactness assumption: κ = limn→∞ 1

n log

• ic(Lσn−1ω ◦ · · · ◦ Lω)
• is smaller than

λ = limn→∞ 1

n log

• Lσn−1ω ◦ · · · ◦ Lω
• .

Conclusion (invertible case): There exist λ1 ≥ λ2 ≥ . . .; λi → κ and equivariant finite-dimensional subspaces V1(ω), V2(ω), . . . and an equivariant infinite-dimensional space R(ω) with X = V1(ω) ⊕ V2(ω) ⊕ . . . ⊕ R(ω). Points in Vi(ω) expand at rate λi; points in R(ω) expand at rate ≤ κ.

Anthony Quas Multiplicative Ergodic Theorems

Black boxes

Thieullen created a black box that does: invertible → non-invertible.

Anthony Quas Multiplicative Ergodic Theorems

Black boxes

Thieullen created a black box that does: invertible → non-invertible. We’ve created a black box that does: non-invertible → semi-invertible.

Anthony Quas Multiplicative Ergodic Theorems

Applications to Perron-Frobenius operators

Theorem (Froyland,Lloyd,Q 2010)

The Perron-Frobenius operators of random one-dimensional expanding maps (with at most countably many maps) satisfy the conclusions of the MET. The ‘essential spectral radius’, κ, in BV is given by lim

n→∞

• min

x

• F (n)

ω

′(x) −1/n .

Anthony Quas Multiplicative Ergodic Theorems

Theorem (Gonz´ alez-Tokman, Q 2011)

The Perron-Frobenius operators of random one-dimensional expanding maps acting on a ‘fractional Sobolev space’ satisfy the conclusions of the MET.

Anthony Quas Multiplicative Ergodic Theorems

Theorem (Gonz´ alez-Tokman, Q 2011)

The Perron-Frobenius operators of random one-dimensional expanding maps acting on a ‘fractional Sobolev space’ satisfy the conclusions of the MET.

Theorem (Gonz´ alez-Tokman, Q 2011)

The Perron-Frobenius operators of random higher-dimensional expanding maps all in a neighbourhood of a generic map acting on a fractional Sobolev space satisfy conclusions of the MET.

Anthony Quas Multiplicative Ergodic Theorems

Stability results

Theorem (Froyland, Gonz´ alez-Tokman, Q 2013)

Let R be a random one-dimensional system of expanding maps acting on BV. Let Rk be a family of small perturbations of R (e.g. Ulam approximations, approximations to the dynamical system, Fourier actions).

Anthony Quas Multiplicative Ergodic Theorems

Stability results

Theorem (Froyland, Gonz´ alez-Tokman, Q 2013)

Let R be a random one-dimensional system of expanding maps acting on BV. Let Rk be a family of small perturbations of R (e.g. Ulam approximations, approximations to the dynamical system, Fourier actions). Then the top subspace for Rk converges in probability to the top subspace for R.

Anthony Quas Multiplicative Ergodic Theorems

Theorem (Froyland, Gonz´ alez-Tokman, Q 2013)

Let (σ, Ω) be an ergodic mpt; let A : Ω → Md×d(R) be a family of matrices.

Anthony Quas Multiplicative Ergodic Theorems

Theorem (Froyland, Gonz´ alez-Tokman, Q 2013)

Let (σ, Ω) be an ergodic mpt; let A : Ω → Md×d(R) be a family of

• matrices. Let (∆n(ξ)) be a family of iid matrices taking values in

the unit ball of Md×d (independent of Ω). Write ¯ Ω for Ω × Ξ and Aǫ

¯ ω = Aω + ǫ∆0(ξ).

Anthony Quas Multiplicative Ergodic Theorems

Theorem (Froyland, Gonz´ alez-Tokman, Q 2013)

Let (σ, Ω) be an ergodic mpt; let A : Ω → Md×d(R) be a family of

• matrices. Let (∆n(ξ)) be a family of iid matrices taking values in

the unit ball of Md×d (independent of Ω). Write ¯ Ω for Ω × Ξ and Aǫ

¯ ω = Aω + ǫ∆0(ξ).

Then λǫ

i → λi as ǫ → 0 and the Oseledets subspaces converge in

probability to the unperturbed subspaces.

Anthony Quas Multiplicative Ergodic Theorems