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 R d of the same dimension, we define ∠ ( U , V ) = d H ( U ∩ B , V ∩ B ), where d H 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 s j ( A ) denote the j th singular value of the matrix A and let Ξ j ( A ) denote log s 1 ( A ) + . . . + log s j ( A ). Note that Ξ j ( A ) = log � Λ j A � , so that Ξ j ( AB ) ≤ Ξ j ( A ) + Ξ j ( 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 of 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 of 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 ) L g ( 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 ) L g ( x ) dx . � � f ◦ T n ( x ) g ( x ) dx = f ( x ) L n g ( 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 ) L g ( x ) dx . � � f ◦ T n ( x ) g ( x ) dx = f ( x ) L n g ( 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 ( L g )( 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 ) L g ( x ) dx . � � f ◦ T n ( x ) g ( x ) dx = f ( x ) L n g ( 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 ( L g )( 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 · L n g . L is an averaging operator. We can hope L n g → 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 operator 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 operator on Banach spaces that measure the smoothness of functions: BV , Sobolev spaces, C k ( X ). Never quasi-compact on L p 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 operator on Banach spaces that measure the smoothness of functions: BV , Sobolev spaces, C k ( X ). Never quasi-compact on L p 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 ) = A σ n − 1 ω · · · A ω ω Anthony Quas Multiplicative Ergodic Theorems
Oseledets(invertible) If σ is invertible and the matrices A ω are invertible, then there exist λ 1 > λ 2 > . . . > λ k and V 1 ( ω ) , . . . , V k ( ω ) such that: Anthony Quas Multiplicative Ergodic Theorems
Oseledets(invertible) If σ is invertible and the matrices A ω are invertible, then there exist λ 1 > λ 2 > . . . > λ k and V 1 ( ω ) , . . . , V k ( ω ) such that: 1. R d = V 1 ( ω ) ⊕ . . . ⊕ V k ( ω ); Splitting Anthony Quas Multiplicative Ergodic Theorems
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