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Spectral Theory of Dynamical Systems

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Spectral theory of dynamical systems is a study of special unitary representations, called Koopman representations (see theglossary). Invariants of such representations are called spectral invariants of measure‐preserving systems. Together with the entropy, theyconstitute the most important invariants used in the study of measure‐theoretic intrinsic properties and classification problems of dynamicalsystems as well as in applications. Spectral theory was originated by von Neumann, Halmos and Koopman in the 1930s. In this article we will focus onrecent progresses in the spectral theory of finite measure‐preserving dynamical systems.

Introduction

Throughout \( { {\mathbb{A}} } \) denotesa non-compact l.c.s.c. Abelian group (\( { {\mathbb{A}} }\) will be most often \( { {\mathbb{Z}} }\) or \( { {\mathbb{R}} }\)). The assumption of second countability implies that \( { {\mathbb{A}} } \) is metrizable, σ-compact and the space \( { C_0({\mathbb{A}}) } \)is...

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Abbreviations

Spectral decomposition of a unitary representation:

If \( {\cal U}=(U_a)_{a\in{\mathbb{A}}} \) is a continuous unitary representation of a locally compact second countable (l.c.s.c.) Abelian group \( {\mathbb{A}} \) in a separable Hilbert space H then a decomposition \( H=\bigoplus_{i=1}^\infty {\mathbb{A}}(x_i) \) is called spectral if \( \sigma_{x_1}\gg\sigma_{x_2}\gg\ldots \) (such a sequence of measures is also called spectral ); here \( {\mathbb{A}}(x):=\overline{span}\{U_ax\colon a\in{\mathbb{A}}\} \) is called the cyclic space generated by \( x\in H \) and \( \sigma_x \) stands for the spectral measure of x.

Maximal spectral type and the multiplicity function of \( { {\cal U} } \) :

The maximal spectral type \( { \sigma_{{\cal U}} } \) of \( { {\cal U} } \) is the type of \( { \sigma_{x_1} } \) in any spectral decomposition of H; the multiplicity function \( { M_{{\cal U}}\colon \widehat{\mathbb{A}}\to\{1,2,\ldots\}\cup\{+\infty\} } \) is defined \( { \sigma_{{\cal U}} } \)-a.e. and \( { M_{{\cal U}}(\chi)=\sum_{i=1}^\infty 1_{Y_i}(\chi) } \), where \( { Y_1=\widehat{\mathbb{A}} } \) and \( { Y_i=\text{supp}\: {{\text{d}}\sigma_{x_i}}/{{\text{d}}\sigma_{x_1}} } \) for \( { i\geq2 } \).

A representation \( { {\cal U} } \) is said to have simple spectrum if H is reduced to a single cyclic space. The multiplicity is uniform if there is only one essential value of \( { M_{{\cal U}} } \). The essential supremum of \( { M_{{\cal U}} } \) is called the maximal spectral multiplicity. \( { {\cal U} } \) is said to have discrete spectrum if H has an orthonormal base consisting of eigenvectors of \( { {\cal U} } \); \( { {\cal U} } \) has singular ( Haar , absolutely continuous ) spectrum if the maximal spectral type of \( { {\cal U} } \) is singular with respect to (equivalent to, absolutely continuous with respect to) a Haar measure of \( { \widehat{\mathbb{A}} } \).

Koopman representation of a dynamical system \( { {\cal T} } \) :

Let \( {\mathbb{A}} \) be a l.c.s.c. (and not compact) Abelian group and \( {\cal T}\colon a\mapsto T_a \) a representation of \( {\mathbb{A}} \) in the group \( Aut{(X,{\cal B},\mu)} \) of (measure‐preserving) automorphisms of a standard probability Borel space \( {(X,{\cal B},\mu)} \). The Koopman representation \( {\cal U}={\cal U}_{{\cal T}} \) of \( {\cal T} \) in \( L^2{(X,{\cal B},\mu)} \) is defined as the unitary representation \( a\mapsto U_{T_a}\in U( L^2{(X,{\cal B},\mu)}) \), where \( U_{T_a}(f)=f\circ T_a \).

Ergodicity, weak mixing, mild mixing, mixing and rigidity of \( { {\cal T} } \) :

A measure‐preserving \( { {\mathbb{A}} } \)-action \( {\cal T}=(T_a)_{a\in{\mathbb{A}}} \) is called ergodic if \( { \chi_0\equiv1\in\widehat{\mathbb{A}} } \) is a simple eigenvalue of \( { {\cal U}_{{\cal T}} } \). It is weakly mixing if \( { {\cal U}_{{\cal T}} } \) has a continuous spectrum on the subspace \( { L^2_0{(X,{\cal B},\mu)} } \) of zero mean functions. \( { {\cal T} } \) is said to be rigid if there is a sequence \( { (a_n) } \) going to infinity in \( { {\mathbb{A}} } \) such that the sequence \( { (U_{T_{a_n}}) } \) goes to the identity in the strong (or weak) operator topology; \( { {\cal T} } \) is said to be mildly mixing if it has no non-trivial rigid factors. We say that \( { {\cal T} } \) is mixing if the operator equal to zero is the only limit point of \( { \{U_{T_a}|_{L^2_0{(X,{\cal B},\mu)}}\colon a\in{\mathbb{A}}\} } \) in the weak operator topology.

Spectral disjointness:

Two \( { {\mathbb{A}} } \)-actions \( { {\cal S} } \) and \( { {\cal T} } \) are called spectrally disjoint if the maximal spectral types of their Koopman representations \( { {\cal U}_{{\cal T}} } \) and \( { {\cal U}_{{\cal S}} } \) on the corresponding \( { L^2_0 } \)-spaces are mutually singular.

SCS property:

We say that a Borel measure σ on \( \smash{\widehat{\mathbb{A}}} \) satisfies the strong convolution singularity property (SCS property) if, for each \( n\geq1 \), in the disintegration (given by the map \( (\chi_1,\ldots,\chi_n)\mapsto\chi_1\cdot\ldots\cdot\chi_n \)) \( \sigma^{\otimes n}=\int_{\widehat{\mathbb{A}}}\nu_\chi\,d\sigma^{(n)}(\chi) \) the conditional measures \( \nu_\chi \) are atomic with exactly \( n! \) atoms (\( \sigma^{(n)} \) stands for the nth convolution \( \sigma\ast\ldots\ast\sigma \)). An \( {\mathbb{A}} \)-action \( {\cal T} \) satisfies the SCS property if the maximal spectral type of \( {\cal U}_{{\cal T}} \) on \( L^2_0 \) is a type of an SCS measure.

Kolmogorov group property:

An \( { {\mathbb{A}} } \)-action \( { {\cal T} } \) satisfies the Kolmogorov group property if \( { \sigma_{{\cal U}_{{\cal T}}}\ast\sigma_{{\cal U}_{{\cal T}}}\ll\sigma_{{\cal U}_{{\cal T}}} } \).

Weighted operator:

Let T be an ergodic automorphism of \( { {(X,{\cal B},\mu)} } \) and \( { \xi\colon X\to{\mathbb{T}} } \) be a measurable function. The (unitary) operator \( { V=V_{\xi,T} } \) acting on \( { L^2{(X,{\cal B},\mu)} } \) by the formula \( { V_{\xi,T}(f)(x)=\xi(x)f(Tx) } \) is called a  weighted operator .

Induced automorphism :

Assume that T is an automorphism of a standard probability Borel space \( {(X,{\cal B},\mu)} \). Let \( A\in{\cal B} \), \( \mu(A) > 0 \). The induced automorphism \( T_A \) is defined on the conditional space \( (A,{\cal B}_A,\mu_A) \), where \( {\cal B}_A \) is the trace of \( {\cal B} \) on A, \( \mu_A(B)=\mu(B)/\mu(A) \) for \( B\in{\cal B}_A \) and \( T_A(x)=T^{k_A(x)}x \), where \( k_A(x) \) is the smallest \( k \geq 1 \) for which \( T^kx\in A \).

AT property of an automorphism:

An automorphism T of a standard probability Borel space \( { {(X,{\cal B},\mu)} } \) is called approximatively transitive (AT for short) if for every \( { \varepsilon > 0 } \) and every finite set \( { f_1,\ldots,f_n } \) of non-negative \( { L^1 } \)-functions on \( { {(X,{\cal B},\mu)} } \) we can find \( { f\in L^1{(X,{\cal B},\mu)} } \) also non-negative such that \( { \|f_i-\sum_{j}\alpha_{ij}f\circ T^{n_j}\|_{L_1} < \varepsilon } \) for all \( { i=1,\ldots, n } \) (for some \( { \alpha_{ij}\geq 0 } \), \( { n_j\in{\mathbb{N}} } \)).

Cocycles and group extensions:

If T is an ergodic automorphism, G is a l.c.s.c. Abelian group and \( { \varphi\colon X\to G } \) is measurable then the pair \( { (T,\varphi) } \) generates a  cocycle \( { \varphi^{(\cdot)}(\cdot)\colon {\mathbb{Z}}\times X\to G } \), where

$$ \varphi^{(n)}(x)=\left\{\begin{array}{l@{\kern2mm}ll} \varphi(x)+\ldots+\varphi(T^{n-1}x)& \mbox{for}&n > 0\:,\\ 0&\mbox{for}&n=0\:,\\ -(\varphi(T^nx)+\ldots+\varphi(T^{-1}x))&\mbox{for}&n < 0\:.\end{array}\right. $$

(That is \( { (\varphi^{(n)}) } \) is a standard 1-cocycle in the algebraic sense for the \( { {\mathbb{Z}} } \)-action \( { n(f)=f\circ T^n } \) on the group of measurable functions on X with values in G; hence the function \( { \varphi\colon X\to G } \) itself is often called a cocycle.)

Assume additionally that G is compact. Using the cocycle φ we define a  group extension \( T_{\varphi} \) on \( (X\times G,{\cal B}\otimes{\cal B}(G),\mu\otimes \lambda_G) \) (\( \lambda_G \) stands for Haar measure of G), where \( T_{\varphi}(x,g)=(Tx,\varphi(x)+g) \).

Special flow:

Given an ergodic automorphism T on a standard probability Borel space \( { {(X,{\cal B},\mu)} } \) and a positive integrable function \( { f\colon X\to{\mathbb{R}}^+ } \) we put

$$ \begin{aligned} X^f &=\{(x,t)\in X\,\times\,{\mathbb{R}}\colon 0\leq t < f(x)\}\:, \\ {\cal B}^f&={\cal B}\otimes{\cal B}({\mathbb{R}})|_{X^f}\:, \end{aligned} $$

and define \( { \mu^f } \) as normalized \( { \mu\otimes\lambda_{{\mathbb{R}}}|_{X^f} } \). By a  special flow we mean the \( { {\mathbb{R}} } \)-action \( { T^f=(T^f_t)_{t\in{\mathbb{R}}} } \) under which a point \( { (x,s)\in X^f } \) moves vertically with the unit speed, and once it reaches the graph of f, it is identified with \( { (Tx,0) } \).

Markov operator:

A linear operator \( J\colon L^2{(X,{\cal B},\mu)}\to L^2{(Y,{\cal C},\nu)} \) is called Markov if it sends non-negative functions to non-negative functions and \( { J1=J^\ast1=1 } \).

Unitary actions on Fock spaces:

If H is a separable Hilbert space then by \( H^{\odot n} \) we denote the subspace of n‑tensors of \( H^{\otimes n} \) symmetric under all permutations of coordinates, \( n\geq1 \); then the Hilbert space \( F(H):=\bigoplus_{n=0}^\infty H^{\odot n} \) is called a  symmetric Fock space . If \( V\in U(H) \) then \( F(V):=\bigoplus_ {n=0}^\infty V^{\odot n}\in U(F(H)) \) where \( { V^{\odot n} = V^{\otimes n} | H^{\odot n} } \).

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Acknowledgments

Research supported by the EU Program Transfer of Knowledge “Operator Theory Methods for Differential Equations” TODEQ and thePolish Ministry of Science and High Education.

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    Mariusz Lemańczyk

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Lemańczyk, M. (2009). Spectral Theory of Dynamical Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_511

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