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Spectral Theory of Dynamical Systems

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Maximal Spectral Type of a Koopman Representation, Alexeyev's Theorem

Spectral Theory of Weighted Operators

The Multiplicity Function

Rokhlin Cocycles

Rank-1 and Related Systems

Spectral Theory of Dynamical Systems of Probabilistic Origin

Inducing and Spectral Theory

Special Flows and Flows on Surfaces, Interval Exchange Transformations

Future Directions

Acknowledgments

Bibliography

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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 \({\mathbb{A}}(x):=\overline{\operatorname{span}}\{U_ax\colon a\in{\mathbb{A}}\}\) 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 \(\mathcal{T}\) :

Let \({\mathbb{A}}\) be a l.c.s.c. (and not compact) Abelian group and \(\mathcal{T}\colon a\mapsto T_a \) a representation of \({\mathbb{A}}\) in the group \( \operatorname{Aut}{(X,{\cal B},\mu)}\) of (measure‐preserving) automorphisms of a standard probability Borel space \({(X,{\cal B},\mu)}\). The Koopman representation \({\cal U}={\cal U}_{\mathcal{T}}\) of \(\mathcal{T}\) 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 \(\mathcal{T}\) :

A measure‐preserving \({{\mathbb{A}}}\)-action \(\mathcal{T}=(T_a)_{a\in{\mathbb{A}}}\) is called ergodic if \({\chi_0\equiv1\in\widehat{\mathbb{A}}}\) is a simple eigenvalue of \({{\cal U}_{\mathcal{T}}}\). It is weakly mixing if \({{\cal U}_{\mathcal{T}}}\) has a continuous spectrum on the subspace \({L^2_0{(X,{\cal B},\mu)}}\) of zero mean functions. \(\mathcal{T}\) 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; \(\mathcal{T}\) is said to be mildly mixing if it has no non-trivial rigid factors. We say that \(\mathcal{T}\) 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 \(\mathcal{T}\) are called spectrally disjoint if the maximal spectral types of their Koopman representations \({{\cal U}_{\mathcal{T}}}\) 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 \( \smash{(\chi_1,\ldots,\chi_n)\mapsto\chi_1\cdot\ldots\cdot\chi_n} \)) \( \smash{\sigma^{\otimes n}=\int_{\widehat{\mathbb{A}}}\nu_\chi\,d\sigma^{(n)}(\chi)} \) 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 \(\mathcal{T}\) satisfies the SCS property if the maximal spectral type of \({\cal U}_{\mathcal{T}}\) on \( L^2_0 \) is a type of an SCS measure.

Kolmogorov group property:

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

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 the Polish Ministry of Science and High Education.

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  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Mariusz Lemańczyk

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Lemańczyk, M. (2012). Spectral Theory of Dynamical Systems. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_104

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