Dynamical systems and ergodic theory Definition

Spectral theory of dynamical systems

Builds on Probability space Measure-preserving transformation Koopman operator Unitary operator Eigenvalue

Definition

Let $(X, \mathcal{B}, \mu)$ and $(Y, \mathcal{C}, \nu)$ be probability spaces, and let $T \colon X \to X$ and $S \colon Y \to Y$ be invertible measure-preserving transformations with measurable measure-preserving inverses. Let $U_T$ and $U_S$ be their Koopman operators on $L^2(\mu)$ and $L^2(\nu)$ .

$T$ and $S$ are spectrally isomorphic if there is a unitary operator $W \colon L^2(\mu) \to L^2(\nu)$ with $W U_T = U_S W$ .
A spectral invariant is a property of such transformations that holds for $S$ whenever it holds for $T$ and $T$ , $S$ are spectrally isomorphic.
$T$ has discrete spectrum if $L^2(\mu)$ has an orthonormal basis of eigenvectors of $U_T$ .

The spectral theory of dynamical systems is the study of transformations up to spectral isomorphism, through the spectrum and the spectral measures of $U_T$ .

Remarks

Isomorphic transformations are spectrally isomorphic: an isomorphism $\varphi$ with $\varphi \circ T = S \circ \varphi$ gives $W f = f \circ \varphi^{-1}$ .

Example of a spectral invariant: ergodicity, since $T$ is ergodic if and only if the eigenvalue $1$ of $U_T$ has multiplicity $1$ . Weak mixing is another: it holds if and only if $1$ is a simple eigenvalue and $U_T$ has no other eigenvalue. An irrational rotation $x \mapsto x + \alpha \bmod 1$ has discrete spectrum, with basis $e^{2\pi i k x}$ and eigenvalues $e^{2\pi i k \alpha}$ , $k \in \Z$ .

Non-example: the Kolmogorov–Sinai entropy is not a spectral invariant. Two Bernoulli shifts with different entropies, such as the $(1/2, 1/2)$ and the $(1/3, 1/3, 1/3)$ shifts, are spectrally isomorphic. Both Koopman operators have countable Lebesgue spectrum on the orthogonal complement of the constants.

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Eigenvalue concept
Koopman operator concept
Measure-preserving transformation concept
Probability space concept
Unitary operator concept
Spectral theory of dynamical systems definition

Tag	`PV2DGD23AY`
URI	`m2://PV2DGD23AY`
Kind	definition
Writing	W0 stub
MSC	37A30, 37A35, 37A05
Defines	Spectral theory of dynamical systems
Uses	Probability space, Measure-preserving transformation, Koopman operator, Unitary operator, Eigenvalue

Spectral theory of dynamical systems

Definition

Remarks

Reading path

Further reading

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