KoopmanVonNeumannTheory

Koopman-von Neumann theory extends the study of dynamical systems, ${T^t}_{t \in \mathbb{R}}$, where $T^t: X \rightarrow X$ are transformations on a measure space $(X, \mathcal{B}, \mu)$, by analyzing the action of the transformations on functions in $L^2(X, \mu)$ (the space of square-integrable functions).

The Koopman operator, $U^t$, associated with a dynamical system is defined by:

$$U^t f(x) = f(T^t x)$$

Here, $f$ is an observable of the system, a real or complex-valued function on $X$, and $x$ represents a point in the state space $X$.

Stone’s theorem ensures that for any strongly continuous one-parameter unitary group ${U^t}_{t \in \mathbb{R}}$, there exists a unique self-adjoint operator $A$ such that:

$$U^t = e^{itA}$$

This links the study of the spectrum of the self-adjoint operator $A$ to the spectral properties of the dynamical system, allowing for a detailed analysis of the system's ergodic properties.

In this framework, ergodicity is related to the spectrum of the Koopman operator. For instance, if the spectrum is purely point, and each eigenfunction is integrable, the system is said to be uniquely ergodic, implying that there exists a unique invariant measure under which the time averages of observables converge to their space averages.

Koopman-von Neumann theory also encompasses the spectral analysis of the Koopman operator, allowing for the exploration of eigenfunctions, eigenvalues, and the continuous spectrum. This spectrum provides profound insights into the dynamical properties of the system, like recurrence, stability, and ergodicity, and it serves as a bridge to link deterministic dynamics with statistical properties, permitting advanced analysis on non-linear and complex systems.