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Reproducing kernel Hilbert space compactification of unitary evolution groups. (English) Zbl 1473.37101

Summary: A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator \(W_\tau\) on a reproducing kernel Hilbert space (RKHS). The operator \(W_\tau\) is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, \( W_\tau\) generates a unitary evolution group \(\{e^{tW_\tau}\}_{t\in\mathbb{R}}\) on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of \(W_\tau\) as \(\tau\to 0^+\), as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rössler system, are presented.

MSC:

37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
81V45 Atomic physics

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