What Is Ergodic Theory?

Eisner, Tanja; Farkas, Bálint; Haase, Markus; Nagel, Rainer

doi:10.1007/978-3-319-16898-2_1

Cited by 14 publications

(49 citation statements)

References 20 publications

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“…The Koopman operator theory 7 – 13 is already widely used to analyze data and provide models for complex dynamic processes. Mathematically, the Koopman operator 14 is a linear representation on an observables space of an appropriate group action on state space. As it is a linear representation, it is natural that key objects of analysis will be its eigenvalues and eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition

Hogg¹,

Fonoberova²,

Mezić

2020

Sci Rep

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Sea ice cover in the Arctic and Antarctic is an important indicator of changes in the climate, with important environmental, economic and security consequences. The complexity of the spatio-temporal dynamics of sea ice makes it difficult to assess the temporal nature of the changes—e.g. linear or exponential—and their precise geographical loci. In this study, Koopman Mode Decomposition (KMD) is applied to satellite data of sea ice concentration for the Northern and Southern hemispheres to gain insight into the temporal and spatial dynamics of the sea ice behavior and to predict future sea ice behavior. We observe spatial modes corresponding to the mean and annual variation of Arctic and Antarctic sea ice concentration and observe decreases in the mean sea ice concentration from early to later periods, as well as corresponding shifts in the locations that undergo significant annual variation in sea ice concentration. We discover exponentially decaying spatial modes in both hemispheres and discuss their precise spatial extent, and also perform predictions of future sea ice concentration. The Koopman operator-based, data-driven decomposition technique gives insight into spatial and temporal dynamics of sea ice concentration not apparent in traditional approaches.

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Section: Introductionmentioning

confidence: 99%

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman mode decomposition

Hogg¹,

Fonoberova²,

Mezić

2020

Sci Rep

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“…The following characterization of an operator having discrete spectrum can be found in [EFHN15,Theorem,16.36].…”

Section: 2mentioning

confidence: 99%

“…Therefore, minimal group rotations can be seen as the canonical representatives of minimal systems with discrete spectrum. Moreover, the Pontryagin duality theorem shows that (G, a) and (G * * , δ a ) are isomorphic which has two consequences: On the one hand, G * ∼ = G * (a) via χ → χ(a) and G * (a) = σ p (T ϕa ) where T ϕa denotes the Koopman operator of ϕ a , see [EFHN15,Propositions 14.22 and 14.24]. In particular, σ p (T ϕa ) is a subgroup of T and for the canonical inclusion ι :…”

Section: Realization and Uniquenessmentioning

confidence: 99%

“…So for ν-almost Proposition 2.7 instead of Theorem 2.8, one can continue the proof of Theorem 3.10 analogously. By von Neumann's theorem [EFHN15,Theorem 7.20] (X, ϕ) is then not only Markov-isomorphic but pointisomorphic to the rotation on a compact group rotation bundle.…”

Section: Realization and Uniquenessmentioning

confidence: 99%

“…Notation. Our notation and terminology are that of [EFHN15] in general. We abbreviate a probability space (X, Σ, µ) by writing X := (X, Σ, µ) and if ϕ : X → X is measurable and measure-preserving, we call (X, ϕ) a measure-preserving dynamical system.…”

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confidence: 99%

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On the isomorphism problem for non-minimal transformations with discrete spectrum

Edeko¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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The article presents a new perspective on the isomorphism problem for non-ergodic measure-preserving dynamical systems with discrete spectrum which is based on the connection between ergodic theory and topological dynamics constituted by topological models. By first solving the isomorphism problem for a certain class of topological dynamical systems, it is shown that the measure-preserving case can in fact be deduced from the topological one via the construction of topological models. As a byproduct, a new characterization of mean ergodicity for topological dynamical systems is obtained.

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