Compact operator semigroups applied to dynamical systems

Kreidler, Henrik

doi:10.1007/s00233-018-9958-x

Cited by 4 publications

(9 citation statements)

References 33 publications

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“…Section 3 classifies and strengthens the corresponding results contained in the recent papers [15,19,20]. Theorem 3.3 establishes that if (Ω, ϕ) ∈ D tm , then every ergodic sequence V contains a convergent subsequence; in particular, there exists a convergent subsequence of Cesáro means.…”

Section: Introductionmentioning

confidence: 61%

“…In the tame case, one can say much more about the convergence of ergodic means. Independently, the equivalence (AEN) ⇔ (single m in o) for (Ω, ϕ) ∈ D tm was established in [15,Theorem 5.10]. Theorem 3.4, in particular, ensures the uniquely ergodicity of minimal tame semicascades, a fact obtained for a wider class of tame systems as early as in [9,13].…”

Section: Convergence Of Ergodic Meansmentioning

confidence: 94%

“…According to [19,Theorem 3.2], all ergodic nets (1) converge if and only if Ker K c consists of a single element, which is necessarily the zero element of the semigroup K c . The paper [15] uses a slightly different (wider and more traditional) definition of ergodic nets; namely, it is assumed that V α ∈ co K 0 = K c in (1). Nevertheless, the condition card Ker K c = 1 also implies the convergence of all nets of this kind [15, Theorem 4.3].…”

Section: Preliminariesmentioning

confidence: 99%

“…. Properties (c) and (e) as equivalent definitions of a tame dynamical system arose in [15,Proposition 3.11] and [20,Theorem 3.4], respectively; the remaining properties can be found in [5,6]. Essentially, the semigroups E(Ω, ϕ) and K c (Ω, ϕ) in conditions (a) and (c) are sequentially compact.…”

Section: Preliminariesmentioning

confidence: 99%

“…The interest in such objects is due to the relatively simple topology of their enveloping semigroups often combined with a pretty complex phase dynamics. A number of assertions on the convergence of generalized ergodic means for (Ω, ϕ) ∈ D tm were established in [15,20], where the paper [15] deals with the more general case in which arbitrary amenable operator semigroups act on X. There are reasons to believe that the tame-untame dichotomy is somehow related to the absence or existence of chaotic phase dynamics.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Ergodic Properties of Tame Dynamical Systems

Romanov

2019

Math Notes

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We study the problem on the weak-star decomposability of a topological N 0 -dynamical system (Ω, ϕ), where ϕ is an endomorphism of a metric compact set Ω, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup E(Ω, ϕ) consists of B 1 -transformations Ω → Ω), we show that (i) the desired decomposition exists for an appropriate choice of the generalized sequential averaging method; (ii) every sequence of weighted ergodic means for the shift operator x → x • ϕ, x ∈ C(Ω), contains a pointwise convergent subsequence. We also discuss the relationship between the statistical properties of (Ω, ϕ) and the mutual structure of minimal sets and ergodic measures.2010 Mathematics Subject Classification: Primary 37A30, 47A35; Secondary 20M20

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Section: Introductionmentioning

confidence: 61%

Section: Convergence Of Ergodic Meansmentioning

confidence: 94%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ergodic Properties of Tame Dynamical Systems

Romanov

2019

Math Notes

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Diffeomorphically Learning Stable Koopman Operators

Bevanda

Beier

Sebastian

et al. 2022

IEEE Control Syst. Lett.

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Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear dynamical systems. This makes system analysis and long-term predictions simple -involving only matrix multiplications. However, the transformation to a linear system is generally non-trivial and unknown, requiring learning-based approaches. While there exists a variety of approaches, they usually lack crucial learning-theoretic guarantees, such that the behavior of the obtained models with increasing data and dimensionality is often unclear. We address the aforementioned by deriving a novel reproducing kernel Hilbert space (RKHS) that solely spans transformations into linear dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization risk bounds under weaker assumptions than existing work. Our numerical experiments indicate advantages over state-of-the-art statistical learning approaches for Koopman-based predictors.

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On discrete semi-flows with universally measurable Ellis semigroup

Romanov¹

2017

ijma

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We show that, in the framework of ZFC axioms, any continuous compact discrete semi-flow whose Ellis semigroup consists of universally measurable transformations is tame, and the union of its minimal sets is dense in the minimal attraction center. At the same time, under ZFC, the Ellis semigroups of a broad class of discrete dynamical systems contain transformations that are not universally measurable.

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Compact operator semigroups applied to dynamical systems

Cited by 4 publications

References 33 publications

Ergodic Properties of Tame Dynamical Systems

Ergodic Properties of Tame Dynamical Systems

Diffeomorphically Learning Stable Koopman Operators

On discrete semi-flows with universally measurable Ellis semigroup

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