On the Ergodic Theory    of Equations of Mathematical Physics

Козлов, В. В.

doi:10.1134/s1061920821010088

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…Mapping ( 15) is an infinitedimensional variant of the Kronecker-Weyl mapping. Its properties are quite similar to the ergodic properties of continuous Kronecker-Weyl flows on infinitedimensional tori (they are discussed in [8,9], where further references can be found).…”

Section: Discrete Spectrummentioning

confidence: 65%

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Symplectic Geometry of the Koopman Operator

Козлов

2021

Dokl. Math.

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We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.

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Section: Discrete Spectrummentioning

confidence: 65%

“…To prove the involution of these invariants, we need to check the equality (9) Since the operators , and commute with each other, both sides of ( 9) are symmetric with respect to k and l. This proves equality (7).…”

Section: Quadratic Invariantsmentioning

confidence: 99%

Symplectic Geometry of the Koopman Operator

Козлов

2021

Dokl. Math.

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On a Paradoxical Property of the Shift Mapping on an Infinite-Dimensional Tori

Glyzin,

Kolesov

2024

Dokl. Math.

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On a paradoxical property of the shifting mapping on an infinite-dimensional tori

Glyzin,

Kolesov

2024

Doklady Rossijskoj akademii nauk. Matematika, informatika, proc

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An infinite-dimensional torus T∞=lp/2πZ∞, where lp, p ≥ 1 – space of sequences, Z∞ – natural integer lattice in lp, is considered. We study the classical question in the theory of dynamical systems about the behavior of trajectories of a shift mapping on the specified torus. More precisely, some sufficient conditions are proposed that guarantee the emptiness of the ω-limit and α-limit sets of any of the shift mapping onto T∞.

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On the Ergodic Theory of Equations of Mathematical Physics

Cited by 3 publications

References 16 publications

Symplectic Geometry of the Koopman Operator

Symplectic Geometry of the Koopman Operator

On a Paradoxical Property of the Shift Mapping on an Infinite-Dimensional Tori

On a paradoxical property of the shifting mapping on an infinite-dimensional tori

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