Generic mixing transformations are rank 1

Bashtanov, A. I.

doi:10.1134/s0001434613010227

Cited by 13 publications

(7 citation statements)

References 10 publications

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“…The space of mixing actions. Bashtanov's result [3] together with the well-known facts shows that the generic mixing transformation has no factor and commutes only with its powers. So for the space Mix there are no analogues of the mentioned sophisticated results by Ageev, Glasner-Weiss, King, de la Rue-de Sam Lazaro, and Eremenko-Stepin.…”

Section: Closing Remarksmentioning

confidence: 89%

Entropy invariants of generic actions

Ryzhikov¹

2020

Preprint

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We show that the typical dynamical system sometimes begins to behave like a non-deterministic system with a small classical entropy, and this behavior lasts an extremely long time, until the system starts decreasing entropy. Then again it will become almost non-deterministic for a very very long time, but with more smaller classical entropy. Playing on this fact and considering sigma-compact families of measurepreserving zero-entropy transformations, for example, the rectangle exchange transformations, we choose the Kushnirenko entropy so that it is equal to zero for the transformations under consideration, but is infinite for the generic transformation.

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Section: Closing Remarksmentioning

confidence: 89%

Entropy invariants of generic actions

Ryzhikov¹

2020

Preprint

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“…Bashtanov [31], [32] proved that the conjugacy classes (of mixing automorphisms) are dense in this topology. Hence, properties like to have trivial centralizer and no (non-trivial) factors are typical in this topology.…”

Section: Markov Operators Joinings and Koopman Representations Disjoi...mentioning

confidence: 94%

Spectral theory of dynamical systems

Kanigowski¹,

Lemańczyk²

2020

Preprint

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1 Definition of the subject Spectral theory of dynamical systems is a study of special unitary representations, called Koopman representations (see the glossary). Invariants of such representations are called spectral invariants of measure-preserving systems. Together with the entropy, they consitute the most important invariants used in the study of measure-theoretic intrinsic properties and classification problems of dynamical systems as well as in applications. Spectral theory was originated by von Neumann, Halmos and Koopman in the 1930s. In this article we will focus on recent progresses in the spectral theory of finite measure-preserving dynamical systems.

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“…A "typical" automorphism of a standard probability Borel space has the rank one property. The "typicality" of rank one is still true in the Alpern-Tikhonov topology we mentioned in section "Maximal Spectral Type of a Koopman Representation: Alexeyev's Theorem" by Bashtanov (2013). Baxter (1971) showed that the maximal spectral type of such a T is realized by a characteristic function.…”

Section: Rank-1 and Related Systemsmentioning

confidence: 99%

Spectral Theory of Dynamical Systems

Kanigowski¹,

Lemańczyk²

2020

Encyclopedia of Complexity and Systems Science

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Spectral theory of dynamical systems is a study of special unitary representations, called Koopman representations (see Section "Glossary and Notation"). Invariants of such representations are called spectral invariants of measure-preserving systems. Together with the entropy, they constitute the most important invariants used in the study of measure-theoretic intrinsic properties and classification problems of dynamical systems as well as in applications. Spectral theory was originated by von Neumann, Halmos, and Koopman in the 1930s. In this article, we will focus on recent progresses in the spectral theory of finite measure-preserving dynamical systems.

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Generic mixing transformations are rank 1

Cited by 13 publications

References 10 publications

Entropy invariants of generic actions

Entropy invariants of generic actions

Spectral theory of dynamical systems

Spectral Theory of Dynamical Systems

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