Measure-preserving rank one transformations

Ryzhikov, V. V.

doi:10.1090/mosc/310

Cited by 8 publications

(4 citation statements)

References 62 publications

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“…This process of cutting, adding and stacking continues to infinity and we obtain an invertible transformation of X that preserves the Lebesgue measure of intervals. See, for example, [10], [11], where numerous applications including ones for Gaussian and Poisson suspensions have been indicated.…”

Section: Poisson Suspensions With Completely Positive P -Entropymentioning

confidence: 99%

Spectra of Self-Similar Ergodic Actions

Ryzhikov¹

2023

Math Notes

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Answering Vershik's question we show that quasi-similarity does not conserve the entropy, proving quasisimilarity of all Bernoulli actions of the groups with an element of infinite order. We produce joining-stable invariants considering zero P -entropy systems and prove an analog of Pinsker's theorem establishing the disjointness of such systems with the actions of comletely positive P -entropy. Then we apply Poisson suspensions to get a class of the corresponding examples.

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Section: Poisson Suspensions With Completely Positive P -Entropymentioning

confidence: 99%

Spectra of Self-Similar Ergodic Actions

Ryzhikov¹

2023

Math Notes

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“…There are many applications of such weak limits, see, for example, [14], [19], and [20], where the typicalness of the limits aΘ + (1 − a)I has been proven. 1.2.…”

Section: Generic Properties Of Transformationsmentioning

confidence: 99%

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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“…V. V. Ryzhikov [18] generalized the Chacon transformation by using different size of spacers. The Chacon transformation is rank-one, and the examples and properties of rank-one transformations are studied in [19]. We might ask if our result applies to generalized Chacon transformations as well.…”

Section: Introductionmentioning

confidence: 97%

Size of exceptional sets in weakly mixing systems

Park¹,

Park²

2023

Preprint

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For any weakly mixing system (X, B, µ, T ) and A, B ∈ B, there is a density zero set JA,B such that µ(A ∩ T −n B) converges to µ(A)µ(B) for n / ∈ J. In this paper, we study bounds on the size of this exceptional set J. First, we show that, given the rate of weak mixing, we can find an upper bound on the size of J. We use this property to show the existence of an exceptional set J of an interval exchange system with eitherdepending on whether or not the transformation is of rotation class. We also explicitly construct an exceptional set J for the Chacon transformation and give upper and lower bounds for its size. More precisely, we show that there is a constant C > 0 such that for any increasing function h : R+ → R+ diverging to infinity, there is an exceptional set n) . For the lower bound, we prove that for any t > 0, there exists Lebesgue measurable sets A, B ⊆ [0, 1] and C, N > 0 such that |JA,B ∩ [0, n]| ≥ C(log n) t for every n > N .

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Measure-preserving rank one transformations

Cited by 8 publications

References 62 publications

Spectra of Self-Similar Ergodic Actions

Spectra of Self-Similar Ergodic Actions

Entropy invariants of generic actions

Size of exceptional sets in weakly mixing systems

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