Stochastic intertwinings and multiple mixing of dynamical systems

Ryzhikov, V. V.

doi:10.1007/bf02259620

Cited by 20 publications

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References 26 publications

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“…The purpose of the first part of the paper (Sections 2 and 3) is to extend (1.2) to measures ρ singular with respect to μ. Section 2 contains some abstract results based on Ryzhikov's ideas from [29]. In Section 3, these results are used to prove that if the joint action of the geodesic and the horocycle flow on ρ is strongly continuous, then (1.2) holds (see Theorem 3.3, Corollary 3.5 and Theorem 3.9).…”

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“…Nevertheless, as it was proved by Ryzhikov [27], the absence of mixing for T implies zero Lebesgue measure of I(T ) and zero (two-dimensional) Lebesgue measure of I aut (T ). Furthermore, if T is additionally rigid (that is, T tn → Id for some t n → ∞), then T and T s are disjoint in the sense of Furstenberg for almost every s ∈ R, and T s is disjoint from T t for almost every (s, t) ∈ R 2 with respect to the Lebesgue measure (see [29]). In Section 4, using results of Section 2 we extend the disjointness result (see Theorem 4.2) to the class of weakly mixing flows for which there exist t n → ∞, 0 < λ 1 and a probability Borel measure P on R such that Some applications of Theorem 4.2 for special flows over interval exchange transformations are shown in Section 5.…”

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confidence: 99%

“…Take an arbitrary countable multiplicative subgroup G ⊂ R * . The example (pointed in Section 6) of weakly mixing flow T with the MSJ property and such T t and T s are disjoint for distinct s and t allows us to construct (the idea of this construction comes from [29]) a self-similar flow T G such that I(T G ) = G and T G s is disjoint from T G for all s / ∈ G. If G ⊂ R * is uncountable, then the problem of realization remains open. Section 7 uses the results of Section 6 and Appendix B in an essential way.…”

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On the self-similarity problem for ergodic flows

Frączek

Lemańczyk

2009

Proceedings of the London Mathematical Society

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Given an ergodic flow (Tt) t∈R we study the problem of its self-similarities, that is, we want to describe the set of s ∈ R for which the original flow is isomorphic to the flow (Tst) t∈R . The problem is examined in some classes of special flows over irrational rotations and over interval exchange transformations. In particular, translation flows on translation surfaces are considered: we prove that under the weak mixing condition the set of self-similarities has Lebesgue measure zero. For von Neumann special flows over irrational rotations given by Diophantine numbers, this set is shown to be equal to {1}, while for horocycle flows a weak convergence in case of some singular (with respect to the volume measure) measures is shown to give rise to some new equidistribution result. The problem of self-similarity is also studied from the spectral point of view, especially in the class of Gaussian systems.Contents SI(T ) = {s ∈ R * : T and T s are spectrally isomorphic}.Clearly, SI(T ) is a multiplicative subgroup of R * , it contains I(T ) and since T preserves the space of real-valued functions, also −1 ∈ SI(T ). If T is spectrally self-similar, that is, SI(T ) = {−1, 1} and SI(T ) has positive Lebesgue measure, then T has pure Lebesgue spectrum (see Proposition 8.6). On the other side, if T has singular continuous spectrum, then SI(T ) has zero Lebesgue measure and T s is spectrally disjoint from T for almost all s (Theorem 8.4). Moreover, T s and T t are spectrally disjoint for almost all (s, t) ∈ R 2 . The spectral disjointness results were inspired by a discussion of the second author with B. Host in 1994. † In Section 9, we construct ergodic flows that are not self-similar in the unitary category (see Theorem 9.4 and Remark 9.5). For this purpose, we deal with Gaussian systems, that is dynamical systems that are completely determined by the spectral measure of the underlying Gaussian process. A construction of a measure that is supported by a set, which emulates a classical Kronecker set yields a Gaussian flow T with simple spectrum such that SI(T ) = {−1, 1} and T s is spectrally disjoint from T for s = ±1. Moreover, for some countable multiplicative symmetric subgroups G ⊂ R * a modification of the construction yields † B. Host showed Lemma 8.2 below for continuous singular measures on R.

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On the self-similarity problem for ergodic flows

Frączek

Lemańczyk

2009

Proceedings of the London Mathematical Society

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“…The properties S(n -1, n) were studied in [1][2][3][4][5][6]. The properties S(n -1, n) were studied in [1][2][3][4][5][6].…”

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“…The properties S(n -1, n) were studied in [1][2][3][4][5][6]. It is known (see [4,6]) that i) all (even) properties S(2p -1,2p), p > 1, are equivalent;ii) the property S(3, 4) implies the property S(2q, 2q + 1) for any q = 1,2, ....In this paper, we consider an example of noncommutative action ~ having the property S(2q, 2q + 1) but not having the property S(2p-1,2p). It is known (see [4,6]) that i) all (even) properties S(2p -1,2p), p > 1, are equivalent;…”

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Even and odd primality of dynamical systems with invariant measure

Ryzhikov

1996

Math Notes

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KEY WOADS: dynamical system, invariant measure, multiple mixing, joining theory.In this paper, by a dynamical system we mean an action ~ = {Tg : g E G} of a group G in a Lebesgue space (X, g), g(X) = 1 such that ~ preserves the measure g. We say that the measure u given on the Cartesian cube X" belongs to the class M(n -1, n), n > 2, if its projections on the (n -1)-dimensional faces of the cube X" are equal to p,,-l. We say that an action q~ has the property S(n -1, n) (or belongs to the class S(n -1, n)) if g" is the only measure of the class M(n -1, n) that is invariant under the action 9 | q~ | | q~ (n times). The properties S(n -1, n) were studied in [1][2][3][4][5][6]. The interest in these properties arises from some problems of joining theory (see [3]) and from the investigation of multiple mixing (if a commuting mixing system of the class S(n, n + 1) intermixes with multiplicity n -1, then it also intermixes with multiplicity n, (see [5]).) It is known (see [4,6]) that i) all (even) properties S(2p -1,2p), p > 1, are equivalent;ii) the property S(3, 4) implies the property S(2q, 2q + 1) for any q = 1,2, ....In this paper, we consider an example of noncommutative action ~ having the property S(2q, 2q + 1) but not having the property S(2p-1,2p). In this case we say that the operation 'IJ has the odd tensor primality. This terminology can be explained by -the following. The dynamical system 9 corresponds to the group ~ of unitary operators acting in the space L2(X, g). The odd tensor primality means that does not allow nontrivial Markov intertwinings with even tensor degree of the system ~.Let us define the action ~. First, note that the automorphisms of the group X = Z2 x Z2 x Z2 x -.. and the shifts on X preserve the Haar measure /~ on X. For a permutation a of the set of natural numbers N, we define an automorphism T~, of the group X by setting T~,({zi}) = {z~,(0}. We assign a shift S~(x) = z + a to the sequence a = {ai} E X ; here ' +' is the group operation in X. In the role of ~I', we consider the action generated by all T,, and S,,(z).

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Spectral Theory of Dynamical Systems

Kanigowski

Lemańczyk

2023

Encyclopedia of Complexity and Systems Science Series

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Stochastic intertwinings and multiple mixing of dynamical systems

Cited by 20 publications

References 26 publications

On the self-similarity problem for ergodic flows

On the self-similarity problem for ergodic flows

Even and odd primality of dynamical systems with invariant measure

Spectral Theory of Dynamical Systems

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