Generalizations of theorems on mixing flows with non-degenerate saddle points on a 2-torus

Кочергин, Андрей Васильевич

doi:10.1070/sm2004v195n09abeh000843

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…The question about mixing of such flows, risen in the same paper [Arn91], was answered by Sinai and Khanin in [SK92], where it was proved that, under a generic diophantine condition on the rotation angle, suspension flows with asymmetric singularities over a rotation are strongly mixing (see also [Kha96]). The diophantine condition of [SK92] was weakened by Kochergin in a series of works ([Koc03b,Koc04a,Koc04b,Koc04c]).…”

Section: Motivation and Main Referencesmentioning

confidence: 99%

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Ulcigrai¹

2007

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385706000836How to cite this article: CORINNA ULCIGRAI (2007). Mixing of asymmetric logarithmic suspension ows over interval exchange transformations.Abstract. We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

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Section: Motivation and Main Referencesmentioning

confidence: 99%

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Ulcigrai¹

2007

Ergod. Th. Dynam. Sys.

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“…The question posed by Arnold was answered by Sinai and Khanin [25], who proved that, under a full-measure Diophantine condition on the rotation angle, the flow is mixing. This condition was weakened by Kochergin [12,13,14,15]. The presence of singularities in the roof function is necessary, as well as the asymmetry condition: in this setting, mixing does not occur for functions of bounded variation or, assuming a full-measure Diophantine condition on the rotation angle, for functions with symmetric logarithmic singularities; see the results by Kochergin in [8] and [11] respectively.…”

Section: Introductionmentioning

confidence: 99%

Quantitative Mixing for Locally Hamiltonian Flows with Saddle Loops on Compact Surfaces

Ravotti

2017

Ann. Henri Poincaré

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Abstract. Given a compact surface M with a smooth area form ω, we consider an open and dense subset of the set of smooth closed 1-forms on M with isolated zeros which admit at least one saddle loop homologous to zero and we prove that almost every element in the former induces a mixing flow on each minimal component. Moreover, we provide an estimate of the speed of the decay of correlations for smooth functions with compact support on the complement of the set of singularities. This result is achieved by proving a quantitative version for the case of finitely many singularities of a theorem by Ulcigrai (Ergod Theory Dyn Syst 27(3): 2007), stating that any suspension flow with one asymmetric logarithmic singularity over almost every interval exchange transformation is mixing. In particular, the quantitative mixing estimate we prove applies to asymmetric logarithmic suspension flows over rotations, which were shown to be mixing by Sinai and Khanin.

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“…In the non-symmetric logarithmic case the special flow built over any irrational rotation is mixing (see [21]), and hence the method of proving the absence of self-similarity presented in Section 6 fails. In [26] de la Rue and de Sam Lazaro have shown that a typical automorphisms of a standard Borel probability space is embeddable in a measurable flow; that is, a typical automorphism T is isomorphic to the time-1 map T 1 of a measurable flow (T t ) t∈R .…”

Section: Open Problemsmentioning

confidence: 99%

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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Generalizations of theorems on mixing flows with non-degenerate saddle points on a 2-torus

Cited by 6 publications

References 17 publications

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Quantitative Mixing for Locally Hamiltonian Flows with Saddle Loops on Compact Surfaces

On the self-similarity problem for ergodic flows

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