On the self-similarity problem for smooth flows on orientable surfaces

Kułaga, Joanna

doi:10.1017/s0143385711000459

Cited by 12 publications

(9 citation statements)

References 25 publications

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“…Weak mixing follows immediately from [31], as the class of IETs considered there includes all IETs of bounded type. Moreover, under the additional assumption that the base IET satisfies so-called balanced partition lengths condition, such flows are not partially rigid [21]. In Section 3 we show that for an IET this additional condition is equivalent to being of bounded type, thus making the proof of Corollary 1.5 complete.…”

Section: Main Results and Its Consequencesmentioning

confidence: 92%

“…Definition 3.2 (cf. [21]). We say that the interval exchange transformation T = T π,λ has balanced partition lenghts whenever there exists c > 0 such that for any n ∈ N the following two conditions hold:…”

Section: Iets Of Bounded Typementioning

confidence: 99%

“…Remark 3.1. In the original definition of balanced partition lenghts in [21] the quantifiers were different: in (ii) instead of 0 ≤ j ≤ n − 1 only j = 0 was considered.…”

Section: Iets Of Bounded Typementioning

confidence: 99%

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Ratner’s property and mild mixing for smooth flows on surfaces

Kanigowski¹,

Kułaga-Przymus²

2015

Ergod. Th. Dynam. Sys.

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Let T = (T f t ) t∈R be a special flow built over an IET T : T → T of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T . We show that T satisfies so-called switchable Ratner's property which was introduced in [4]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [31] and not mixing [32]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows which are mildly mixing and not mixing.

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Section: Main Results and Its Consequencesmentioning

confidence: 92%

Section: Iets Of Bounded Typementioning

confidence: 99%

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Ratner’s property and mild mixing for smooth flows on surfaces

Kanigowski¹,

Kułaga-Przymus²

2015

Ergod. Th. Dynam. Sys.

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“…The following result shows that the discontinuities for iterations of IETs of periodic type are well distributed. Proposition 3.3 (see [22]). For every IET T of periodic type there exists c ≥ 1 such that for every n ≥ 1 we have…”

Section: Ergodicity Of Piecewise Linear Cocyclesmentioning

confidence: 99%

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

Conze

Frączek

2011

Advances in Mathematics

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International audienceWe consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincaré sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surface

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“…14 In [25] it is shown that on on each compact orientable surface of genus at least 2 there is a smooth (non-singular) non-self-similar flow. It is unknown whether these constructions are non-reversible.…”

Section: 2mentioning

confidence: 99%

Non-reversibility and self-joinings of higher orders for ergodic flows

Frączek

Kułaga-Przymus

Lemańczyk

2014

JAMA

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By studying the weak closure of multidimensional off-diagonal self-joinings we provide a criterion for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid automorphisms. In particular, we apply the criterion to special flows over irrational rotations, providing a large class of non-reversible flows, including some analytic reparametrizations of linear flows on T 2 , so called von Neumann's flows and some special flows with piecewise polynomial roof functions. A topological counterpart is also developed with the full solution of the problem of the topological self-similarity of continuous special flows over irrational rotations. This yields examples of continuous special flows over irrational rotations without topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities. ContentsClearly, J / ∈ SL 2 (R). However, if Γ satisfies Γ = J −1 ΓJ then J will also act on Γ\P SL 2 (R):and since J yields an order two map, we obtain that in this case the horocycle flow is reversible. It follows that I((h t ) t∈R ) = R * .Corollary 1.1. In the modular case Γ := P SL 2 (Z) ⊂ P SL 2 (R), the horocycle flow (h t ) t∈R is reversible.There are even cocompact lattices Γ which are not "compatible" with the matrix J. In this case a deep theory of Ratner [37] implies that in particular (h t ) t∈R is not measure-theoretically isomorphic to its inverse. 5 Notice that the same argument works for an arbitrary Koopman representation Ut = U Tt . In other words, an arbitrary Koopman representation is unitarily reversible.

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On the self-similarity problem for smooth flows on orientable surfaces

Abstract: On each compact connected orientable surface of genus greater than one we construct a class of flows without self-similarities.

Cited by 12 publications

References 25 publications

Ratner’s property and mild mixing for smooth flows on surfaces

Ratner’s property and mild mixing for smooth flows on surfaces

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

Non-reversibility and self-joinings of higher orders for ergodic flows

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