Time-changes of horocycle flows

Forni, Giovanni; Ulcigrai, Corinna

doi:10.3934/jmd.2012.6.251

Cited by 36 publications

(64 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this proof of mixing, shearing is the key phenomenon. We show that the speed of decay of correlations can be reduced to the speed of equidistribution of the flow by an argument in the spirit of Marcus [17], using a bootstrap trick inspired by [5]. The geometric mechanism is the following: each horizontal segment {(x, y) : x ∈ J ∈ P f (t)} in X(t) gets sheared along the flow direction and approximates a long segment of an orbit of the flow φ t , see Figure 3.…”

Section: Final Partition and Mixing Setmentioning

confidence: 99%

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

Ravotti

2017

Ann. Henri Poincaré

View full text Add to dashboard Cite

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.

show abstract

Section: Final Partition and Mixing Setmentioning

confidence: 99%

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

Ravotti

2017

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…For time changes of horocycle flows, polynomial decay of correlations, as well as the Lebesgue spectral property, were proved in [11]. For time changes of nilflows, even for Heisenberg nilflows of bounded type, it is unclear whether the spectrum has an absolutely continuous component.…”

Section: Introductionmentioning

confidence: 99%

Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

Forni¹,

Kanigowski²

2019

Journal De L’École Polytechnique — Mathématiques

Self Cite

View full text Add to dashboard Cite

We study time changes of bounded type Heisenberg nilflows (φ t ) acting on the Heisenberg nilmanifold M . We show that for every positive τ ∈ W s (M ), s > 7/2, every non-trivial time change (φ τ t ) enjoys the Ratner property. As a consequence every mixing time change is mixing of all orders. Moreover we show that for every τ ∈ W s (M ), s > 9/2 and every p, q ∈ N, p = q, (φ τ pt ) and (φ τ qt ) are disjoint. As a consequence Sarnak Conjecture on Möbius disjointness holds for all such time changes.

show abstract

“…A natural question is which ergodic properties persist under perturbations: ergodicity is preserved by time-changes, but mixing is more delicate. The case of time-changes of the horocycle flow and of unipotent flows on semisimple Lie groups have been studied by many authors, including Marcus [12], Forni and Ulcigrai [7], Tiedra de Aldecoa [16], and Simonelli [14]; in this paper, building on a previous work by Avila, Forni and Ulcigrai [1], we address the question of mixing for time-changes of nilflows. The simplest non-abelian nilpotent group is the Heisenberg group H consisting of 3 × 3 upper triangular unipotent matrices, which is 3-dimensional and 2-step nilpotent.…”

Section: Introductionmentioning

confidence: 99%

Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows

Ravotti¹

2018

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We consider suspension flows over uniquely ergodic skew-translations on a d-dimensional torus T d , for d ≥ 2. We prove that there exists a set R of smooth functions, which is dense in the space C (T d ) of continuous functions, such that every roof function in R which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uniquely ergodic nilflow on a quasi-abelian filiform nilmanifold, there exists a dense subspace of smooth time-changes in which mixing occurs if and only if the time-change is not cohomologous to a constant. This generalizes a theorem by Avila, Forni and Ulcigrai (J. Diff. Geom., 2011) for the classical Heisenberg group.In this section, we present the general structure of the proof of Theorem 1.1-(a), stating some intermediate results, which are proved in later sections.Let T be a uniquely ergodic skew-translation as in (1.1). If we denote by E j the image of the linear map (A − Id) j , we have a filtration of R d into rational subspacesUp to a linear isomorphism, we can assume that the basis {e 1 , . . . , e d } of Z d is adapted to the filtration above, in particular {e d0+1 , . . . , e d } is a basis of E k , where d − d 0 = dim E k . Since T is not a rotation, k ≥ 1 and 1 ≤ d 0 ≤ d − 1. We remark that w ∈ E j for j ≥ 1 if and only if there exists v ∈ E j−1 such that v(A − Id) = w, i.e. vA = v + w. In particular,

show abstract

Time-changes of horocycle flows

Cited by 36 publications

References 20 publications

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

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

Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows

Contact Info

Product

Resources

About