The cohomological equation for Roth-type interval exchange maps

Marmi, Stefano; Moussa, Pierre; Yoccoz, Jean-Christophe

doi:10.1090/s0894-0347-05-00490-x

Cited by 124 publications

(244 citation statements)

References 31 publications

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“…By proposition 1.3.1 in [9], if the i.e.m. satisfies condition (A), then for all ε > 0 there is C ε > 0 such that for all n 0…”

Section: Aq1mentioning

confidence: 90%

“…This property is fundamental in order to be able to group together several iterations of V to obtain the accelerated Zorich continued fraction algorithm introduced in [9].…”

Section: Continued Fraction Algorithms For Iemsmentioning

confidence: 99%

“…Please clarify whether edits to the sentence In [9] it......condition holds.' retain the intended meaning.…”

Section: Aq1mentioning

confidence: 99%

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The recurrence time for interval exchange maps

Kim¹,

Marmi²

2008

Nonlinearity

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“…By proposition 1.3.1 in [9], if the i.e.m. satisfies condition (A), then for all ε > 0 there is C ε > 0 such that for all n 0…”

Section: Aq1mentioning

confidence: 90%

“…This property is fundamental in order to be able to group together several iterations of V to obtain the accelerated Zorich continued fraction algorithm introduced in [9].…”

Section: Continued Fraction Algorithms For Iemsmentioning

confidence: 99%

See 1 more Smart Citation

The recurrence time for interval exchange maps

Kim¹,

Marmi²

2008

Nonlinearity

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“…There are other definitions of Diophantine type given by the size of continued fraction matrices in the Rauzy-Vecch induction algorithm [16,18]. For example Roth type Diophantine condition form a full measure set [21] and can be used for obtaining Hölder estimates for the solution of the cohomological equation [22]. See also [13] and [17] for more discussion on the size of the continued fraction matrices.…”

Section: 3mentioning

confidence: 99%

Long hitting time for translation flows and L-shaped billiards

Kim¹,

Marchese²,

Marmi³

2019

Journal of Modern Dynamics

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We consider the flow in direction θ on a translation surface and we study the asymptotic behavior for r → 0 of the time needed by orbits to hit the r -neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction θ.In higher genus, we consider a generalized geometric notion of Diophantine type of a direction θ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any square-tiled surface with the same topology the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and Diophantine type subsists. Our results apply to L-shaped billiards.

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“…Now we begin an account of the dynamics of the Teichmüller flow, viewed through the lens of Veech’s zippered rectangles construction. We draw in the following sections from the sources [AGY06, Via08] that both build on work of Marmi, Moussa and Yoccoz [MMY05].…”

Section: Introductionmentioning

confidence: 99%

On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

Magee¹

2019

Compositio Math.

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J.-C. Yoccoz proposed a natural extension of Selberg's Eigenvalue Conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg's 3 16 Theorem to moduli spaces of abelian differentials on surfaces of genus ≥ 2.• There is an action of SL 2 (R) on M. The restriction of the SL 2 (R) action to the one parameter diagonal subgroup gives a flow on M called the Teichmüller flow that generalizes the geodesic flow on the unit tangent bundle of X.• There is a unique probability measure ν M on M that is SL 2 (R)-invariant, ergodic for the Teichmüller flow, and in the Lebesgue class with respect to a natural affine orbifold structure on M. This is due to works of Masur [Mas82] and Veech [Vee82].• The space SO(2)\M is locally foliated by H and hence it is possible to define a foliated Laplacian ∆ M on SO(2)\M generalizing ∆ X . This operator has a simple eigenvalue at zero and by a result of Avila and Gouëzel [AG13], its spectrum below 1 4 has no accumulation points other than possibly at 1 4 . Each of these objects lifts to M(q), so there is an SL 2 (R) action on M(q) preserving a finite measure ν M(q) , and a foliated Laplacian ∆ M(q) whose spectrum below 1 4 does not accumulate 2 away from 1 4 . Hence we can write λ 1 (M(q)) for the infimum of the non-zero spectrum 3 of ∆ M(q) . The following extension of Selberg's conjecture to genus g ≥ 2 was proposed by Yoccoz 4 . Conjecture 1.3 (Yoccoz). For all q ≥ 2, and any connected component M of a stratum, A. λ 1 (M(q)) ≥ 1 4 .B. The measure on the unitary dual of SL 2 (R) that decomposes L 2 (M(q), ν M(q) ) is supported away from complementary series representations.The main theorem of this paper gives an approximation to Conjecture 1.3.Theorem 1.4. For any connected component M of a stratum, there exists ǫ, η > 0 and Q 0 ∈ Z + such that for all q coprime to Q 0 the following hold.B. The measure on the unitary dual of SL 2 (R) that decomposes L 2 (M(q), ν M(q) ) is supported away from complementary series representations Comp u with u ∈ (1 − η, 1).C. The Teichmüller flow on M(q) has exponential decay of correlations on compactly supported C 1 observables with a rate of decay that is independent of q.2 By [AG13, Remark 2.4] this result also applies to M(q).3 In contrast to the situation with X, where it is known [Sel56] that there are infinitely many eigenvalues of ∆X , we do not know whether ∆M or ∆ M(q) have any non-zero eigenvalues. 4 The formulation of the conjecture appears in print in [AG13], although Avila and Gouëzel stopped short of making the conjecture because of lack of evidence. We learned from C. Matheus that Yoccoz had made this conjecture in private.

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The cohomological equation for Roth-type interval exchange maps

Cited by 124 publications

References 31 publications

The recurrence time for interval exchange maps

The recurrence time for interval exchange maps

Long hitting time for translation flows and L-shaped billiards

On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

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