Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials

Bufetov, Alexander I.

doi:10.1090/s0894-0347-06-00528-5

Cited by 51 publications

(84 citation statements)

References 23 publications

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“…However, in the case of the Rauzy-Veech algorithm the measure has infinite mass whereas the invariant measure for the Zorich algorithm has finite mass. The ergodic properties of these renormalisation dynamics in parameter space have been studied in detail ( [Vee84a], [Vee84b], [Zor97], [Zor99], [AGY06], [Buf06], [AB07], [Yoc10]). …”

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confidence: 99%

Linearization of generalized interval exchange maps

2012

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A standard interval exchange map is a one-to-one map of the interval that is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type, which is almost surely satisfied in parameter space. Let T0 be a standard interval exchange map of restricted Roth type, and let r be an integer 2. We prove that, amongst C r+3 deformations of T0 that are C r+3 tangent to T0 at the singularities, those that are conjugated to T0 by a C r -diffeomorphism close to the identity form a C 1 -submanifold of codimension (g − 1)(2r + 1) + s.Here, g is the genus and s is the number of marked points of the translation surface obtained by suspension of T0. Both g and s can be computed from the combinatorics of T0.

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confidence: 99%

Linearization of generalized interval exchange maps

2012

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“…Condition (3.3), a variant of the exponential estimate for return times of the Teichmüller flow into compact sets, is proved for an arbitrary genus g 2 in [12,Prop. 11.3] by modifying an argument from [6]. Theorem 1.1 and Theorem 1.3 are proved completely.…”

Section: Proof Of Propositionmentioning

confidence: 99%

Hölder regularity for the spectrum of translation flows

Bufetov¹,

Solomyak²

2021

Journal De L’École Polytechnique — Mathématiques

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“…Katok (1980) [10], W. Veech (1982) [20], H. Masur (1982) [13], M. Viana (2006) [21], A.I. Bufetov( 2006) [4]. Starting already from the work of W. Veech the main object of interest here was shifted from a pure dynamical point of view to connections with geometry, in particular with Teichmuller flows.…”

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confidence: 99%

Invariant measures of torus piecewise isometries

Blank¹

2021

Preprint

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By now, we have learned quite well how to study hyperbolic (locally expanding/contracting or both) chaotic dynamical systems, thanks to a large extent to the development of the so called operator approach. Contrary to this almost nothing is known about piecewise isometries, except for a special case of one-dimensional interval exchange mappings. The last case is fundamentally different from the general situation in the presence of an invariant measure (Lebesgue measure), which helps a lot in the analysis. We start by showing that already the restriction of the rotation of the plane to a torus demonstrates a number of rather unexpected properties. Our main results describe sufficient conditions for the existence/absence of invariant measures of torus piecewise isometries. Technically these results are based on the approximation of the maps under study by weakly periodic ones.

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Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials

Cited by 51 publications

References 23 publications

Linearization of generalized interval exchange maps

Linearization of generalized interval exchange maps

Hölder regularity for the spectrum of translation flows

Invariant measures of torus piecewise isometries

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