Quantitative Recurrence and Large Deviations for Teichmuller Geodesic Flow

Athreya, Jayadev S.

doi:10.1007/s10711-006-9058-z

Cited by 37 publications

(94 citation statements)

References 24 publications

(21 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is covered in z X D by a union of graphs of Möbius transformations ‫ވ‬ ! ‫.ވ‬ We introduce these curves in Section 2.3 and show that (1)(2)(3) .P D / D 5 2 .X D /:…”

Section: Siegel-veech Constantsmentioning

confidence: 99%

“…(1-11) These together with (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) give us enough equations to solve for a. Theorem 1.3 follows from (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) by pairing OE! 1 with both sides as in the proof of Theorem 1.1.…”

Section: Siegel-veech Constantsmentioning

confidence: 99%

“…i are the 2-forms on X D defined by (2)(3)(4)(5)(6). The volume of X D was calculated by Siegel when D is a fundamental discriminant.…”

Section: Baily-borel Compactificationmentioning

confidence: 99%

See 2 more Smart Citations

Euler characteristics of Teichmüller curves in genus two

Bainbridge

2007

Geom. Topol.

119

View full text Add to dashboard Cite

We calculate the Euler characteristics of all of the Teichmüller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves are naturally embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmüller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies.The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmüller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmüller curves.We apply these results to calculate the Siegel-Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich-Zorich cocycle for any ergodic, SL 2 .‫-/ޒ‬invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich). The Jacobian Jac.X / has real multiplication by O D , the unique real quadratic order of discriminant D .There is an Abelian differential ! on X which is an eigenform for real multiplication and has a double zero. where the top map is an embedding, and the vertical map is a two-to-one map sending an Abelian surface A to the unique Riemann surface X 2 M 2 such that Jac.X / Š A.The curve W D is not a Shimura curve on X D ; however, it is a Teichmüller curve, a curve which is isometrically immersed in the moduli space M 2 with respect to the Teichmüller metric. In fact, the curves W D with D nonsquare are all but one of the primitive Teichmüller curves in M 2 , where a Teichmüller curve is said to be primitive if it does not arise from a Teichmüller curve of lower genus by a certain branched covering construction. The curves W D arise from the study of billiards in certain L-shaped polygons, and the study of these curves has applications to the dynamics of billiards in these polygons. Euler characteristics

show abstract

“…It is covered in z X D by a union of graphs of Möbius transformations ‫ވ‬ ! ‫.ވ‬ We introduce these curves in Section 2.3 and show that (1)(2)(3) .P D / D 5 2 .X D /:…”

Section: Siegel-veech Constantsmentioning

confidence: 99%

Section: Siegel-veech Constantsmentioning

confidence: 99%

See 1 more Smart Citation

Euler characteristics of Teichmüller curves in genus two

Bainbridge

2007

Geom. Topol.

119

View full text Add to dashboard Cite

show abstract

“…The first results on exponential decay for the probabilities of return times were obtained by Jayadev Athreya [30]. In his approach, Athreya used the dynamics of SL(2, R)-action, which allowed him to obtain optimal exponents.…”

Section: Proposition 15 There Exists a Constant C Such That The Follmentioning

confidence: 99%

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

Bufetov¹

2006

J. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…We note that (1.2) is an immediate consequence of Theorem 1.1, which is a bit more precise. In order to prove Theorem 1.1 one needs Corollary 1.3 and certain recurrence results for geodesics, which are based on [4].…”

Section: Introductionmentioning

confidence: 99%