Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-7c2ld Total loading time: 0.279 Render date: 2021-12-09T12:03:38.443Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Veech’s dichotomy and the lattice property

Published online by Cambridge University Press:  15 September 2008

JOHN SMILLIE
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY, USA (email: smillie@math.cornell.edu)
BARAK WEISS
Affiliation:
Department of Mathematics, Ben Gurion University, Be’er Sheva, 84105, Israel (email: barakw@math.bgu.ac.il)

Abstract

Veech showed that if a translation surface has a stabilizer which is a lattice in SL(2,ℝ), then any direction for the corresponding constant slope flow is either completely periodic or uniquely ergodic. We show that the converse does not hold: there are translation surfaces that satisfy Veech’s dichotomy but for which the corresponding stabilizer subgroup is not a lattice. The construction relies on work of Hubert and Schmidt.

Type
Research Article
Copyright
Copyright © 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bers, L.. Fiber spaces over Teichmüller spaces. Acta Math. 130 (1973), 89126.CrossRefGoogle Scholar
[2] Cheung, Y., Hubert, P. and Masur, H.. Topological dichotomy and strict ergodicity for translation surfaces. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2006.Google Scholar
[3] Earle, C.. Teichmüller Theory (Discrete Groups and Automorphic Functions). Ed. W. J. Harvey. Academic Press, London, New York, 1977.Google Scholar
[4] Earle, C. and Eells, J.. A fibre bundle description of Teichmüller theory. J. Differential Geom. 3 (1969), 1943.CrossRefGoogle Scholar
[5] Earle, C. and Fowler, R.. Holomorphic families of open Riemann surfaces. Math. Ann. 270(2) (1985), 249273.CrossRefGoogle Scholar
[6] Fox, R. H. and Kershner, R. B.. Concerning the transitive properties of geodesics on a rational polyhedron. Duke Math. J. 2(1) (1936), 147150.CrossRefGoogle Scholar
[7] Farkas, H. and Kra, I.. Riemann Surfaces, 2nd edn. Springer, Berlin, 1992.CrossRefGoogle ScholarPubMed
[8] Gutkin, E.. Billiards on almost integrable polyhedral surfaces. Ergod. Th. & Dynam. Sys. 4 (1984), 569584.CrossRefGoogle Scholar
[9] Gutkin, E., Hubert, P. and Schmidt, T.. Affine diffeomorphisms of translation surfaces: periodic points, fuchsian groups, and arithmeticity. Ann. Sci. École Norm. Sup. 36 (2003), 847866.CrossRefGoogle Scholar
[10] Gutkin, E. and Judge, C.. Affine maps of translation surfaces: geometry and arithmetic. Duke Math. J. 103 (2000), 191213.Google Scholar
[11] Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers. Clarendon Press, Oxford, 1938.Google Scholar
[12] Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.Google Scholar
[13] Hubert, P. and Schmidt, T.. Infinitely generated Veech groups. Duke Math. J. 123(1) (2004), 4969.Google Scholar
[14] Masur, H.. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66 (1992), 387442.CrossRefGoogle Scholar
[15] Masur, H. and Tabachnikov, S.. Rational billiards and flat structures. Handbook of Dynamical Systems, Volume 1. Eds. B. Hasselblatt and A. Katok. Elsevier, Amsterdam, 2001, pp. 10151089.Google Scholar
[16] Natanzon, S. M.. The topological structure of the space of holomorphic morphisms of Riemann surfaces. Russian Math. Surveys 53 (1998), 398400 (Translation from Russian).CrossRefGoogle Scholar
[17] Thurston, W.. The geometry and topology of 3-manifolds, Chapter 13. Available at:http://www.msri.org/communications/books/gt3m/PS.Google Scholar
[18] Veech, W. A.. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3) (1989), 553583.CrossRefGoogle Scholar
[19] Veech, W. A.. The billiard in a regular polygon. Geom. Funct. Anal. 2 (1992), 341379.CrossRefGoogle Scholar
[20] Vorobets, Ya. B.. Planar structures and billiards in rational polygons: the Veech alternative. Russian Math. Surveys 51(5) (1996), 779817 (Translation from Russian).CrossRefGoogle Scholar
5
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Veech’s dichotomy and the lattice property
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Veech’s dichotomy and the lattice property
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Veech’s dichotomy and the lattice property
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *