Hostname: page-component-6bb9c88b65-2jdt9 Total loading time: 0 Render date: 2025-07-22T15:47:18.561Z Has data issue: false hasContentIssue false
  • English
  • Français

On unipotent flows in ℋ(1,1)

Published online by Cambridge University Press:  23 June 2009

KARIANE CALTA  and
KEVIN WORTMAN
KARIANE CALTA
Affiliation:
Department of Mathematics, Rockefeller Hall, 124 Raymond Avenue, Vassar College, Poughkeepsie, NY 12604-0257, USA (email: kacalta@vassar.edu)
KEVIN WORTMAN
Affiliation:
Department of Mathematics, University of Utah, 155 S 1400 E Room 233, Salt Lake City, UT 84112-0090, USA (email: wortman@math.utah.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum ℋ(1,1).

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 379 - 398
Copyright
Copyright © Cambridge University Press 2009

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.)

Article purchase

Temporarily unavailable

References

[1]Calta, K.. Veech surfaces and complete periodicity in genus two. J. Amer. Math. Soc. 17 (2004), 871–908.CrossRefGoogle Scholar
[2]Eskin, A., Marklof, J. and Morris, D.. Unipotent flows on the space of branched covers of Veech surfaces. Ergod. Th. & Dynam. Sys. 26 (2005), 129–162.CrossRefGoogle Scholar
[3]Masur, H., Hubert, P., Schmidt, T. and Zorich, A.. Problems on billiards, flat surfaces, and translation surfaces. Problems on mapping class groups and related topics. Ed. B. Farb. Proc. Symp. Pure Applied Math. 74 (2006).CrossRefGoogle Scholar
[4]Margulis, G. A. and Tomanov, G. M.. Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116(13) (1994), 347–392.CrossRefGoogle Scholar
[5]Masur, H. and Tabachnikov, S.. Rational billiards and flat structures. Handbook of Dynamical Systems, Vol. 1A. North-Holland, Amsterdam, 2002, pp. 1015–1089.CrossRefGoogle Scholar
[6]McMullen, C.. Billiards and Teichmüller curves on Hilbert modular surfaces. J. Amer. Math. Soc. 16 (2003), 857–885.CrossRefGoogle Scholar
[7]McMullen, C.. Dynamics of SL 2(ℝ) over moduli space in genus two. Ann. of Math. (2) 165 (2007), 397–456.CrossRefGoogle Scholar
[8]McMullen, C.. Teichmüller curves in genus two: the decagon and beyond. J. Reine Angew Math. 582 (2005), 173–200.CrossRefGoogle Scholar
[9]McMullen, C.. Teichmüller geodesics of infinite complexity. Acta. Math. 191 (2003), 191–223.CrossRefGoogle Scholar
[10]Minsky, Y. and Weiss, B.. Nondivergence of horocyclic flows on moduli space. J. Reine Angew. Math. 552 (2002), 131–177.Google Scholar
[11]Morris, D.. Ratner’s Theorem on Unipotent flows (Chicago Lecture Series in Mathematics). The University of Chicago Press, Chicago, 2005.Google Scholar
[12]Ratner, M.. Strict measure rigidity for unipotent subgroups of solvable groups. Invent. Math. 101 (1990), 449–482.CrossRefGoogle Scholar
[13]Ratner, M.. On measure rigidity of unipotent subgroups of semisimple groups. Acta Math. 165 (1990), 229–309.Google Scholar
[14]Ratner, M.. Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63 (1991), 235–280.Google Scholar
[15]Ratner, M.. On Raghunathan’s measure conjecture. Ann. of Math. (2) 134 (1991), 545–607.CrossRefGoogle Scholar
[16]Ratner, M.. Raghunathan’s conjectures for SL(2,R). Israel J. Math. 80 (1992), 1–31.CrossRefGoogle Scholar
[17]Veech, W.. Teichmüller curves in moduli space, Eisenstein series, and an application to triangular billiards. Invent. Math. 97 (1989), 553–583.CrossRefGoogle Scholar