Abstract
In this paper, there are two sections. In Section 7, we simplifythe eigenvalue-based surplus shortline method for arbitrary finite polysquaretranslation surfaces. This makes it substantially simpler to determine the irregularityexponents of some infinite orbits, and quicker to find the escape rate toinfinity of some orbits in some infinite models. In Section 8, our primary goalis to extend the surplus shortline method, both this eigenvalue-based version aswell as the eigenvalue-free version, for application to a large class of 2-dimensionalflat dynamical systems beyond polysquares, including all Veech surfaces, and establishtime-quantitative equidistribution and time-quantitative superdensity ofsome infinite orbits in these new systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Avila and V. Delecroix, Weak mixing directions in non-arithmetic Veech surfaces,J. Amer. Math. Soc., 29 (2016), 1167–1208.
J. Beck, M. Donders and Y. Yang, Quantitative behavior of non-integrable systems. I,Acta Math. Hungar., 161 (2020), 66–184.
J. Beck, M. Donders and Y. Yang, Quantitative behavior of non-integrable systems. II,Acta Math. Hungar., 162 (2020), 220–324.
J. Beck, W. W. L. Chen and Y. Yang, Quantitative behavior of non-integrable systems.III, Acta Math. Hungar. (to appear).
M. Boshernitzan, A condition for minimal interval exchange maps to be uniquelyergodic, Duke Math. J., 52 (1985), 723–752.
M. Boshernitzan, A condition for unique ergodicity of minimal symbolic flows, ErgodicTheory Dynam. Systems, 12 (1992), 425–428.
K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc.,17 (2004), 871–908.
Y. Cheung and H. Masur, MinimalE. Gutkin, Billiards on almost integrable polyhedral surfaces, Ergodic Theory Dynam.Systems, 4 (1984), 569–584. non-ergodic directions on genus-2 translation surfaces,Ergodic Theory Dynam. Systems, 26 (2006), 341–351.
E. Gutkin, Billiards on almost integrable polyhedral surfaces, Ergodic Theory Dynam.Systems, 4 (1984), 569–584.
P. Hubert and T. A. Schmidt, An introduction to Veech surfaces, in: Handbook ofDynamical Systems, B. Hasselblatt, A. Katok, eds., vol. 1B, Elsevier (2006),pp. 501–526.
C. T. McMullen, Billiards and Teichm¨uller curves on Hilbert modular surfaces,J. Amer. Math. Soc., 16 (2003), 857–885.
C. T. McMullen, Teichm¨uller curves in genus two: the decagon and beyond, J. ReineAngew. Math., 582 (2005), 173–199.
C. T. McMullen, Teichm¨uller curves in genus two: torsion divisors and ratios of sines,Invent. Math., 165 (2006), 651–672.
C. T. McMullen, Prym varieties and Teichm¨uller curves, Duke Math. J., 133 (2006),569–590.
C. T. McMullen, R. E. Mukamel and A. Wright, Cubic curves and totally geodesicsubvarieties of moduli space, Ann. of Math., 185 (2017), 957–990.
W. A. Veech, Boshernitzan’s criterion for unique ergodicity of an interval exchangetransformation, Ergodic Theory Dynam. Systems, 7 (1987), 149–153.
W. A. Veech, Teichm¨uller curves in moduli space, Eisenstein series and an applicationto triangle billiards, Invent. Math., 97 (1989), 553–583.
W. A. Veech, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341–379.
Acknowledgements
The authors thank the referee for very careful reading of the manuscript and for valuable and insightful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Beck, J., Chen, W.W.L. & Yang, Y. Quantitative Behavior Of Non-Integrable Systems. IV. Acta Math. Hungar. 167, 1–160 (2022). https://doi.org/10.1007/s10474-022-01240-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-022-01240-3