Hostname: page-component-848d4c4894-89wxm Total loading time: 0 Render date: 2024-07-07T15:14:39.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Blocking: new examples and properties of products

Published online by Cambridge University Press:  01 April 2009

PILAR HERREROS
PILAR HERREROS*
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA (email: pherrero@math.upenn.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We say that a pair of points x and y is secure if there exists a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena in both the manifold and the billiard table settings. In approaching this, we study whether a product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security which requires that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we construct a compact C1 surface which contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, and generates billiard tables with similar blocking behavior.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 569 - 578
Copyright
Copyright © Cambridge University Press 2008

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]Burns, K. and Gutkin, E.. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete Contin. Dyn. Sys. A 21 (2008), 403–413.CrossRefGoogle Scholar
[2]Cheung, Y.. Hausdorff dimension of the set of nonergodic directions. Ann. of Math. (2) 158 (2003), 661–678.CrossRefGoogle Scholar
[3]Gutkin, E.. Blocking of billiard orbits and security for polygons and flat surfaces. Geom. Funct. Anal. 15 (2005), 83–105.Google Scholar
[4]Gutkin, E.. Insecure configurations in lattice translation surfaces, with applications to polygonal billiards. Discrete Contin. Dyn. Syst. A 16 (2006), 367–382.Google Scholar
[5]Gutkin, E. and Schröder, V.. Connecting geodesics and security of configurations in compact locally symmetric spaces. Geom. Dedicata 118 (2006), 185–208.CrossRefGoogle Scholar
[6]Hiemer, P. and Snurnikov, V.. Polygonal billiards with small obstacles. J. Stat. Phys. 90(1–2) (1998), 453–466.Google Scholar
[7]Hubert, P., Schmoll, M. and Troubetzkoy, S.. Modular fibers and illumination problems. Int. Math. Res. Not. 2008 (2008), 42pp.Google Scholar
[8]Lafont, J.-F. and Schmidt, B.. Blocking light in compact Riemannian manifolds. Geom. Topol. 11 (2007), 867–887.CrossRefGoogle Scholar
[9]Monteil, T.. On the finite blocking property. Ann. Inst. Fourier 55 (2005), 1195–1217.Google Scholar
[10]Monteil, T.. Finite blocking property vs pure periodicity. arXiv:math.DS/0406506.Google Scholar
[11]Schmoll, M.. On the asymptotic quadratic growth rate of saddle connections and periodic orbits on marked flat tori. Geom. Funct. Anal. 12(3) (2002), 622–649.CrossRefGoogle Scholar
[12]Schmidt, B. and Souto, J.. Chords, light, and another synthetic characterization of the round sphere. arXiv:math.GT/0704.3642.Google Scholar
[13]Tabachnikov, S.. Birkhoff billiards are insecure. Discrete Contin. Dyn. Syst. to appear.Google Scholar