Insecure configurations in lattice translation surfaces, with applications to polygonal billiards
University of California and IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil 22460-320, Brazil
If $X$ is an insecure space, it is natural to ask how big the set of insecure configurations is. We investigate this problem for flat surfaces, in particular for translation surfaces and polygons, from the viewpoint of measure theory.
Here is a sample of our results. Let $X$ be a lattice translation surface or a lattice polygon. Then the following dichotomy holds: i) The surface (polygon) $X$ is arithmetic. Then all configurations in $X$ are secure; ii) The surface (polygon) $X$ is nonarithmetic. Then almost all configurations in $X$ are insecure.
Francesca Alessio, Piero Montecchiari, Andres Zuniga. Prescribed energy connecting orbits for gradient systems. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4895-4928. doi: 10.3934/dcds.2019200
Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20.
Thomas Dauer, Marlies Gerber. Generic absence of finite blocking for interior points of Birkhoff billiards. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4871-4893. doi: 10.3934/dcds.2016010
Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020
Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993
Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.
Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 403-413. doi: 10.3934/dcds.2008.21.403
Flaviano Battelli, Michal Fečkan. On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3043-3061. doi: 10.3934/dcdsb.2017162
Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315
Alexander Gorodnik, Frédéric Paulin. Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. Journal of Modern Dynamics, 2014, 8 (1) : 25-59. doi: 10.3934/jmd.2014.8.25
2020 Impact Factor: 1.392
[Back to Top]