
Previous Article
Every ergodic measure is uniquely maximizing
 DCDS Home
 This Issue

Next Article
Tiling Abelian groups with a single tile
Insecure configurations in lattice translation surfaces, with applications to polygonal billiards
1.  University of California and IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil 22460320, 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.
[1] 
Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems  A, 2014, 34 (7) : 27512778. doi: 10.3934/dcds.2014.34.2751 
[2] 
Michael Dellnitz, O. Junge, B Thiere. The numerical detection of connecting orbits. Discrete & Continuous Dynamical Systems  B, 2001, 1 (1) : 125135. doi: 10.3934/dcdsb.2001.1.125 
[3] 
Vito Mandorino. Connecting orbits for families of Tonelli Hamiltonians. Journal of Modern Dynamics, 2012, 6 (4) : 499538. doi: 10.3934/jmd.2012.6.499 
[4] 
Francesca Alessio, Piero Montecchiari, Andres Zuniga. Prescribed energy connecting orbits for gradient systems. Discrete & Continuous Dynamical Systems  A, 2019, 39 (8) : 48954928. doi: 10.3934/dcds.2019200 
[5] 
Alexey Glutsyuk, Yury Kudryashov. No planar billiard possesses an open set of quadrilateral trajectories. Journal of Modern Dynamics, 2012, 6 (3) : 287326. doi: 10.3934/jmd.2012.6.287 
[6] 
Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 1320. 
[7] 
Panayotis Smyrnelis. Connecting orbits in Hilbert spaces and applications to P.D.E. Communications on Pure & Applied Analysis, 2020, 19 (5) : 27972818. doi: 10.3934/cpaa.2020122 
[8] 
Paul Wright. Differentiability of Hausdorff dimension of the nonwandering set in a planar open billiard. Discrete & Continuous Dynamical Systems  A, 2016, 36 (7) : 39934014. doi: 10.3934/dcds.2016.36.3993 
[9] 
Thomas Dauer, Marlies Gerber. Generic absence of finite blocking for interior points of Birkhoff billiards. Discrete & Continuous Dynamical Systems  A, 2016, 36 (9) : 48714893. doi: 10.3934/dcds.2016010 
[10] 
Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114118. 
[11] 
Nuno Luzia. Measure of full dimension for some nonconformal repellers. Discrete & Continuous Dynamical Systems  A, 2010, 26 (1) : 291302. doi: 10.3934/dcds.2010.26.291 
[12] 
Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete & Continuous Dynamical Systems  A, 2008, 21 (2) : 403413. doi: 10.3934/dcds.2008.21.403 
[13] 
Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems  A, 2007, 18 (4) : 817827. doi: 10.3934/dcds.2007.18.817 
[14] 
Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287296. doi: 10.3934/proc.2015.0287 
[15] 
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) : 30433061. doi: 10.3934/dcdsb.2017162 
[16] 
Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for nonconformal repellers. Discrete & Continuous Dynamical Systems  A, 2017, 37 (11) : 57635780. doi: 10.3934/dcds.2017250 
[17] 
Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143153. doi: 10.3934/jmd.2017007 
[18] 
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  A, 2015, 35 (8) : 33153326. doi: 10.3934/dcds.2015.35.3315 
[19] 
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) : 2559. doi: 10.3934/jmd.2014.8.25 
[20] 
Leonardo Manuel Cabrer, Daniele Mundici. Classifying GL$(n,\mathbb{Z})$orbits of points and rational subspaces. Discrete & Continuous Dynamical Systems  A, 2016, 36 (9) : 47234738. doi: 10.3934/dcds.2016005 
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]