# American Institute of Mathematical Sciences

July  2013, 7(3): 395-427. doi: 10.3934/jmd.2013.7.395

## Winning games for bounded geodesics in moduli spaces of quadratic differentials

 1 Mathematics Department, 155 S 1400 E Room 233, University of Utah, Salt Lake City, UT 84112-0090, United States 2 Mathematics Department, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, United States 3 Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Il 60637, United States

Received  December 2012 Published  December 2013

We prove that the set of bounded geodesics in Teichmüller space is a winning set for Schmidt's game. This is a notion of largeness in a metric space that can apply to measure $0$ and meager sets. We prove analogous closely related results on any Riemann surface, in any stratum of quadratic differentials, on any Teichmüller disk and for intervals exchanges with any fixed irreducible permutation.
Citation: Jonathan Chaika, Yitwah Cheung, Howard Masur. Winning games for bounded geodesics in moduli spaces of quadratic differentials. Journal of Modern Dynamics, 2013, 7 (3) : 395-427. doi: 10.3934/jmd.2013.7.395
##### References:

show all references

##### References:
 [1] Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429 [2] C. Alonso-González, M. I. Camacho, F. Cano. Topological classification of multiple saddle connections. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 395-414. doi: 10.3934/dcds.2006.15.395 [3] Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004 [4] Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739 [5] Alexandre A. P. Rodrigues. Moduli for heteroclinic connections involving saddle-foci and periodic solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3155-3182. doi: 10.3934/dcds.2015.35.3155 [6] Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575 [7] Weisheng Wu. Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3463-3481. doi: 10.3934/dcds.2016.36.3463 [8] Michael Magee, Rene Rühr. Counting saddle connections in a homology class modulo $\boldsymbol q$ (with an appendix by Rodolfo Gutiérrez-Romo). Journal of Modern Dynamics, 2019, 15: 237-262. doi: 10.3934/jmd.2019020 [9] Oğul Esen, Partha Guha. On the geometry of the Schmidt-Legendre transformation. Journal of Geometric Mechanics, 2018, 10 (3) : 251-291. doi: 10.3934/jgm.2018010 [10] Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201 [11] Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431 [12] Krystyna Kuperberg. 2-wild trajectories. Conference Publications, 2005, 2005 (Special) : 518-523. doi: 10.3934/proc.2005.2005.518 [13] Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433 [14] Gabriel P. Paternain. Transparent connections over negatively curved surfaces. Journal of Modern Dynamics, 2009, 3 (2) : 311-333. doi: 10.3934/jmd.2009.3.311 [15] Felicia Maria G. Magpantay, Xingfu Zou. Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections. Mathematical Biosciences & Engineering, 2010, 7 (2) : 421-442. doi: 10.3934/mbe.2010.7.421 [16] Maria Grazia Naso. Controllability to trajectories for semilinear thermoelastic plates. Conference Publications, 2005, 2005 (Special) : 672-681. doi: 10.3934/proc.2005.2005.672 [17] M. L. Bertotti, Sergey V. Bolotin. Chaotic trajectories for natural systems on a torus. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1343-1357. doi: 10.3934/dcds.2003.9.1343 [18] Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117 [19] Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics & Games, 2016, 3 (4) : 303-318. doi: 10.3934/jdg.2016016 [20] Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33

2018 Impact Factor: 0.295