# American Institute of Mathematical Sciences

June  2009, 2(2): 315-324. doi: 10.3934/dcdss.2009.2.315

## An application of topological multiple recurrence to tiling

 1 Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States 2 Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, United States

Received  April 2008 Revised  August 2008 Published  April 2009

We show that given any tiling of Euclidean space, any geometric pattern of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated version of the pattern. The rather simple proof uses Furstenberg's topological multiple recurrence theorem.
Citation: Rafael De La Llave, A. Windsor. An application of topological multiple recurrence to tiling. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315
 [1] Vincent Penné, Benoît Saussol, Sandro Vaienti. Dimensions for recurrence times: topological and dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 783-798. doi: 10.3934/dcds.1999.5.783 [2] Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581 [3] Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137 [4] Serge Troubetzkoy. Recurrence in generic staircases. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047 [5] Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1 [6] Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1 [7] Milton Ko. Rényi entropy and recurrence. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403 [8] Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729 [9] Jie Li, Kesong Yan, Xiangdong Ye. Recurrence properties and disjointness on the induced spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1059-1073. doi: 10.3934/dcds.2015.35.1059 [10] A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587 [11] Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175 [12] Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039 [13] Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259 [14] L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821 [15] Richard D. Neidinger. Efficient recurrence relations for univariate and multivariate Taylor series coefficients. Conference Publications, 2013, 2013 (special) : 587-596. doi: 10.3934/proc.2013.2013.587 [16] Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034 [17] Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017 [18] Jean René Chazottes, E. Ugalde. Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 565-586. doi: 10.3934/dcdsb.2005.5.565 [19] Dang H. Nguyen, George Yin. Recurrence for switching diffusion with past dependent switching and countable state space. Mathematical Control & Related Fields, 2018, 8 (3&4) : 879-897. doi: 10.3934/mcrf.2018039 [20] C. Foias, M. S Jolly, O. P. Manley. Recurrence in the 2-$D$ Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 253-268. doi: 10.3934/dcds.2004.10.253

2019 Impact Factor: 1.233