September  2008, 22(3): 791-806. doi: 10.3934/dcds.2008.22.791

Packings induced by piecewise isometries cannot contain the arbelos

1. 

Mathematics Research Institute, School of Engineering, Computing and Mathematics, Harrison Building, University of Exeter, Exeter, EX4 4QF, United Kingdom, United Kingdom, United Kingdom

Received  May 2007 Revised  May 2008 Published  August 2008

Planar piecewise isometries with convex polygonal atoms that are piecewise irrational rotations can naturally generate a packing of phase space given by periodic cells that are discs. We show that such packings cannot contain certain subpackings of Apollonian packings, namely those belonging to a family of Arbelos subpackings. We do this by showing that the unit complex numbers giving the directions of tangency within such an isometric-generated packing lie in a finitely generated subgroup of the circle group, whereas this is not the case for the Arbelos subpackings. In the opposite direction, we show that, given an arbitrary disc packing of a polygonal region, there is a piecewise isometry whose regular cells approximate the given packing to any specified precision.
Citation: Marcello Trovati, Peter Ashwin, Nigel Byott. Packings induced by piecewise isometries cannot contain the arbelos. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 791-806. doi: 10.3934/dcds.2008.22.791
[1]

Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381

[2]

Heide Gluesing-Luerssen. On isometries for convolutional codes. Advances in Mathematics of Communications, 2009, 3 (2) : 179-203. doi: 10.3934/amc.2009.3.179

[3]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45

[4]

A. A. Pinto, D. Sullivan. The circle and the solenoid. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 463-504. doi: 10.3934/dcds.2006.16.463

[5]

Gérard Cohen, Alexander Vardy. Duality between packings and coverings of the Hamming space. Advances in Mathematics of Communications, 2007, 1 (1) : 93-97. doi: 10.3934/amc.2007.1.93

[6]

Daniel Coronel, Andrés Navas, Mario Ponce. On bounded cocycles of isometries over minimal dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 45-74. doi: 10.3934/jmd.2013.7.45

[7]

Alex Kontorovich. The local-global principle for integral Soddy sphere packings. Journal of Modern Dynamics, 2019, 15: 209-236. doi: 10.3934/jmd.2019019

[8]

Stéphane Sabourau. Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces. Journal of Modern Dynamics, 2013, 7 (2) : 269-290. doi: 10.3934/jmd.2013.7.269

[9]

Jimmy Tseng. On circle rotations and the shrinking target properties. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1111-1122. doi: 10.3934/dcds.2008.20.1111

[10]

Heather Hannah, A. Alexandrou Himonas, Gerson Petronilho. Anisotropic Gevrey regularity for mKdV on the circle. Conference Publications, 2011, 2011 (Special) : 634-642. doi: 10.3934/proc.2011.2011.634

[11]

Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251

[12]

Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719

[13]

Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597

[14]

Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099

[15]

Abdumajid Begmatov, Akhtam Dzhalilov, Dieter Mayer. Renormalizations of circle hoemomorphisms with a single break point. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4487-4513. doi: 10.3934/dcds.2014.34.4487

[16]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098

[17]

Arek Goetz. Dynamics of a piecewise rotation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593

[18]

Shigenori Matsumoto. A generic-dimensional property of the invariant measures for circle diffeomorphisms. Journal of Modern Dynamics, 2013, 7 (4) : 553-563. doi: 10.3934/jmd.2013.7.553

[19]

Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535

[20]

Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]