American Institute of Mathematical Sciences

Advanced
October  2016, 36(10): 5223-5230. doi: 10.3934/dcds.2016027

Strong coincidence and overlap coincidence

Shigeki Akiyama 1,

1. 

Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 350-8571

Received  September 2015 Revised  December 2015 Published  July 2016

We show that strong coincidences of a certain many choices of control points are equivalent to overlap coincidence for the suspension tiling of Pisot substitution. The result is valid for dimension $\ge 2$ as well, under certain topological conditions. This result gives a converse of the paper [2] and elucidates the tight relationship between two coincidences.
Keywords: substitution, Pisot number., pure discrete spectrum, Self-affine tiling, coincidence.
Mathematics Subject Classification: Primary: 52C23; Secondary: 37B50, 11K1.
Citation: Shigeki Akiyama. Strong coincidence and overlap coincidence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5223-5230. doi: 10.3934/dcds.2016027
References:
[1]

S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings,, Adv. Math., 226 (2011), 2855.  doi: 10.1016/j.aim.2010.07.019.  Google Scholar

[2]

S. Akiyama and J.-Y. Lee, Overlap coincidence to strong coincidence in substitution tiling dynamics,, European J. Combin., 39 (2014), 233.  doi: 10.1016/j.ejc.2014.01.009.  Google Scholar

[3]

P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals,, Bull. Belg. Math. Soc. Simon Stevin, 8 (2001), 181.   Google Scholar

[4]

M. Barge and B. Diamond, Coincidence for substitutions of Pisot type,, Bull. Soc. Math. France, 130 (2002), 619.   Google Scholar

[5]

M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions,, Amer. J. Math., 128 (2006), 1219.  doi: 10.1353/ajm.2006.0037.  Google Scholar

[6]

F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (): 221.   Google Scholar

[7]

S. Ito and H. Rao, Atomic surfaces, tilings and coincidence. I. Irreducible case,, Israel J. Math., 153 (2006), 129.  doi: 10.1007/BF02771781.  Google Scholar

[8]

T. Kamae, A topological invariant of substitution minimal sets,, J. Math. Soc. Japan, 24 (1972), 285.  doi: 10.2969/jmsj/02420285.  Google Scholar

[9]

J. C. Lagarias and Y. Wang, Substitution Delone Sets,, Discrete Comput. Geom., 29 (2003), 175.  doi: 10.1007/s00454-002-2820-6.  Google Scholar

[10]

J. Lagarias, Meyer's concept of quasicrystal and quasiregular sets,, Comm. Math. Phys., 179 (1996), 365.  doi: 10.1007/BF02102593.  Google Scholar

[11]

J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets,, J. Geom. Phys., 57 (2007), 2263.  doi: 10.1016/j.geomphys.2007.07.003.  Google Scholar

[12]

J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems,, Discrete Comput. Geom., 29 (2003), 525.  doi: 10.1007/s00454-003-0781-z.  Google Scholar

[13]

J.-Y. Lee and B. Solomyak, Pisot family substitution tilings, discrete spectrum and the Meyer property,, Discr. Conti. Dynam. Sys., 32 (2012), 935.   Google Scholar

[14]

J. Luo, S. Akiyama and J. M. Thuswaldner, On the boundary connectedness of connected tiles,, Math. Proc. Cambridge Phil. Soc., 137 (2004), 397.  doi: 10.1017/S0305004104007625.  Google Scholar

[15]

K. Nakaishi, Pisot conjecture and Rauzy fractals,, preprint., ().   Google Scholar

[16]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis,, Lecture Notes in Mathematics, (1294).   Google Scholar

[17]

G. Rauzy, Nombres algébriques et substitutions,, Bull. Soc. Math. France, 110 (1982), 147.   Google Scholar

[18]

A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractals,, Mém. Soc. Math. Fr. (N.S.), 118 (2009).   Google Scholar

[19]

B. Solomyak, Dynamics of self-similar tilings,, Ergodic Theory Dynam. Systems, 17 (1997), 695.  doi: 10.1017/S0143385797084988.  Google Scholar

show all references

References:
[1]

S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings,, Adv. Math., 226 (2011), 2855.  doi: 10.1016/j.aim.2010.07.019.  Google Scholar

[2]

S. Akiyama and J.-Y. Lee, Overlap coincidence to strong coincidence in substitution tiling dynamics,, European J. Combin., 39 (2014), 233.  doi: 10.1016/j.ejc.2014.01.009.  Google Scholar

[3]

P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals,, Bull. Belg. Math. Soc. Simon Stevin, 8 (2001), 181.   Google Scholar

[4]

M. Barge and B. Diamond, Coincidence for substitutions of Pisot type,, Bull. Soc. Math. France, 130 (2002), 619.   Google Scholar

[5]

M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions,, Amer. J. Math., 128 (2006), 1219.  doi: 10.1353/ajm.2006.0037.  Google Scholar

[6]

F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (): 221.   Google Scholar

[7]

S. Ito and H. Rao, Atomic surfaces, tilings and coincidence. I. Irreducible case,, Israel J. Math., 153 (2006), 129.  doi: 10.1007/BF02771781.  Google Scholar

[8]

T. Kamae, A topological invariant of substitution minimal sets,, J. Math. Soc. Japan, 24 (1972), 285.  doi: 10.2969/jmsj/02420285.  Google Scholar

[9]

J. C. Lagarias and Y. Wang, Substitution Delone Sets,, Discrete Comput. Geom., 29 (2003), 175.  doi: 10.1007/s00454-002-2820-6.  Google Scholar

[10]

J. Lagarias, Meyer's concept of quasicrystal and quasiregular sets,, Comm. Math. Phys., 179 (1996), 365.  doi: 10.1007/BF02102593.  Google Scholar

[11]

J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets,, J. Geom. Phys., 57 (2007), 2263.  doi: 10.1016/j.geomphys.2007.07.003.  Google Scholar

[12]

J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems,, Discrete Comput. Geom., 29 (2003), 525.  doi: 10.1007/s00454-003-0781-z.  Google Scholar

[13]

J.-Y. Lee and B. Solomyak, Pisot family substitution tilings, discrete spectrum and the Meyer property,, Discr. Conti. Dynam. Sys., 32 (2012), 935.   Google Scholar

[14]

J. Luo, S. Akiyama and J. M. Thuswaldner, On the boundary connectedness of connected tiles,, Math. Proc. Cambridge Phil. Soc., 137 (2004), 397.  doi: 10.1017/S0305004104007625.  Google Scholar

[15]

K. Nakaishi, Pisot conjecture and Rauzy fractals,, preprint., ().   Google Scholar

[16]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis,, Lecture Notes in Mathematics, (1294).   Google Scholar

[17]

G. Rauzy, Nombres algébriques et substitutions,, Bull. Soc. Math. France, 110 (1982), 147.   Google Scholar

[18]

A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractals,, Mém. Soc. Math. Fr. (N.S.), 118 (2009).   Google Scholar

[19]

B. Solomyak, Dynamics of self-similar tilings,, Ergodic Theory Dynam. Systems, 17 (1997), 695.  doi: 10.1017/S0143385797084988.  Google Scholar

[1]

Jeong-Yup Lee, Boris Solomyak. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 935-959. doi: 10.3934/dcds.2012.32.935
[2]

Marcy Barge, Sonja Štimac, R. F. Williams. Pure discrete spectrum in substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 579-597. doi: 10.3934/dcds.2013.33.579
[3]

Marcy Barge. Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1159-1173. doi: 10.3934/dcds.2016.36.1159
[4]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015
[5]

Jeanette Olli. Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4173-4186. doi: 10.3934/dcds.2013.33.4173
[6]

Rui Pacheco, Helder Vilarinho. Statistical stability for multi-substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4579-4594. doi: 10.3934/dcds.2013.33.4579
[7]

Younghwan Son. Substitutions, tiling dynamical systems and minimal self-joinings. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4855-4874. doi: 10.3934/dcds.2014.34.4855
[8]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[9]

Qixuan Wang, Hans G. Othmer. The performance of discrete models of low reynolds number swimmers. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1303-1320. doi: 10.3934/mbe.2015.12.1303
[10]

Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107
[11]

Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010
[12]

Nikolai Edeko. On the isomorphism problem for non-minimal transformations with discrete spectrum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6001-6021. doi: 10.3934/dcds.2019262
[13]

Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785
[14]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[15]

Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040
[16]

Grzegorz Graff, Jerzy Jezierski. Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds. Conference Publications, 2011, 2011 (Special) : 523-532. doi: 10.3934/proc.2011.2011.523
[17]

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
[18]

S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361
[19]

Xiangying Meng, Gemma Huguet, John Rinzel. Type III excitability, slope sensitivity and coincidence detection. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2729-2757. doi: 10.3934/dcds.2012.32.2729
[20]

C. T. Cremins, G. Infante. A semilinear $A$-spectrum. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 235-242. doi: 10.3934/dcdss.2008.1.235

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences