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Strong coincidence and overlap coincidence

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  • 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.
    Mathematics Subject Classification: Primary: 52C23; Secondary: 37B50, 11K16.


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  • [1]

    S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings, Adv. Math., 226 (2011), 2855-2883.doi: 10.1016/j.aim.2010.07.019.


    S. Akiyama and J.-Y. Lee, Overlap coincidence to strong coincidence in substitution tiling dynamics, European J. Combin., 39 (2014), 233-243.doi: 10.1016/j.ejc.2014.01.009.


    P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, 8 (2001), 181-207.


    M. Barge and B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France, 130 (2002), 619-626.


    M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer. J. Math., 128 (2006), 1219-1282.doi: 10.1353/ajm.2006.0037.


    F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (1977/78), 221-239.


    S. Ito and H. Rao, Atomic surfaces, tilings and coincidence. I. Irreducible case, Israel J. Math., 153 (2006), 129-155.doi: 10.1007/BF02771781.


    T. Kamae, A topological invariant of substitution minimal sets, J. Math. Soc. Japan, 24 (1972), 285-306.doi: 10.2969/jmsj/02420285.


    J. C. Lagarias and Y. Wang, Substitution Delone Sets, Discrete Comput. Geom., 29 (2003), 175-209.doi: 10.1007/s00454-002-2820-6.


    J. Lagarias, Meyer's concept of quasicrystal and quasiregular sets, Comm. Math. Phys., 179 (1996), 365-376.doi: 10.1007/BF02102593.


    J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets, J. Geom. Phys., 57 (2007), 2263-2285.doi: 10.1016/j.geomphys.2007.07.003.


    J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems, Discrete Comput. Geom., 29 (2003), 525-560.doi: 10.1007/s00454-003-0781-z.


    J.-Y. Lee and B. Solomyak, Pisot family substitution tilings, discrete spectrum and the Meyer property, Discr. Conti. Dynam. Sys., 32 (2012), 935-959.


    J. Luo, S. Akiyama and J. M. Thuswaldner, On the boundary connectedness of connected tiles, Math. Proc. Cambridge Phil. Soc., 137 (2004), 397-410.doi: 10.1017/S0305004104007625.


    K. Nakaishi, Pisot conjecture and Rauzy fractals, preprint.


    M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987.


    G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France, 110 (1982), 147-178.


    A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. (N.S.), 118 (2009), p140.


    B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695-738.doi: 10.1017/S0143385797084988.

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