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June  2019, 39(6): 3149-3177. doi: 10.3934/dcds.2019130

On substitution tilings and Delone sets without finite local complexity

1. 

Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701, Korea

2. 

KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Korea

3. 

Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

* Corresponding author: Jeong-Yup Lee

Received  April 2018 Revised  November 2018 Published  February 2019

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [29] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

Citation: Jeong-Yup Lee, Boris Solomyak. On substitution tilings and Delone sets without finite local complexity. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3149-3177. doi: 10.3934/dcds.2019130
References:
[1]

S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot substitution conjecture, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savinien), Progr. Math., 309, Birkhäuser/Springer, Basel, (2015), 33–72. doi: 10.1007/978-3-0348-0903-0_2.

[2] M. Baake and U. Grimm, Aperiodic Order, Vol. 1. A Mathematical Invitation. With a Foreword by Roger Penrose, Encyclopedia of Mathematics and its Applications, 149. Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025256.
[3] M. Baake and U. Grimm, Aperiodic Order, Vol. 2. Crystallography and Almost Periodicity. With a foreword by Jeffrey C. Lagarias., Encyclopedia of Mathematics and its Applications, 166. Cambridge University Press, Cambridge, 2017. 
[4]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergod. Th. & Dynam. Sys., 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.

[5]

M. Baake and D. Lenz, Spectral notions of aperiodic order, Discrete Contin. Dyn. Syst., 10 (2017), 161-190.  doi: 10.3934/dcdss.2017009.

[6]

M. Baake and R. V. Moody, Self-similar measures for quasi-crystals, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph series, 13, AMS, Providence RI, (2000), 1–42.

[7]

J. Bellissard, D. J. L. Herrmann and M. Zarrouati, Hulls of aperiodic solids and gap labeling theorems, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, 13, AMS, Providence RI (2000), 207–258.

[8]

D. Damanik and D. Lenz, Linear repetitivity. I. Uniform subadditive ergodic theorems and applications, Discrete Comput. Geom., 26 (2001), 411-428.  doi: 10.1007/s00454-001-0033-z.

[9]

L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math., 239 (2002), 118-126. 

[10]

J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.  doi: 10.1090/S0002-9939-1962-0139135-6.

[11]

N. P. Frank, Tilings with infinite local complexity, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz, J. Savinien), Progr. Math., 309, Birkhäuser/Springer, Basel, (2015), 223–257. doi: 10.1007/978-3-0348-0903-0_7.

[12]

N. P. Frank and E. A. Jr Robinson, Generalized β-expansions, substitution tilings, and local finiteness, Trans. Amer. Math. Soc., 360 (2008), 1163-1177.  doi: 10.1090/S0002-9947-07-04527-8.

[13]

N. P. Frank and L. Sadun, Topology of (some) tiling spaces without finite local complexity, Discrete Contin. Dyn. Syst., 23 (2009), 847-865.  doi: 10.3934/dcds.2009.23.847.

[14]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $ {\mathbb R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.

[15]

N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, Topology Proc., 43 (2014), 235-276. 

[16]

D. Frettlöh and C. Richard, Dynamical properties of almost repetitive Delone sets, Discrete Contin. Dyn. Syst., 34 (2014), 531-556.  doi: 10.3934/dcds.2014.34.531.

[17]

A. Hof, On diffraction by aperiodic structures, Comm. Math. Phys., 169 (1995), 25-43.  doi: 10.1007/BF02101595.

[18]

R. Kenyon, Self-replicating tilings, Symbolic dynamics and its application (New Haven, CT, 1991), Contemp. Math., 135, Amer. Math. Soc., Providence, RI, (1992), 239–263. doi: 10.1090/conm/135/1185093.

[19]

R. Kenyon, Rigidity of planar tilings, Invent. Math., 107 (1992), 637-651.  doi: 10.1007/BF01231905.

[20]

R. Kenyon, Inflationary tilings with similarity structure, Comment. Math. Helv., 69 (1994), 169-198.  doi: 10.1007/BF02564481.

[21]

R. Kenyon, The construction of self-similar tilings, Geometric and Funct. Anal., 6 (1996), 471-488.  doi: 10.1007/BF02249260.

[22]

I. Környei, On a theorem of Pisot, Publ. Math. Debrecen, 34 (1987), 169-179. 

[23]

J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, Vol. 13, AMS, Providence, RI, (2000), 61–93.

[24]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergod. Th. & Dynam. Sys., 23 (2003), 831-867.  doi: 10.1017/S0143385702001566.

[25]

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

[26]

J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003–1018. doi: 10.1007/s00023-002-8646-1.

[27]

J.-Y. LeeR. 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.

[28]

J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comput. Geom., 39 (2008), 319-338.  doi: 10.1007/s00454-008-9054-1.

[29]

J.-Y. Lee and B. Solomyak, Pisot family self-affine tilings, discrete spectrum, and the Meyer property, Discrete Contin. Dyn. Syst., 32 (2012), 935-959.  doi: 10.3934/dcds.2012.32.935.

[30]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in Operator Algebras and Mathematical Physics (eds. J. M. Combes, G. Elliott, G. Nenciu, H. Siedentop and S. Stratila) Theta, Bucharest, (2003), 267–285.

[31]

C. Mauduit, Caractérisation des ensembles normaux subsitutifs, Invent. Math., 95 (1989), 133-147.  doi: 10.1007/BF01394146.

[32]

D. Mauldin and S. Williams, Hausdorff dimension in graph direct constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.  doi: 10.1090/S0002-9947-1988-0961615-4.

[33]

B. Mossé, Puissances de mots et reconnaisabilité des points fixes d'une substitution, Theor. Comp. Sci., 99 (1992), 327-334.  doi: 10.1016/0304-3975(92)90357-L.

[34]

P. Müller and C. Richard, Ergodic properties of randomly coloured point sets, Canad. J. Math., 65 (2013), 349-402.  doi: 10.4153/CJM-2012-009-7.

[35]

B. Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc., 351 (1999), 3315-3349.  doi: 10.1090/S0002-9947-99-02360-0.

[36]

M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, 2nd edition. Lecture Notes in Math., 1294, Springer, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.

[37]

C. Radin, Space tilings and substitutions, Geom. Dedicata, 55 (1995), 257-264.  doi: 10.1007/BF01266317.

[38]

C. Radin and M. Wolff, Space tilings and local isomorphism, Geom. Dedicata, 42 (1992), 355-360.  doi: 10.1007/BF02414073.

[39]

E. A. Robinson, Jr., Symbolic dynamics and tilings of $ {\mathbb R}^d$, in Symbolic Dynamics and Its Applications, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, (2004), 81–119. doi: 10.1090/psapm/060/2078847.

[40]

B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. & Dynam. Sys., 17 (1997), 695-738. Corrections to 'Dynamics of self-similar tilings', Ibid. 19 (1999), 1685. doi: 10.1017/S0143385797084988.

[41]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279.  doi: 10.1007/PL00009386.

[42]

B. Solomyak, Eigenfunctions for substitution tiling systems, in Probability and Number Theory-Kanazawa 2005, Adv. Stud. Pure Math., 49 (2007), 433–454.

[43]

W. Thurston, Groups, Tilings, and Finite State Automata, AMS lecture notes, 1989.

[44]

P. Walters, An Introduction to Ergodic Theory, Springer, 1982.

show all references

References:
[1]

S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot substitution conjecture, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savinien), Progr. Math., 309, Birkhäuser/Springer, Basel, (2015), 33–72. doi: 10.1007/978-3-0348-0903-0_2.

[2] M. Baake and U. Grimm, Aperiodic Order, Vol. 1. A Mathematical Invitation. With a Foreword by Roger Penrose, Encyclopedia of Mathematics and its Applications, 149. Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025256.
[3] M. Baake and U. Grimm, Aperiodic Order, Vol. 2. Crystallography and Almost Periodicity. With a foreword by Jeffrey C. Lagarias., Encyclopedia of Mathematics and its Applications, 166. Cambridge University Press, Cambridge, 2017. 
[4]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergod. Th. & Dynam. Sys., 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.

[5]

M. Baake and D. Lenz, Spectral notions of aperiodic order, Discrete Contin. Dyn. Syst., 10 (2017), 161-190.  doi: 10.3934/dcdss.2017009.

[6]

M. Baake and R. V. Moody, Self-similar measures for quasi-crystals, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph series, 13, AMS, Providence RI, (2000), 1–42.

[7]

J. Bellissard, D. J. L. Herrmann and M. Zarrouati, Hulls of aperiodic solids and gap labeling theorems, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, 13, AMS, Providence RI (2000), 207–258.

[8]

D. Damanik and D. Lenz, Linear repetitivity. I. Uniform subadditive ergodic theorems and applications, Discrete Comput. Geom., 26 (2001), 411-428.  doi: 10.1007/s00454-001-0033-z.

[9]

L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math., 239 (2002), 118-126. 

[10]

J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.  doi: 10.1090/S0002-9939-1962-0139135-6.

[11]

N. P. Frank, Tilings with infinite local complexity, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz, J. Savinien), Progr. Math., 309, Birkhäuser/Springer, Basel, (2015), 223–257. doi: 10.1007/978-3-0348-0903-0_7.

[12]

N. P. Frank and E. A. Jr Robinson, Generalized β-expansions, substitution tilings, and local finiteness, Trans. Amer. Math. Soc., 360 (2008), 1163-1177.  doi: 10.1090/S0002-9947-07-04527-8.

[13]

N. P. Frank and L. Sadun, Topology of (some) tiling spaces without finite local complexity, Discrete Contin. Dyn. Syst., 23 (2009), 847-865.  doi: 10.3934/dcds.2009.23.847.

[14]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $ {\mathbb R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.

[15]

N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, Topology Proc., 43 (2014), 235-276. 

[16]

D. Frettlöh and C. Richard, Dynamical properties of almost repetitive Delone sets, Discrete Contin. Dyn. Syst., 34 (2014), 531-556.  doi: 10.3934/dcds.2014.34.531.

[17]

A. Hof, On diffraction by aperiodic structures, Comm. Math. Phys., 169 (1995), 25-43.  doi: 10.1007/BF02101595.

[18]

R. Kenyon, Self-replicating tilings, Symbolic dynamics and its application (New Haven, CT, 1991), Contemp. Math., 135, Amer. Math. Soc., Providence, RI, (1992), 239–263. doi: 10.1090/conm/135/1185093.

[19]

R. Kenyon, Rigidity of planar tilings, Invent. Math., 107 (1992), 637-651.  doi: 10.1007/BF01231905.

[20]

R. Kenyon, Inflationary tilings with similarity structure, Comment. Math. Helv., 69 (1994), 169-198.  doi: 10.1007/BF02564481.

[21]

R. Kenyon, The construction of self-similar tilings, Geometric and Funct. Anal., 6 (1996), 471-488.  doi: 10.1007/BF02249260.

[22]

I. Környei, On a theorem of Pisot, Publ. Math. Debrecen, 34 (1987), 169-179. 

[23]

J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, Vol. 13, AMS, Providence, RI, (2000), 61–93.

[24]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergod. Th. & Dynam. Sys., 23 (2003), 831-867.  doi: 10.1017/S0143385702001566.

[25]

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

[26]

J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003–1018. doi: 10.1007/s00023-002-8646-1.

[27]

J.-Y. LeeR. 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.

[28]

J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comput. Geom., 39 (2008), 319-338.  doi: 10.1007/s00454-008-9054-1.

[29]

J.-Y. Lee and B. Solomyak, Pisot family self-affine tilings, discrete spectrum, and the Meyer property, Discrete Contin. Dyn. Syst., 32 (2012), 935-959.  doi: 10.3934/dcds.2012.32.935.

[30]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in Operator Algebras and Mathematical Physics (eds. J. M. Combes, G. Elliott, G. Nenciu, H. Siedentop and S. Stratila) Theta, Bucharest, (2003), 267–285.

[31]

C. Mauduit, Caractérisation des ensembles normaux subsitutifs, Invent. Math., 95 (1989), 133-147.  doi: 10.1007/BF01394146.

[32]

D. Mauldin and S. Williams, Hausdorff dimension in graph direct constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.  doi: 10.1090/S0002-9947-1988-0961615-4.

[33]

B. Mossé, Puissances de mots et reconnaisabilité des points fixes d'une substitution, Theor. Comp. Sci., 99 (1992), 327-334.  doi: 10.1016/0304-3975(92)90357-L.

[34]

P. Müller and C. Richard, Ergodic properties of randomly coloured point sets, Canad. J. Math., 65 (2013), 349-402.  doi: 10.4153/CJM-2012-009-7.

[35]

B. Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc., 351 (1999), 3315-3349.  doi: 10.1090/S0002-9947-99-02360-0.

[36]

M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, 2nd edition. Lecture Notes in Math., 1294, Springer, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.

[37]

C. Radin, Space tilings and substitutions, Geom. Dedicata, 55 (1995), 257-264.  doi: 10.1007/BF01266317.

[38]

C. Radin and M. Wolff, Space tilings and local isomorphism, Geom. Dedicata, 42 (1992), 355-360.  doi: 10.1007/BF02414073.

[39]

E. A. Robinson, Jr., Symbolic dynamics and tilings of $ {\mathbb R}^d$, in Symbolic Dynamics and Its Applications, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, (2004), 81–119. doi: 10.1090/psapm/060/2078847.

[40]

B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. & Dynam. Sys., 17 (1997), 695-738. Corrections to 'Dynamics of self-similar tilings', Ibid. 19 (1999), 1685. doi: 10.1017/S0143385797084988.

[41]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279.  doi: 10.1007/PL00009386.

[42]

B. Solomyak, Eigenfunctions for substitution tiling systems, in Probability and Number Theory-Kanazawa 2005, Adv. Stud. Pure Math., 49 (2007), 433–454.

[43]

W. Thurston, Groups, Tilings, and Finite State Automata, AMS lecture notes, 1989.

[44]

P. Walters, An Introduction to Ergodic Theory, Springer, 1982.

Figure 1.  Prototiles of the Frank-Robinson substitution tiling without FLC
Figure 2.  A patch of the tiling from Example 6.4 for $a = 2-\sqrt{2}$. The dots in the figure indicate the representative points of tiles
Figure 3.  Modification of Kenyon's example. The figure shows a patch of the substitution tiling in the case of $a = 2-\sqrt{2}$. The dots in the figure indicate the representative points of tiles
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