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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 & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] M. Baake and D. Lenz, Spectral notions of aperiodic order, Discrete Contin. Dyn. Syst., 10 (2017), 161-190. doi: 10.3934/dcdss.2017009. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math., 239 (2002), 118-126. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, Topology Proc., 43 (2014), 235-276. Google Scholar [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. Google Scholar [17] A. Hof, On diffraction by aperiodic structures, Comm. Math. Phys., 169 (1995), 25-43. doi: 10.1007/BF02101595. Google Scholar [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. Google Scholar [19] R. Kenyon, Rigidity of planar tilings, Invent. Math., 107 (1992), 637-651. doi: 10.1007/BF01231905. Google Scholar [20] R. Kenyon, Inflationary tilings with similarity structure, Comment. Math. Helv., 69 (1994), 169-198. doi: 10.1007/BF02564481. Google Scholar [21] R. Kenyon, The construction of self-similar tilings, Geometric and Funct. Anal., 6 (1996), 471-488. doi: 10.1007/BF02249260. Google Scholar [22] I. Környei, On a theorem of Pisot, Publ. Math. Debrecen, 34 (1987), 169-179. Google Scholar [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. Google Scholar [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. Google Scholar [25] J. C. Lagarias and Y. Wang, Substitution Delone sets, Discrete Comput. Geom., 29 (2003), 175-209. doi: 10.1007/s00454-002-2820-6. Google Scholar [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. Google Scholar [27] 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] C. Mauduit, Caractérisation des ensembles normaux subsitutifs, Invent. Math., 95 (1989), 133-147. doi: 10.1007/BF01394146. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] C. Radin, Space tilings and substitutions, Geom. Dedicata, 55 (1995), 257-264. doi: 10.1007/BF01266317. Google Scholar [38] C. Radin and M. Wolff, Space tilings and local isomorphism, Geom. Dedicata, 42 (1992), 355-360. doi: 10.1007/BF02414073. Google Scholar [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. Google Scholar [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. Google Scholar [41] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279. doi: 10.1007/PL00009386. Google Scholar [42] B. Solomyak, Eigenfunctions for substitution tiling systems, in Probability and Number Theory-Kanazawa 2005, Adv. Stud. Pure Math., 49 (2007), 433–454. Google Scholar [43] W. Thurston, Groups, Tilings, and Finite State Automata, AMS lecture notes, 1989.Google Scholar [44] P. Walters, An Introduction to Ergodic Theory, Springer, 1982. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] M. Baake and D. Lenz, Spectral notions of aperiodic order, Discrete Contin. Dyn. Syst., 10 (2017), 161-190. doi: 10.3934/dcdss.2017009. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] L. Danzer, Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math., 239 (2002), 118-126. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, Topology Proc., 43 (2014), 235-276. Google Scholar [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. Google Scholar [17] A. Hof, On diffraction by aperiodic structures, Comm. Math. Phys., 169 (1995), 25-43. doi: 10.1007/BF02101595. Google Scholar [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. Google Scholar [19] R. Kenyon, Rigidity of planar tilings, Invent. Math., 107 (1992), 637-651. doi: 10.1007/BF01231905. Google Scholar [20] R. Kenyon, Inflationary tilings with similarity structure, Comment. Math. Helv., 69 (1994), 169-198. doi: 10.1007/BF02564481. Google Scholar [21] R. Kenyon, The construction of self-similar tilings, Geometric and Funct. Anal., 6 (1996), 471-488. doi: 10.1007/BF02249260. Google Scholar [22] I. Környei, On a theorem of Pisot, Publ. Math. Debrecen, 34 (1987), 169-179. Google Scholar [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. Google Scholar [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. Google Scholar [25] J. C. Lagarias and Y. Wang, Substitution Delone sets, Discrete Comput. Geom., 29 (2003), 175-209. doi: 10.1007/s00454-002-2820-6. Google Scholar [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. Google Scholar [27] 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] C. Mauduit, Caractérisation des ensembles normaux subsitutifs, Invent. Math., 95 (1989), 133-147. doi: 10.1007/BF01394146. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] C. Radin, Space tilings and substitutions, Geom. Dedicata, 55 (1995), 257-264. doi: 10.1007/BF01266317. Google Scholar [38] C. Radin and M. Wolff, Space tilings and local isomorphism, Geom. Dedicata, 42 (1992), 355-360. doi: 10.1007/BF02414073. Google Scholar [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. Google Scholar [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. Google Scholar [41] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279. doi: 10.1007/PL00009386. Google Scholar [42] B. Solomyak, Eigenfunctions for substitution tiling systems, in Probability and Number Theory-Kanazawa 2005, Adv. Stud. Pure Math., 49 (2007), 433–454. Google Scholar [43] W. Thurston, Groups, Tilings, and Finite State Automata, AMS lecture notes, 1989.Google Scholar [44] P. Walters, An Introduction to Ergodic Theory, Springer, 1982. Google Scholar
Prototiles of the Frank-Robinson substitution tiling without FLC
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
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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