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February  2014, 34(2): 531-556. doi: 10.3934/dcds.2014.34.531

## Dynamical properties of almost repetitive Delone sets

 1 Technische Fakultät, Universität Bielefeld, Universitätsstraße 25, 33501 Bielefeld, Germany 2 Department für Mathematik, Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen,, Germany

Received  October 2012 Revised  April 2013 Published  August 2013

We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense tile orientations.
Citation: Dirk Frettlöh, Christoph Richard. Dynamical properties of almost repetitive Delone sets. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 531-556. doi: 10.3934/dcds.2014.34.531
##### References:
 [1] H. Abels, A. Manoussos and G. Noskov, Proper actions and proper invariant metrics,, J. London Math. Soc. (2), 83 (2011), 619. doi: 10.1112/jlms/jdq091. Google Scholar [2] M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction,, J. Fourier Anal. Appl., 11 (2005), 125. doi: 10.1007/s00041-005-4021-1. Google Scholar [3] M. Baake, M. Schlottmann and P. D. Jarvis, Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability,, J. Phys. A, 24 (1991), 4637. doi: 10.1088/0305-4470/24/19/025. Google Scholar [4] J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling,, Comm. Math. Phys., 261 (2006), 1. doi: 10.1007/s00220-005-1445-z. Google Scholar [5] E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections,, in, 64 (1987), 241. doi: 10.1090/conm/064/881466. Google Scholar [6] J. H. Conway and C. Radin, Quaquaversal tilings and rotations,, Invent. Math., 132 (1998), 179. doi: 10.1007/s002220050221. Google Scholar [7] C. Corduneanu, "Almost Periodic Functions",, Wiley Interscience, (1968). Google Scholar [8] M. I. Cortez and B. Solomyak, Invariant measures for non-primitive tiling substitutions,, J. Anal. Math., 115 (2011), 293. doi: 10.1007/s11854-011-0031-x. Google Scholar [9] D. Damanik and D. Lenz, Linear repetitivity. I. Uniform subadditive ergodic theorems and applications,, Discrete Comput. Geom., 26 (2001), 411. doi: 10.1007/s00454-001-0033-z. Google Scholar [10] L. Danzer, Quasiperiodicity: local and global aspects,, in, 382 (1991), 561. doi: 10.1007/3-540-54040-7_164. Google Scholar [11] D. Frettlöh, Substitution tilings with statistical circular symmetry,, Eur. J. Comb., 29 (2008), 1881. doi: 10.1016/j.ejc.2008.01.006. Google Scholar [12] D. Frettlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets,, Discrete Comput. Geom., 37 (2007), 381. doi: 10.1007/s00454-006-1280-9. Google Scholar [13] N. P. Frank and E. A. Robinson, Generalized $\beta$-expansions, substitution tilings, and local finiteness,, Trans. Amer. Math. Soc., 360 (2008), 1163. doi: 10.1090/S0002-9947-07-04527-8. Google Scholar [14] N. P. Frank and L. Sadun, Topology of some tiling spaces without finite local complexity,, Discrete Contin. Dyn. Syst., 23 (2009), 847. doi: 10.3934/dcds.2009.23.847. Google Scholar [15] N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $\mathbb R^\mathsfd$,, preprint, (). Google Scholar [16] N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity,, preprint, (). Google Scholar [17] W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties,, Bull. Amer. Math. Soc., 50 (1944), 915. doi: 10.1090/S0002-9904-1944-08262-1. Google Scholar [18] B. Grünbaum and G. C. Shephard, "Tilings and Patterns. An Introduction,", A Series of Books in the Mathematical Sciences, (1989). Google Scholar [19] J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals,, Ergodic Theory Dynam. Systems, 23 (2003), 831. doi: 10.1017/S0143385702001566. Google Scholar [20] J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. H. Poincaré, 3 (2002), 1003. doi: 10.1007/s00023-002-8646-1. Google Scholar [21] D. Lenz and C. Richard, Pure point diffraction and cut and project schemes for measures: The smooth case,, Math. Z., 256 (2007), 347. doi: 10.1007/s00209-006-0077-0. Google Scholar [22] W. F. Lunnon and P. A. B. Pleasants, Quasicrystallographic tilings,, J. Math. Pures et Appl. (9), 66 (1987), 217. Google Scholar [23] W. Miller, Jr., "Symmetry Groups and their Applications,", Pure and Applied Mathematics, (1972). Google Scholar [24] M. Morse and G. A. Hedlund, Symbolic dynamics,, Amer. J. Math., 60 (1938), 815. doi: 10.2307/2371264. Google Scholar [25] M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1. doi: 10.2307/2371431. Google Scholar [26] P. Müller and C. Richard, Ergodic properties of randomly coloured point sets,, Canad. J. Math., 65 (2013), 349. doi: 10.4153/CJM-2012-009-7. Google Scholar [27] C. Radin, Space tilings and substitutions,, Geom. Dedicata, 55 (1995), 257. doi: 10.1007/BF01266317. Google Scholar [28] C. Radin and M. Wolff, Space tilings and local isomorphism,, Geom. Dedicata, 42 (1992), 355. doi: 10.1007/BF02414073. Google Scholar [29] E. A. Robinson, Jr., The dynamical properties of Penrose tilings,, Trans. Amer. Math. Soc., 348 (1996), 4447. doi: 10.1090/S0002-9947-96-01640-6. Google Scholar [30] E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbb R^\mathsfd$,, in, 60 (2004), 81. Google Scholar [31] D. J. Rudolph, Markov tilings of $\mathbb R^\mathsfn$ and representations of $\mathbb R^\mathsfn$ actions,, in, 94 (1989), 271. doi: 10.1090/conm/094/1012996. Google Scholar [32] L. Sadun, Some generalizations of the pinwheel tiling,, Discrete Comput. Geom., 20 (1998), 79. doi: 10.1007/PL00009379. Google Scholar [33] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265. doi: 10.1007/PL00009386. Google Scholar [34] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695-738;, Corrections to:, 19 (1999). doi: 10.1017/S0143385797084988. Google Scholar [35] W. Thurston, "Groups, Tilings, and Finite State Automata,", AMS Colloquium Lecture Notes, (1989). Google Scholar [36] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar [37] T. Yokonuma, Discrete sets and associated dynamical systems in a non-commutative setting,, Canad. Math. Bull., 48 (2005), 302. doi: 10.4153/CMB-2005-028-8. Google Scholar

show all references

##### References:
 [1] H. Abels, A. Manoussos and G. Noskov, Proper actions and proper invariant metrics,, J. London Math. Soc. (2), 83 (2011), 619. doi: 10.1112/jlms/jdq091. Google Scholar [2] M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction,, J. Fourier Anal. Appl., 11 (2005), 125. doi: 10.1007/s00041-005-4021-1. Google Scholar [3] M. Baake, M. Schlottmann and P. D. Jarvis, Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability,, J. Phys. A, 24 (1991), 4637. doi: 10.1088/0305-4470/24/19/025. Google Scholar [4] J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling,, Comm. Math. Phys., 261 (2006), 1. doi: 10.1007/s00220-005-1445-z. Google Scholar [5] E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections,, in, 64 (1987), 241. doi: 10.1090/conm/064/881466. Google Scholar [6] J. H. Conway and C. Radin, Quaquaversal tilings and rotations,, Invent. Math., 132 (1998), 179. doi: 10.1007/s002220050221. Google Scholar [7] C. Corduneanu, "Almost Periodic Functions",, Wiley Interscience, (1968). Google Scholar [8] M. I. Cortez and B. Solomyak, Invariant measures for non-primitive tiling substitutions,, J. Anal. Math., 115 (2011), 293. doi: 10.1007/s11854-011-0031-x. Google Scholar [9] D. Damanik and D. Lenz, Linear repetitivity. I. Uniform subadditive ergodic theorems and applications,, Discrete Comput. Geom., 26 (2001), 411. doi: 10.1007/s00454-001-0033-z. Google Scholar [10] L. Danzer, Quasiperiodicity: local and global aspects,, in, 382 (1991), 561. doi: 10.1007/3-540-54040-7_164. Google Scholar [11] D. Frettlöh, Substitution tilings with statistical circular symmetry,, Eur. J. Comb., 29 (2008), 1881. doi: 10.1016/j.ejc.2008.01.006. Google Scholar [12] D. Frettlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets,, Discrete Comput. Geom., 37 (2007), 381. doi: 10.1007/s00454-006-1280-9. Google Scholar [13] N. P. Frank and E. A. Robinson, Generalized $\beta$-expansions, substitution tilings, and local finiteness,, Trans. Amer. Math. Soc., 360 (2008), 1163. doi: 10.1090/S0002-9947-07-04527-8. Google Scholar [14] N. P. Frank and L. Sadun, Topology of some tiling spaces without finite local complexity,, Discrete Contin. Dyn. Syst., 23 (2009), 847. doi: 10.3934/dcds.2009.23.847. Google Scholar [15] N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $\mathbb R^\mathsfd$,, preprint, (). Google Scholar [16] N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity,, preprint, (). Google Scholar [17] W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties,, Bull. Amer. Math. Soc., 50 (1944), 915. doi: 10.1090/S0002-9904-1944-08262-1. Google Scholar [18] B. Grünbaum and G. C. Shephard, "Tilings and Patterns. An Introduction,", A Series of Books in the Mathematical Sciences, (1989). Google Scholar [19] J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals,, Ergodic Theory Dynam. Systems, 23 (2003), 831. doi: 10.1017/S0143385702001566. Google Scholar [20] J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. H. Poincaré, 3 (2002), 1003. doi: 10.1007/s00023-002-8646-1. Google Scholar [21] D. Lenz and C. Richard, Pure point diffraction and cut and project schemes for measures: The smooth case,, Math. Z., 256 (2007), 347. doi: 10.1007/s00209-006-0077-0. Google Scholar [22] W. F. Lunnon and P. A. B. Pleasants, Quasicrystallographic tilings,, J. Math. Pures et Appl. (9), 66 (1987), 217. Google Scholar [23] W. Miller, Jr., "Symmetry Groups and their Applications,", Pure and Applied Mathematics, (1972). Google Scholar [24] M. Morse and G. A. Hedlund, Symbolic dynamics,, Amer. J. Math., 60 (1938), 815. doi: 10.2307/2371264. Google Scholar [25] M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1. doi: 10.2307/2371431. Google Scholar [26] P. Müller and C. Richard, Ergodic properties of randomly coloured point sets,, Canad. J. Math., 65 (2013), 349. doi: 10.4153/CJM-2012-009-7. Google Scholar [27] C. Radin, Space tilings and substitutions,, Geom. Dedicata, 55 (1995), 257. doi: 10.1007/BF01266317. Google Scholar [28] C. Radin and M. Wolff, Space tilings and local isomorphism,, Geom. Dedicata, 42 (1992), 355. doi: 10.1007/BF02414073. Google Scholar [29] E. A. Robinson, Jr., The dynamical properties of Penrose tilings,, Trans. Amer. Math. Soc., 348 (1996), 4447. doi: 10.1090/S0002-9947-96-01640-6. Google Scholar [30] E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbb R^\mathsfd$,, in, 60 (2004), 81. Google Scholar [31] D. J. Rudolph, Markov tilings of $\mathbb R^\mathsfn$ and representations of $\mathbb R^\mathsfn$ actions,, in, 94 (1989), 271. doi: 10.1090/conm/094/1012996. Google Scholar [32] L. Sadun, Some generalizations of the pinwheel tiling,, Discrete Comput. Geom., 20 (1998), 79. doi: 10.1007/PL00009379. Google Scholar [33] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265. doi: 10.1007/PL00009386. Google Scholar [34] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695-738;, Corrections to:, 19 (1999). doi: 10.1017/S0143385797084988. Google Scholar [35] W. Thurston, "Groups, Tilings, and Finite State Automata,", AMS Colloquium Lecture Notes, (1989). Google Scholar [36] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar [37] T. Yokonuma, Discrete sets and associated dynamical systems in a non-commutative setting,, Canad. Math. Bull., 48 (2005), 302. doi: 10.4153/CMB-2005-028-8. Google Scholar
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