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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 and Continuous Dynamical Systems, 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), 619-636. doi: 10.1112/jlms/jdq091.

[2]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction, J. Fourier Anal. Appl., 11 (2005), 125-150. doi: 10.1007/s00041-005-4021-1.

[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), 4637-4654. doi: 10.1088/0305-4470/24/19/025.

[4]

J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys., 261 (2006), 1-41. doi: 10.1007/s00220-005-1445-z.

[5]

E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections, in "The Legacy of Sonya Kovalevskaya" (Cambridge, Mass., and Amherst, Mass., 1985), Contemp. Math., 64, Amer. Math. Soc., Providence, RI, (1987), 241-264. doi: 10.1090/conm/064/881466.

[6]

J. H. Conway and C. Radin, Quaquaversal tilings and rotations, Invent. Math., 132 (1998), 179-188. doi: 10.1007/s002220050221.

[7]

C. Corduneanu, "Almost Periodic Functions", Wiley Interscience, New York, 1968.

[8]

M. I. Cortez and B. Solomyak, Invariant measures for non-primitive tiling substitutions, J. Anal. Math., 115 (2011), 293-342. doi: 10.1007/s11854-011-0031-x.

[9]

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.

[10]

L. Danzer, Quasiperiodicity: local and global aspects, in "Group theoretical methods in physics" (Moscow, 1990), Lecture Notes in Phys., 382, Springer, Berlin (1991), 561-572. doi: 10.1007/3-540-54040-7_164.

[11]

D. Frettlöh, Substitution tilings with statistical circular symmetry, Eur. J. Comb., 29 (2008), 1881-1893. doi: 10.1016/j.ejc.2008.01.006.

[12]

D. Frettlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets, Discrete Comput. Geom., 37 (2007), 381-407. doi: 10.1007/s00454-006-1280-9.

[13]

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

[14]

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.

[15]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $\mathbb R^\mathsfd$, preprint, arXiv:1101.4930.

[16]

N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, preprint, arXiv:1201.3911.

[17]

W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. Amer. Math. Soc., 50 (1944), 915-919. doi: 10.1090/S0002-9904-1944-08262-1.

[18]

B. Grünbaum and G. C. Shephard, "Tilings and Patterns. An Introduction," A Series of Books in the Mathematical Sciences, W. H. Freeman and Company, New York, 1989.

[19]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, 23 (2003), 831-867. doi: 10.1017/S0143385702001566.

[20]

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

[21]

D. Lenz and C. Richard, Pure point diffraction and cut and project schemes for measures: The smooth case, Math. Z., 256 (2007), 347-378. doi: 10.1007/s00209-006-0077-0.

[22]

W. F. Lunnon and P. A. B. Pleasants, Quasicrystallographic tilings, J. Math. Pures et Appl. (9), 66 (1987), 217-263.

[23]

W. Miller, Jr., "Symmetry Groups and their Applications," Pure and Applied Mathematics, Vol. 50, Academic Press, New York-London, 1972.

[24]

M. Morse and G. A. Hedlund, Symbolic dynamics, Amer. J. Math., 60 (1938), 815-866. doi: 10.2307/2371264.

[25]

M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.

[26]

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.

[27]

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

[28]

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

[29]

E. A. Robinson, Jr., The dynamical properties of Penrose tilings, Trans. Amer. Math. Soc., 348 (1996), 4447-4464. doi: 10.1090/S0002-9947-96-01640-6.

[30]

E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbb R^\mathsfd$, in "Symbolic Dynamics and its Applications" Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, (2004), 81-119.

[31]

D. J. Rudolph, Markov tilings of $\mathbb R^\mathsfn$ and representations of $\mathbb R^\mathsfn$ actions, in " Measure and Measurable Dynamics" (Rochester, NY, 1987), Contemp. Math., 94, Amer. Math. Soc., Providence, RI, (1989), 271-290. doi: 10.1090/conm/094/1012996.

[32]

L. Sadun, Some generalizations of the pinwheel tiling, Discrete Comput. Geom., 20 (1998), 79-110. doi: 10.1007/PL00009379.

[33]

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

[34]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695-738; Corrections to: "Dynamics of self-similar tilings," Ergodic Theory Dynam. Systems, 19 (1999), 1685. doi: 10.1017/S0143385797084988.

[35]

W. Thurston, "Groups, Tilings, and Finite State Automata," AMS Colloquium Lecture Notes, Boulder, 1989.

[36]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[37]

T. Yokonuma, Discrete sets and associated dynamical systems in a non-commutative setting, Canad. Math. Bull., 48 (2005), 302-316. doi: 10.4153/CMB-2005-028-8.

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), 619-636. doi: 10.1112/jlms/jdq091.

[2]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction, J. Fourier Anal. Appl., 11 (2005), 125-150. doi: 10.1007/s00041-005-4021-1.

[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), 4637-4654. doi: 10.1088/0305-4470/24/19/025.

[4]

J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys., 261 (2006), 1-41. doi: 10.1007/s00220-005-1445-z.

[5]

E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections, in "The Legacy of Sonya Kovalevskaya" (Cambridge, Mass., and Amherst, Mass., 1985), Contemp. Math., 64, Amer. Math. Soc., Providence, RI, (1987), 241-264. doi: 10.1090/conm/064/881466.

[6]

J. H. Conway and C. Radin, Quaquaversal tilings and rotations, Invent. Math., 132 (1998), 179-188. doi: 10.1007/s002220050221.

[7]

C. Corduneanu, "Almost Periodic Functions", Wiley Interscience, New York, 1968.

[8]

M. I. Cortez and B. Solomyak, Invariant measures for non-primitive tiling substitutions, J. Anal. Math., 115 (2011), 293-342. doi: 10.1007/s11854-011-0031-x.

[9]

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.

[10]

L. Danzer, Quasiperiodicity: local and global aspects, in "Group theoretical methods in physics" (Moscow, 1990), Lecture Notes in Phys., 382, Springer, Berlin (1991), 561-572. doi: 10.1007/3-540-54040-7_164.

[11]

D. Frettlöh, Substitution tilings with statistical circular symmetry, Eur. J. Comb., 29 (2008), 1881-1893. doi: 10.1016/j.ejc.2008.01.006.

[12]

D. Frettlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets, Discrete Comput. Geom., 37 (2007), 381-407. doi: 10.1007/s00454-006-1280-9.

[13]

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

[14]

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.

[15]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $\mathbb R^\mathsfd$, preprint, arXiv:1101.4930.

[16]

N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, preprint, arXiv:1201.3911.

[17]

W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. Amer. Math. Soc., 50 (1944), 915-919. doi: 10.1090/S0002-9904-1944-08262-1.

[18]

B. Grünbaum and G. C. Shephard, "Tilings and Patterns. An Introduction," A Series of Books in the Mathematical Sciences, W. H. Freeman and Company, New York, 1989.

[19]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, 23 (2003), 831-867. doi: 10.1017/S0143385702001566.

[20]

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

[21]

D. Lenz and C. Richard, Pure point diffraction and cut and project schemes for measures: The smooth case, Math. Z., 256 (2007), 347-378. doi: 10.1007/s00209-006-0077-0.

[22]

W. F. Lunnon and P. A. B. Pleasants, Quasicrystallographic tilings, J. Math. Pures et Appl. (9), 66 (1987), 217-263.

[23]

W. Miller, Jr., "Symmetry Groups and their Applications," Pure and Applied Mathematics, Vol. 50, Academic Press, New York-London, 1972.

[24]

M. Morse and G. A. Hedlund, Symbolic dynamics, Amer. J. Math., 60 (1938), 815-866. doi: 10.2307/2371264.

[25]

M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.

[26]

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.

[27]

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

[28]

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

[29]

E. A. Robinson, Jr., The dynamical properties of Penrose tilings, Trans. Amer. Math. Soc., 348 (1996), 4447-4464. doi: 10.1090/S0002-9947-96-01640-6.

[30]

E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbb R^\mathsfd$, in "Symbolic Dynamics and its Applications" Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, (2004), 81-119.

[31]

D. J. Rudolph, Markov tilings of $\mathbb R^\mathsfn$ and representations of $\mathbb R^\mathsfn$ actions, in " Measure and Measurable Dynamics" (Rochester, NY, 1987), Contemp. Math., 94, Amer. Math. Soc., Providence, RI, (1989), 271-290. doi: 10.1090/conm/094/1012996.

[32]

L. Sadun, Some generalizations of the pinwheel tiling, Discrete Comput. Geom., 20 (1998), 79-110. doi: 10.1007/PL00009379.

[33]

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

[34]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695-738; Corrections to: "Dynamics of self-similar tilings," Ergodic Theory Dynam. Systems, 19 (1999), 1685. doi: 10.1017/S0143385797084988.

[35]

W. Thurston, "Groups, Tilings, and Finite State Automata," AMS Colloquium Lecture Notes, Boulder, 1989.

[36]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[37]

T. Yokonuma, Discrete sets and associated dynamical systems in a non-commutative setting, Canad. Math. Bull., 48 (2005), 302-316. doi: 10.4153/CMB-2005-028-8.

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