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August  2021, 41(8): 3781-3796. doi: 10.3934/dcds.2021016

Chaotic Delone sets

Jesús A. Álvarez López 1, , Ramón Barral Lijó 2,, , John Hunton 3, , Hiraku Nozawa 4, and  John R. Parker 5,

1. 

Departamento e Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain
2. 

Research Organization of Science and Technology, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan
3. 

Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK
4. 

Department of Mathematical Sciences, Colleges of Science and Engineering, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan
5. 

Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK

* Corresponding author: Ramón Barral Lijó (ramonbarrallijo@gmail.com)

Received  August 2020 Revised  November 2020 Published  August 2021 Early access  January 2021

We present a definition of chaotic Delone set and establish the genericity of chaos in the space of $ (\epsilon,\delta) $-Delone sets for $ \epsilon\geq \delta $. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.

Keywords: Delone set, chaos, tiling, foliated space, hyperbolic dynamical system, geodesic flow.
Mathematics Subject Classification: 37D45, 52C23, 37B51.
Citation: Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa, John R. Parker. Chaotic Delone sets. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3781-3796. doi: 10.3934/dcds.2021016
References:
[1]

J. A. Álvarez López and A. Candel, Algebraic characterization of quasi-isometric spaces via the Higson compactification, Topology Appl., 158 (2011), 1679-1694.  doi: 10.1016/j.topol.2011.05.036.  Google Scholar

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[3]

D. V. Anosov, Geodesic Flows on Closed {R}iemann Manifolds with Negative Curvature, Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969.  Google Scholar

[4]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra, Ergodic Theory Dynam. Systems, 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.  Google Scholar

[5] M. Baake and U. Grimm, Aperiodic Order, Vol. 1., A Mathematical Invitation. With a foreword by Roger Penrose. Vol. 149 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025256.  Google Scholar
[6]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.  doi: 10.1080/00029890.1992.11995856.  Google Scholar

[7]

R. Barral Lijó and H. Nozawa, Genericity of chaos for colored graphs, preprint (2019), arXiv: 1909.01676. Google Scholar

[8]

J. Belissard, D. Hermann and M. Zarrouati, Hulls of aperiodic solids and gap labeling theorems, in Directions in Mathematical Quasicrystals (eds. R. Baake and R. V. Moody), vol. 13, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/crmm/013.  Google Scholar

[9]

G. Cairns, G. Davis, D. Elton, A. Kolganova and P. Perversi, Chaotic group actions, Enseign. Math. (2), 41 (1995), 123-133.   Google Scholar

[10]

D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc., 8 (1933), 254-260.   Google Scholar

[11]

F. Dal'bo, Remarques sur le spectre des longueurs d'une surface et comptages, Bol. Soc. Brasil. Mat. (N.S.), 30 (1999), 199-221.  doi: 10.1007/BF01235869.  Google Scholar

[12]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley, 1989.  Google Scholar

[13]

A. Forrest, J. Hunton and J. Kellendonk, Topological Invariants for Projection Method Patterns, Mem. Amer. Math. Soc., 159 2002, x+120. doi: 10.1090/memo/0758.  Google Scholar

[14]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodesiques, J. Math. Pures Appl., 4 (1898), 27-73.   Google Scholar

[15]

G. A. Hedlund, On the metrical transitivity of the geodesics on closed surfaces of constant negative curvature, Ann. of Math. (2), 35 (1934), 787-808.  doi: 10.2307/1968495.  Google Scholar

[16]

G. A. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc., 45 (1939), 241-260.  doi: 10.1090/S0002-9904-1939-06945-0.  Google Scholar

[17]

S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 87-132.  doi: 10.1090/S0273-0979-06-01115-3.  Google Scholar

[18]

B. P. Kitchens, Symbolic Dynamics. One-sided, Two-sided and Countable State Markov Shifts, Universitext. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58822-8.  Google Scholar

[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.  Google Scholar

[20]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in Operator Algebras and Mathematical Physics: Conference Proceedings : Constanţa (Romania), July 2-7, 2001 (eds. J. Combes, J. Cuntz, G. Elliott, G. Nenciu, H. Siedentop and S. Stratila), 2003.  Google Scholar

[21]

Robert V. Moody (ed.), The Mathematics of Long-Range Aperiodic Order, vol. 489 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-015-8784-6.  Google Scholar

[22]

H. M. Morse, A one-to-one representation of geodesics on a surface of negative curvature, Amer. J. Math., 43 (1921), 33-51.  doi: 10.2307/2370306.  Google Scholar

[23]

H. M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.  doi: 10.1090/S0002-9947-1921-1501161-8.  Google Scholar

[24]

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

[25]

F. M. Schneider, S. Kerkhoff, M. Behrisch and S. Siegmund, Chaotic actions of topological semigroups, Semigroup Forum, 87 (2013), 590-598.  doi: 10.1007/s00233-013-9517-4.  Google Scholar

show all references

References:
[1]

J. A. Álvarez López and A. Candel, Algebraic characterization of quasi-isometric spaces via the Higson compactification, Topology Appl., 158 (2011), 1679-1694.  doi: 10.1016/j.topol.2011.05.036.  Google Scholar

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[3]

D. V. Anosov, Geodesic Flows on Closed {R}iemann Manifolds with Negative Curvature, Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969.  Google Scholar

[4]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra, Ergodic Theory Dynam. Systems, 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.  Google Scholar

[5] M. Baake and U. Grimm, Aperiodic Order, Vol. 1., A Mathematical Invitation. With a foreword by Roger Penrose. Vol. 149 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025256.  Google Scholar
[6]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.  doi: 10.1080/00029890.1992.11995856.  Google Scholar

[7]

R. Barral Lijó and H. Nozawa, Genericity of chaos for colored graphs, preprint (2019), arXiv: 1909.01676. Google Scholar

[8]

J. Belissard, D. Hermann and M. Zarrouati, Hulls of aperiodic solids and gap labeling theorems, in Directions in Mathematical Quasicrystals (eds. R. Baake and R. V. Moody), vol. 13, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/crmm/013.  Google Scholar

[9]

G. Cairns, G. Davis, D. Elton, A. Kolganova and P. Perversi, Chaotic group actions, Enseign. Math. (2), 41 (1995), 123-133.   Google Scholar

[10]

D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc., 8 (1933), 254-260.   Google Scholar

[11]

F. Dal'bo, Remarques sur le spectre des longueurs d'une surface et comptages, Bol. Soc. Brasil. Mat. (N.S.), 30 (1999), 199-221.  doi: 10.1007/BF01235869.  Google Scholar

[12]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley, 1989.  Google Scholar

[13]

A. Forrest, J. Hunton and J. Kellendonk, Topological Invariants for Projection Method Patterns, Mem. Amer. Math. Soc., 159 2002, x+120. doi: 10.1090/memo/0758.  Google Scholar

[14]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodesiques, J. Math. Pures Appl., 4 (1898), 27-73.   Google Scholar

[15]

G. A. Hedlund, On the metrical transitivity of the geodesics on closed surfaces of constant negative curvature, Ann. of Math. (2), 35 (1934), 787-808.  doi: 10.2307/1968495.  Google Scholar

[16]

G. A. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc., 45 (1939), 241-260.  doi: 10.1090/S0002-9904-1939-06945-0.  Google Scholar

[17]

S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 87-132.  doi: 10.1090/S0273-0979-06-01115-3.  Google Scholar

[18]

B. P. Kitchens, Symbolic Dynamics. One-sided, Two-sided and Countable State Markov Shifts, Universitext. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58822-8.  Google Scholar

[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.  Google Scholar

[20]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in Operator Algebras and Mathematical Physics: Conference Proceedings : Constanţa (Romania), July 2-7, 2001 (eds. J. Combes, J. Cuntz, G. Elliott, G. Nenciu, H. Siedentop and S. Stratila), 2003.  Google Scholar

[21]

Robert V. Moody (ed.), The Mathematics of Long-Range Aperiodic Order, vol. 489 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-015-8784-6.  Google Scholar

[22]

H. M. Morse, A one-to-one representation of geodesics on a surface of negative curvature, Amer. J. Math., 43 (1921), 33-51.  doi: 10.2307/2370306.  Google Scholar

[23]

H. M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.  doi: 10.1090/S0002-9947-1921-1501161-8.  Google Scholar

[24]

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

[25]

F. M. Schneider, S. Kerkhoff, M. Behrisch and S. Siegmund, Chaotic actions of topological semigroups, Semigroup Forum, 87 (2013), 590-598.  doi: 10.1007/s00233-013-9517-4.  Google Scholar

Figure 1.  Construction of $ S_{\ell} $ in $ \mathbb{H}^2 $. The black dots represent points in $ \Gamma x $, the blue area is $ E_{\ell} $, the red dots represent points in $ S_{\ell} $.
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Figure 2.  The disks represent the inverse image of $ \Delta $. The projection of $ k_{1} $ to $ \Sigma $ has one-sided tangency, while the projection of $ k_{2} $ to $ \Sigma $ does not.
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Figure 3.  A 12-gon P
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Figure 4.  A triangle T
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Figure 5.  The picture on the left represents $ T\subset \mathbb{T}^n $; the right one its lift to $ \mathbb{R}^n $ following a grid pattern
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Figure 6.  Approximation of $ S^{+}_{\ell} $ by $ S^{+}_{k} $: The vectors $ \nu_{+}(\ell) $ and $ \nu_{+}(k) $ represent the orientations of the normal bundles of $ \ell $ and $ k $, respectively. Two circles with dotted lines represent the boundary of the $ \rho $-neighbourhoods of $ I $ and $ J $, respectively. The dots represent points in $ \Gamma x $. The blue dots belong to both $ E^{+}_{\ell} $ and $ E^{+}_{k} $. But the black dots do not because they belong to the negative side of the boundary of $ E_{\ell} $ or $ E_{k} $, respectively
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