August  2021, 41(8): 3869-3902. doi: 10.3934/dcds.2021020

Random substitution tilings and deviation phenomena

1. 

University of Denver, Denver, CO, USA

2. 

University of Maryland, College Park, MD, USA

* Corresponding author

Received  May 2020 Revised  October 2020 Published  January 2021

Suppose a set of prototiles allows $ N $ different substitution rules. In this paper we study tilings of $ \mathbb{R}^d $ constructed from random application of the substitution rules. The space of all possible tilings obtained from all possible combinations of these substitutions is the union of all possible tilings spaces coming from these substitutions and has the structure of a Cantor set. The renormalization cocycle on the cohomology bundle over this space determines the statistical properties of the tilings through its Lyapunov spectrum by controlling the deviation of ergodic averages of the $ \mathbb{R}^d $ action on the tiling spaces.

Citation: Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3869-3902. doi: 10.3934/dcds.2021020
References:
[1]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems, 18 (1998), 509-537.  doi: 10.1017/S0143385798100457.  Google Scholar

[2]

M. F. Barnsley and A. Vince, Self-similar tilings of fractal blow-ups, Horizons of Fractal Geometry and Complex Dimensions, 41-62, Contemp. Math., 731, Amer. Math. Soc., Providence, RI, (2019). doi: 10.1090/conm/731/14672.  Google Scholar

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J. BellissardA. Julien and J. Savinien, Tiling groupoids and Bratteli diagrams, Ann. Henri Poincaré, 11 (2010), 69-99.  doi: 10.1007/s00023-010-0034-7.  Google Scholar

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V. Berthé and V. Delecroix, Beyond substitutive dynamical systems: $S$-adic expansions, in Numeration and Substitution 2012, RIMS Kôkyûroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, (2014), 81-123.  Google Scholar

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V. BerthéW. Steiner and J. M. Thuswaldner, Geometry, dynamics, and arithmetic of $S$-adic shifts, Ann. Inst. Fourier (Grenoble), 69 (2019), 1347-1409.  doi: 10.5802/aif.3273.  Google Scholar

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A. I. Bufetov and B. Solomyak, Limit theorems for self-similar tilings, Comm. Math. Phys., 319 (2013), 761-789.  doi: 10.1007/s00220-012-1624-7.  Google Scholar

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S. Cosentino and L. Flaminio, Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds, J. Mod. Dyn., 9 (2015), 305-353.  doi: 10.3934/jmd.2015.9.305.  Google Scholar

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V. DelecroixP. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 1085-1110.   Google Scholar

[9]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

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G. Forni, Asymptotic behaviour of ergodic integrals of 'renormalizable' parabolic flows, in Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 317-326.  Google Scholar

[11]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.  Google Scholar

[12]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn., 8 (2014), 271-436.  doi: 10.3934/jmd.2014.8.271.  Google Scholar

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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

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F. GählerE. E. Kwan and G. R. Maloney, A computer search for planar substitution tilings with $n$-fold rotational symmetry, Discrete Comput. Geom., 53 (2015), 445-465.  doi: 10.1007/s00454-014-9659-5.  Google Scholar

[15]

F. Gähler and G. R. Maloney, Cohomology of one-dimensional mixed substitution tiling spaces, Topology Appl., 160 (2013), 703-719.  doi: 10.1016/j.topol.2013.01.019.  Google Scholar

[16]

C. Godrèche and J. M. Luck, Quasiperiodicity and randomness in tilings of the plane, J. Statist. Phys., 55 (1989), 1-28.  doi: 10.1007/BF01042590.  Google Scholar

[17]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[18]

A. Julien and J. Savinien, Tiling groupoids and Bratteli diagrams II: Structure of the orbit equivalence relation, Ann. Henri Poincaré, 13 (2012), 297-332.  doi: 10.1007/s00023-011-0121-4.  Google Scholar

[19]

J. Kellendonk, Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys., 7 (1995), 1133-1180.  doi: 10.1142/S0129055X95000426.  Google Scholar

[20]

J. Kellendonk and I. F. Putnam, The Ruelle-Sullivan map for actions of $\Bbb R^n$, Math. Ann., 334 (2006), 693-711.  doi: 10.1007/s00208-005-0728-1.  Google Scholar

[21]

K. Lindsey and R. Treviño, Infinite type flat surface models of ergodic systems, Discrete Contin. Dyn. Syst., 36 (2016), 5509-5553.  doi: 10.3934/dcds.2016043.  Google Scholar

[22]

D. Rust, An uncountable set of tiling spaces with distinct cohomology, Topology and its Applications, 205 (2016), 58-81.  doi: 10.1016/j.topol.2016.01.020.  Google Scholar

[23]

D. Rust and T. Spindeler, Dynamical systems arising from random substitutions, Indag. Math. (N.S.), 29 (2018), 1131-1155.  doi: 10.1016/j.indag.2018.05.013.  Google Scholar

[24]

L. Sadun, Pattern-equivariant cohomology with integer coefficients, Ergodic Theory Dynam. Systems, 27 (2007), 1991-1998.  doi: 10.1017/S0143385707000259.  Google Scholar

[25]

L. Sadun, Exact regularity and the cohomology of tiling spaces, Ergodic Theory Dynam. Systems, 31 (2011), 1819-1834.  doi: 10.1017/S0143385710000611.  Google Scholar

[26]

S. Schmieding and R. Treviño, Self affine Delone sets and deviation phenomena, Comm. Math. Phys., 357 (2018), 1071-1112.  doi: 10.1007/s00220-017-3011-x.  Google Scholar

[27]

S. Schmieding and R. Treviño, Traces of random operators associated with self-affine delone sets and Shubin's formula, Ann. Henri Poincaré, 19 (2018), 2575-2597.  doi: 10.1007/s00023-018-0700-8.  Google Scholar

[28]

Y. Solomon, A simple condition for bounded displacement, J. Math. Anal. Appl., 414 (2014), 134-148.  doi: 10.1016/j.jmaa.2013.12.050.  Google Scholar

[29]

R. Treviño, Flat surfaces, Bratteli diagrams and unique ergodicity à la Masur, Israel J. Math., 225 (2018), 35-70.  doi: 10.1007/s11856-018-1636-x.  Google Scholar

show all references

References:
[1]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems, 18 (1998), 509-537.  doi: 10.1017/S0143385798100457.  Google Scholar

[2]

M. F. Barnsley and A. Vince, Self-similar tilings of fractal blow-ups, Horizons of Fractal Geometry and Complex Dimensions, 41-62, Contemp. Math., 731, Amer. Math. Soc., Providence, RI, (2019). doi: 10.1090/conm/731/14672.  Google Scholar

[3]

J. BellissardA. Julien and J. Savinien, Tiling groupoids and Bratteli diagrams, Ann. Henri Poincaré, 11 (2010), 69-99.  doi: 10.1007/s00023-010-0034-7.  Google Scholar

[4]

V. Berthé and V. Delecroix, Beyond substitutive dynamical systems: $S$-adic expansions, in Numeration and Substitution 2012, RIMS Kôkyûroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, (2014), 81-123.  Google Scholar

[5]

V. BerthéW. Steiner and J. M. Thuswaldner, Geometry, dynamics, and arithmetic of $S$-adic shifts, Ann. Inst. Fourier (Grenoble), 69 (2019), 1347-1409.  doi: 10.5802/aif.3273.  Google Scholar

[6]

A. I. Bufetov and B. Solomyak, Limit theorems for self-similar tilings, Comm. Math. Phys., 319 (2013), 761-789.  doi: 10.1007/s00220-012-1624-7.  Google Scholar

[7]

S. Cosentino and L. Flaminio, Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds, J. Mod. Dyn., 9 (2015), 305-353.  doi: 10.3934/jmd.2015.9.305.  Google Scholar

[8]

V. DelecroixP. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 1085-1110.   Google Scholar

[9]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[10]

G. Forni, Asymptotic behaviour of ergodic integrals of 'renormalizable' parabolic flows, in Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 317-326.  Google Scholar

[11]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.  Google Scholar

[12]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn., 8 (2014), 271-436.  doi: 10.3934/jmd.2014.8.271.  Google Scholar

[13]

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

[14]

F. GählerE. E. Kwan and G. R. Maloney, A computer search for planar substitution tilings with $n$-fold rotational symmetry, Discrete Comput. Geom., 53 (2015), 445-465.  doi: 10.1007/s00454-014-9659-5.  Google Scholar

[15]

F. Gähler and G. R. Maloney, Cohomology of one-dimensional mixed substitution tiling spaces, Topology Appl., 160 (2013), 703-719.  doi: 10.1016/j.topol.2013.01.019.  Google Scholar

[16]

C. Godrèche and J. M. Luck, Quasiperiodicity and randomness in tilings of the plane, J. Statist. Phys., 55 (1989), 1-28.  doi: 10.1007/BF01042590.  Google Scholar

[17]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[18]

A. Julien and J. Savinien, Tiling groupoids and Bratteli diagrams II: Structure of the orbit equivalence relation, Ann. Henri Poincaré, 13 (2012), 297-332.  doi: 10.1007/s00023-011-0121-4.  Google Scholar

[19]

J. Kellendonk, Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys., 7 (1995), 1133-1180.  doi: 10.1142/S0129055X95000426.  Google Scholar

[20]

J. Kellendonk and I. F. Putnam, The Ruelle-Sullivan map for actions of $\Bbb R^n$, Math. Ann., 334 (2006), 693-711.  doi: 10.1007/s00208-005-0728-1.  Google Scholar

[21]

K. Lindsey and R. Treviño, Infinite type flat surface models of ergodic systems, Discrete Contin. Dyn. Syst., 36 (2016), 5509-5553.  doi: 10.3934/dcds.2016043.  Google Scholar

[22]

D. Rust, An uncountable set of tiling spaces with distinct cohomology, Topology and its Applications, 205 (2016), 58-81.  doi: 10.1016/j.topol.2016.01.020.  Google Scholar

[23]

D. Rust and T. Spindeler, Dynamical systems arising from random substitutions, Indag. Math. (N.S.), 29 (2018), 1131-1155.  doi: 10.1016/j.indag.2018.05.013.  Google Scholar

[24]

L. Sadun, Pattern-equivariant cohomology with integer coefficients, Ergodic Theory Dynam. Systems, 27 (2007), 1991-1998.  doi: 10.1017/S0143385707000259.  Google Scholar

[25]

L. Sadun, Exact regularity and the cohomology of tiling spaces, Ergodic Theory Dynam. Systems, 31 (2011), 1819-1834.  doi: 10.1017/S0143385710000611.  Google Scholar

[26]

S. Schmieding and R. Treviño, Self affine Delone sets and deviation phenomena, Comm. Math. Phys., 357 (2018), 1071-1112.  doi: 10.1007/s00220-017-3011-x.  Google Scholar

[27]

S. Schmieding and R. Treviño, Traces of random operators associated with self-affine delone sets and Shubin's formula, Ann. Henri Poincaré, 19 (2018), 2575-2597.  doi: 10.1007/s00023-018-0700-8.  Google Scholar

[28]

Y. Solomon, A simple condition for bounded displacement, J. Math. Anal. Appl., 414 (2014), 134-148.  doi: 10.1016/j.jmaa.2013.12.050.  Google Scholar

[29]

R. Treviño, Flat surfaces, Bratteli diagrams and unique ergodicity à la Masur, Israel J. Math., 225 (2018), 35-70.  doi: 10.1007/s11856-018-1636-x.  Google Scholar

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