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  August 2021 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

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