October  2013, 33(10): 4579-4594. doi: 10.3934/dcds.2013.33.4579

Statistical stability for multi-substitution tiling spaces

1. 

Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, Covilhã, 6200-001, Portugal, Portugal

Received  July 2012 Revised  January 2013 Published  April 2013

Given a finite set $\{S_1\dots,S_k \}$ of substitution maps acting on a certain finite number (up to translations) of tiles in $\mathbb{R}^d$, we consider the multi-substitution tiling space associated to each sequence $\bar a\in \{1,\ldots,k\}^{\mathbb{N}}$. The action by translations on such spaces gives rise to uniquely ergodic dynamical systems. In this paper we investigate the rate of convergence for ergodic limits of patches frequencies and prove that these limits vary continuously with $\bar a$.
Citation: Rui Pacheco, Helder Vilarinho. Statistical stability for multi-substitution tiling spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4579-4594. doi: 10.3934/dcds.2013.33.4579
References:
[1]

F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynamical Systems, 20 (2000), 1061-1078. doi: 10.1017/S0143385700000584.

[2]

S. Ferenczi, Rank and symbolic complexity subshift factors, Ergodic Theory Dynamical Systems, 16 (1996), 663-682. doi: 10.1017/S0143385700009032.

[3]

N. P. Frank, A primer of substitution tilings of the Euclidean plane, Expositiones Mathematicae, 26 (2008), 295-326. doi: 10.1016/j.exmath.2008.02.001.

[4]

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

[5]

F. Gähler and G. Maloney, Cohomology of one-dimensional mixed substitution tiling spaces, preprint, arXiv:1112.1475.

[6]

C. P. M. Geerse and A. Hof, Lattice gas models on self-similar aperiodic tilings, Rev. Math. Phys., 3 (1991), 163-221. doi: 10.1142/S0129055X91000072.

[7]

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

[8]

Grünbaum and G. C. Shephard, "Tilings and Patterns," Freeman, New York, 1986.

[9]

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

[10]

R. Pacheco and H. Vilarinho, Metrics on tiling spaces, local isomorphism and an application of Brown's lemma, preprint, arXiv:1202.4902. doi: 10.1007/s00605-013-0484-3.

[11]

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

[12]

E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbbmathbb{R}^{d}$, Proc. Sympos. Appl. Math. Amer. Math. Soc., 60 (2004), 81-119.

[13]

D. Ruelle, "Statistical Mechanics: Rigorous Results," W. A. Benjamin, Inc., New York - Amsterdam, 1969.

[14]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory and Dynamical Systems, 17 (1997), 695-738. Errata: Ergodic Theory and Dynamical Systems, 19 (1999), 1685. doi: 10.1017/S0143385797084988.

show all references

References:
[1]

F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynamical Systems, 20 (2000), 1061-1078. doi: 10.1017/S0143385700000584.

[2]

S. Ferenczi, Rank and symbolic complexity subshift factors, Ergodic Theory Dynamical Systems, 16 (1996), 663-682. doi: 10.1017/S0143385700009032.

[3]

N. P. Frank, A primer of substitution tilings of the Euclidean plane, Expositiones Mathematicae, 26 (2008), 295-326. doi: 10.1016/j.exmath.2008.02.001.

[4]

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

[5]

F. Gähler and G. Maloney, Cohomology of one-dimensional mixed substitution tiling spaces, preprint, arXiv:1112.1475.

[6]

C. P. M. Geerse and A. Hof, Lattice gas models on self-similar aperiodic tilings, Rev. Math. Phys., 3 (1991), 163-221. doi: 10.1142/S0129055X91000072.

[7]

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

[8]

Grünbaum and G. C. Shephard, "Tilings and Patterns," Freeman, New York, 1986.

[9]

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

[10]

R. Pacheco and H. Vilarinho, Metrics on tiling spaces, local isomorphism and an application of Brown's lemma, preprint, arXiv:1202.4902. doi: 10.1007/s00605-013-0484-3.

[11]

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

[12]

E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbbmathbb{R}^{d}$, Proc. Sympos. Appl. Math. Amer. Math. Soc., 60 (2004), 81-119.

[13]

D. Ruelle, "Statistical Mechanics: Rigorous Results," W. A. Benjamin, Inc., New York - Amsterdam, 1969.

[14]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory and Dynamical Systems, 17 (1997), 695-738. Errata: Ergodic Theory and Dynamical Systems, 19 (1999), 1685. doi: 10.1017/S0143385797084988.

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