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

Statistical stability for multi-substitution tiling spaces

Rui Pacheco 1, and  Helder Vilarinho 1,

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 $\bar a$.
Keywords: tiling spaces, invariant measures, dynamical systems, Multi-substitutions, statistical stability..
Mathematics Subject Classification: Primary: 37A15, 37A25, 52C2.
Citation: Rui Pacheco, Helder Vilarinho. Statistical stability for multi-substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4579-4594. doi: 10.3934/dcds.2013.33.4579
References:
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W. H. Gottschalk, Orbit-closure decomposition and almost periodic properties,, Bull. Amer. Math. Soc., 50 (1944), 915. doi: 10.1090/S0002-9904-1944-08262-1. Google Scholar

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Grünbaum and G. C. Shephard, "Tilings and Patterns,", Freeman, (1986). Google Scholar

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

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

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E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbbR^d$,, Proc. Sympos. Appl. Math. Amer. Math. Soc., 60 (2004), 81. Google Scholar

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B. Solomyak, Dynamics of self-similar tilings,, Ergodic Theory and Dynamical Systems, 17 (1997), 695. doi: 10.1017/S0143385797084988. Google Scholar

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. doi: 10.1017/S0143385700000584. Google Scholar

[2]

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

[3]

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

[4]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $\mathbbR^d$,, preprint, (). Google Scholar

[5]

F. Gähler and G. Maloney, Cohomology of one-dimensional mixed substitution tiling spaces,, preprint, (). Google Scholar

[6]

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

[7]

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

[8]

Grünbaum and G. C. Shephard, "Tilings and Patterns,", Freeman, (1986). Google Scholar

[9]

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

[10]

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

[11]

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

[12]

E. A. Robinson, Jr., Symbolic dynamics and tilings of $\mathbbR^d$,, Proc. Sympos. Appl. Math. Amer. Math. Soc., 60 (2004), 81. Google Scholar

[13]

D. Ruelle, "Statistical Mechanics: Rigorous Results,", W. A. Benjamin, (1969). Google Scholar

[14]

B. Solomyak, Dynamics of self-similar tilings,, Ergodic Theory and Dynamical Systems, 17 (1997), 695. doi: 10.1017/S0143385797084988. Google Scholar

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