# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. 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

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
 [1] Younghwan Son. Substitutions, tiling dynamical systems and minimal self-joinings. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4855-4874. doi: 10.3934/dcds.2014.34.4855 [2] Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285 [3] Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124 [4] Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211 [5] Siniša Slijepčević. Stability of invariant measures. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345 [6] Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761 [7] Giovanni Panti. Dynamical properties of logical substitutions. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 237-258. doi: 10.3934/dcds.2006.15.237 [8] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [9] Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639 [10] Victor Magron, Marcelo Forets, Didier Henrion. Semidefinite approximations of invariant measures for polynomial systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6745-6770. doi: 10.3934/dcdsb.2019165 [11] P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785 [12] Alfredo Marzocchi, Sara Zandonella Necca. Attractors for dynamical systems in topological spaces. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 585-597. doi: 10.3934/dcds.2002.8.585 [13] Xiaoming Wang. Numerical algorithms for stationary statistical properties of dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4599-4618. doi: 10.3934/dcds.2016.36.4599 [14] Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161 [15] Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211 [16] Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323 [17] Marcy Barge, Sonja Štimac, R. F. Williams. Pure discrete spectrum in substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 579-597. doi: 10.3934/dcds.2013.33.579 [18] Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445 [19] Oliver Jenkinson. Optimization and majorization of invariant measures. Electronic Research Announcements, 2007, 13: 1-12. [20] Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135

2018 Impact Factor: 1.143