# American Institute of Mathematical Sciences

November  2014, 34(11): 4855-4874. doi: 10.3934/dcds.2014.34.4855

## Substitutions, tiling dynamical systems and minimal self-joinings

 1 Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 7610001, Israel

Received  September 2013 Revised  March 2014 Published  May 2014

We investigate substitution subshifts and tiling dynamical systems arising from the substitutions (1) $\theta: 0 \rightarrow 001, 1 \rightarrow 11001$ and (2) $\eta: 0 \rightarrow 001, 1 \rightarrow 11100$. We show that the substitution subshifts arising from $\theta$ and $\eta$ have minimal self-joinings and are mildly mixing. We also give a criterion for 1-dimensional tiling systems arising from $\theta$ or $\eta$ to have minimal self-joinings. We apply this to obtain examples of mildly mixing 1-dimensional tiling systems.
Citation: Younghwan Son. Substitutions, tiling dynamical systems and minimal self-joinings. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4855-4874. doi: 10.3934/dcds.2014.34.4855
##### References:
 [1] D. Berend and C. Radin, Are there chaotic tilings? Comm. Math. Phys., 152 (1993), 215-219. doi: 10.1007/BF02098297. [2] A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces, Ergodic Theory Dynamical Systems, 23 (2003), 1043-1057. doi: 10.1017/S0143385702001633. [3] F. M. Dekking and M. Keane, Mixing properties of substitutions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 42 (1978), 23-33. doi: 10.1007/BF00534205. [4] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981. [5] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [6] E. Glasner, B. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math., 78 (1992), 131-142. doi: 10.1007/BF02801575. [7] K. Jacobs and M. Keane, $0-1$ Sequences of Toeplitz type, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 13 (1969), 123-131. doi: 10.1007/BF00537017. [8] A. del Junco and K. Park, An example of a measure-preserving flow with minimal self-joinings, J. d'Analyse Math., 42 (1982/83), 199-209. doi: 10.1007/BF02786879. [9] A. del Junco, M. Rahe and L. Swanson, Chacon's automorphism has minimal self joinins, J. d'Analyse Math., 37 (1980), 276-284. doi: 10.1007/BF02797688. [10] A. del Junco and D. J. Rudolph, A rank one, rigid, simple, prime map, Ergodic Theory and Dynamical Systems, 7 (1987), 229-247. doi: 10.1017/S0143385700003977. [11] S. Kakutani, Strictly ergodic symbolic dynamical systems, in Proceedings of 6th Berkeley Symposium on Mathematical Statistics and Probability. (eds. L. M. LeCam, J. Neyman, and E. L. Scott) University of California Press, Berkeley, (1972), 319-326. [12] A. B. Katok, Ya. G. Sinai and A. M. Stepin, Theory of dynamical systems and general transformation groups with invariant measure, Mathematical analysis, 13 (1975), 129-262. doi: 10.1007/BF01223133. [13] J. King, The commutant is the weak closure of the powers, for rank one transformations, Ergodic Theory and Dynamical Systems, 6 (1986), 363-384. doi: 10.1017/S0143385700003552. [14] J. King, Ergodic properties where order 4 implies infinite order, Israel J. Math., 80 (1992), 65-86. doi: 10.1007/BF02808154. [15] J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58, (1952), 116-136. doi: 10.1090/S0002-9904-1952-09580-X. [16] K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1983. [17] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, $2^{nd}$ edition. Lecture Notes in Mathematics, 1294. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6. [18] E. A. Robinson, Symbolic dynamics and tilings of $\mathbbmathbb{R}^{d}$, Symbolic dynamics and its applications, in Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, (2004), 81-119. doi: 10.1090/psapm/060/2078847. [19] D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. d'Analyse Math., 35 (1979), 97-122. doi: 10.1007/BF02791063. [20] D. J. Rudolph, Fundamentals of measurable dynamics - Ergodic theory on Lebesque spaces, Oxford University Press, 1990. [21] V. V. Ryzhikov, Self-joinings of commutative actions with an invariant measure, Mat. Zametki, 83 (2008), 723-726. doi: 10.1134/S0001434608050179. [22] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory and Dynamical Systems, 17 (1997), 695-738. doi: 10.1017/S0143385797084988.

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##### References:
 [1] D. Berend and C. Radin, Are there chaotic tilings? Comm. Math. Phys., 152 (1993), 215-219. doi: 10.1007/BF02098297. [2] A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces, Ergodic Theory Dynamical Systems, 23 (2003), 1043-1057. doi: 10.1017/S0143385702001633. [3] F. M. Dekking and M. Keane, Mixing properties of substitutions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 42 (1978), 23-33. doi: 10.1007/BF00534205. [4] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981. [5] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [6] E. Glasner, B. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math., 78 (1992), 131-142. doi: 10.1007/BF02801575. [7] K. Jacobs and M. Keane, $0-1$ Sequences of Toeplitz type, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 13 (1969), 123-131. doi: 10.1007/BF00537017. [8] A. del Junco and K. Park, An example of a measure-preserving flow with minimal self-joinings, J. d'Analyse Math., 42 (1982/83), 199-209. doi: 10.1007/BF02786879. [9] A. del Junco, M. Rahe and L. Swanson, Chacon's automorphism has minimal self joinins, J. d'Analyse Math., 37 (1980), 276-284. doi: 10.1007/BF02797688. [10] A. del Junco and D. J. Rudolph, A rank one, rigid, simple, prime map, Ergodic Theory and Dynamical Systems, 7 (1987), 229-247. doi: 10.1017/S0143385700003977. [11] S. Kakutani, Strictly ergodic symbolic dynamical systems, in Proceedings of 6th Berkeley Symposium on Mathematical Statistics and Probability. (eds. L. M. LeCam, J. Neyman, and E. L. Scott) University of California Press, Berkeley, (1972), 319-326. [12] A. B. Katok, Ya. G. Sinai and A. M. Stepin, Theory of dynamical systems and general transformation groups with invariant measure, Mathematical analysis, 13 (1975), 129-262. doi: 10.1007/BF01223133. [13] J. King, The commutant is the weak closure of the powers, for rank one transformations, Ergodic Theory and Dynamical Systems, 6 (1986), 363-384. doi: 10.1017/S0143385700003552. [14] J. King, Ergodic properties where order 4 implies infinite order, Israel J. Math., 80 (1992), 65-86. doi: 10.1007/BF02808154. [15] J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58, (1952), 116-136. doi: 10.1090/S0002-9904-1952-09580-X. [16] K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1983. [17] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, $2^{nd}$ edition. Lecture Notes in Mathematics, 1294. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6. [18] E. A. Robinson, Symbolic dynamics and tilings of $\mathbbmathbb{R}^{d}$, Symbolic dynamics and its applications, in Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, (2004), 81-119. doi: 10.1090/psapm/060/2078847. [19] D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. d'Analyse Math., 35 (1979), 97-122. doi: 10.1007/BF02791063. [20] D. J. Rudolph, Fundamentals of measurable dynamics - Ergodic theory on Lebesque spaces, Oxford University Press, 1990. [21] V. V. Ryzhikov, Self-joinings of commutative actions with an invariant measure, Mat. Zametki, 83 (2008), 723-726. doi: 10.1134/S0001434608050179. [22] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory and Dynamical Systems, 17 (1997), 695-738. doi: 10.1017/S0143385797084988.
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