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

Substitutions, tiling dynamical systems and minimal self-joinings

Younghwan Son 1,

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.
Keywords: Minimal self-joinings, tiling dynamics, weak mixing., mild mixing, substitution dynamics.
Mathematics Subject Classification: Primary: 37A25, 37B10; Secondary: 52C2.
Citation: Younghwan Son. Substitutions, tiling dynamical systems and minimal self-joinings. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

F. M. Dekking and M. Keane, Mixing properties of substitutions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 42 (1978), 23-33. doi: 10.1007/BF00534205.  Google Scholar

[4]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[5]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. del Junco and K. Park, An example of a measure-preserving flow with minimal self-joinings,, J. d'Analyse Math., 42 (): 199.  doi: 10.1007/BF02786879.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. King, Ergodic properties where order 4 implies infinite order, Israel J. Math., 80 (1992), 65-86. doi: 10.1007/BF02808154.  Google Scholar

[15]

J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58, (1952), 116-136. doi: 10.1090/S0002-9904-1952-09580-X.  Google Scholar

[16]

K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1983.  Google Scholar

[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.  Google Scholar

[18]

E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, 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.  Google Scholar

[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.  Google Scholar

[20]

D. J. Rudolph, Fundamentals of measurable dynamics - Ergodic theory on Lebesque spaces, Oxford University Press, 1990.  Google Scholar

[21]

V. V. Ryzhikov, Self-joinings of commutative actions with an invariant measure, Mat. Zametki, 83 (2008), 723-726. doi: 10.1134/S0001434608050179.  Google Scholar

[22]

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

show all references

References:
[1]

D. Berend and C. Radin, Are there chaotic tilings? Comm. Math. Phys., 152 (1993), 215-219. doi: 10.1007/BF02098297.  Google Scholar

[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.  Google Scholar

[3]

F. M. Dekking and M. Keane, Mixing properties of substitutions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 42 (1978), 23-33. doi: 10.1007/BF00534205.  Google Scholar

[4]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[5]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. del Junco and K. Park, An example of a measure-preserving flow with minimal self-joinings,, J. d'Analyse Math., 42 (): 199.  doi: 10.1007/BF02786879.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. King, Ergodic properties where order 4 implies infinite order, Israel J. Math., 80 (1992), 65-86. doi: 10.1007/BF02808154.  Google Scholar

[15]

J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc., 58, (1952), 116-136. doi: 10.1090/S0002-9904-1952-09580-X.  Google Scholar

[16]

K. Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1983.  Google Scholar

[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.  Google Scholar

[18]

E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, 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.  Google Scholar

[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.  Google Scholar

[20]

D. J. Rudolph, Fundamentals of measurable dynamics - Ergodic theory on Lebesque spaces, Oxford University Press, 1990.  Google Scholar

[21]

V. V. Ryzhikov, Self-joinings of commutative actions with an invariant measure, Mat. Zametki, 83 (2008), 723-726. doi: 10.1134/S0001434608050179.  Google Scholar

[22]

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

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