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March  2018, 38(3): 1033-1062. doi: 10.3934/dcds.2018044

Dynamics in dimension zero A survey

Tomasz Downarowicz 1, and  Olena Karpel 2,

1. 

Faculty of Mathematics and Faculty of Fundamental Problems of Technology, Wroclaw University of Technology, Wroclaw, Poland
2. 

Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine, Current address: Department of Dynamical Systems, Institute of Mathematics of Polish Academy of Sciences, Wroclaw, Poland

Received  November 2016 Revised  October 2017 Published  December 2017

Fund Project: The second author is supported by the NCN (National Science Center, Poland) Grant 2013/08/A/ST1/00275.

The goal of this paper is to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as array representation, inverse limit representation, or Bratteli-Vershik representation. We describe how one can switch from one representation to another. We also briefly review some more recent related notions: symbolic extensions, symbolic extensions with an embedding, and uniform generators. We devote a great deal of attention to marker techniques and we use them to prove two types of results: one concerning entropy and vertical data compression, and another, about the existence of isomorphic minimal models for aperiodic systems. We also introduce so-called decisiveness of Bratteli-Vershik systems and give for it a sufficient condition.

Keywords: Zero-dimensional dynamical systems, symbolic extension, Bratteli-Vershik systems.
Mathematics Subject Classification: Primary: 37B05, 37B10; Secondary: 37B40.
Citation: Tomasz Downarowicz, Olena Karpel. Dynamics in dimension zero A survey. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1033-1062. doi: 10.3934/dcds.2018044
References:
[1]

M. Adamska, S. Bezuglyi, O. Karpel and J. Kwiatkowski, Subdiagrams and invariant measures on Bratteli diagrams, Ergodic Theory Dynam. Syst., 37 (2017), 2417-2452.  doi: 10.1017/etds.2016.8.  Google Scholar

[2]

V. Bergelson, Ergodic Ramsey theory - an update, Ergodic Theory of $ \mathbb{Z}^d$-actions, London Math. Soc. Lecture Note Series, 228 (1996), 1-61.  Google Scholar

[3]

V. Bergelson, Minimal idempotents and ergodic Ramsey theory, Topics in dynamics and ergodic theory, London Math. Soc. Lecture Note Series 310 (2003), Cambridge Univ. Press, Cambridge, 8-39  Google Scholar

[4]

S. Bezuglyi, A. H. Dooley and J. Kwiatkowski, Topologies on the group of Borel automorphisms of a standard Borel space, Topol. Methods Nonlinear Anal., 27 (2006), 333-385.   Google Scholar

[5]

S. Bezuglyi and O. Karpel, Bratteli diagrams: Structure, measures, dynamics, Contemp. Math., 669 (2016), 1-36.   Google Scholar

[6]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Syst., 30 (2010), 973-1007.  doi: 10.1017/S0143385709000443.  Google Scholar

[7]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Finite rank Bratteli diagrams: Structure of invariant measures, Trans. Amer. Math. Soc., 365 (2013), 2637-2679.   Google Scholar

[8]

S. Bezuglyi, J. Kwiatkowski and R. Yassawi, Perfect orderings on finite rank Bratteli diagrams, Canad. J. Math., 66 (2014), 57-101.  doi: 10.4153/CJM-2013-041-6.  Google Scholar

[9]

S. Bezuglyi and R. Yassawi, Orders that yield homeomorphisms on Bratteli diagrams, Dynamical Systems, 32 (2017), 249-282.  doi: 10.1080/14689367.2016.1197888.  Google Scholar

[10]

M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Sys., 3 (1983), 541-557.   Google Scholar

[11]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Inventiones Math., 156 (2004), 119-161.  doi: 10.1007/s00222-003-0335-2.  Google Scholar

[12]

M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Mathematicum, 14 (2002), 713-757.   Google Scholar

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O. Bratteli, Inductive limits of finite-dimensional $ C^{*}$-algebras, Trans. Amer. Math. Soc., 171 (1972), 195-234.   Google Scholar

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D. Burguet, Embedding asymptotically expansive systems, Monatsh Math., 184 (2017), 21-49.  doi: 10.1007/s00605-017-1079-1.  Google Scholar

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D. Burguet and T. Downarowicz, Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits, preprint, arXiv:1705.08829. Google Scholar

[16]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.  doi: 10.1007/BF02773637.  Google Scholar

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P. Collet and J. -P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Modern Birkhäuser Classics, Birkhäuser Basel, 2009.  Google Scholar

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P. Domínguez, A. Hernández and G. Sienra, Totally disconnected Julia set for different classes of meromorphic functions, Conformal Geometry and Dynamics, An Electronic Journal of the Amer. Math. Soc. 18(2014), 1-7. doi: 10.1090/S1088-4173-2014-00258-6.  Google Scholar

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T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Isr. J. Math., 74 (1991), 241-256.  doi: 10.1007/BF02775789.  Google Scholar

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T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.  doi: 10.1007/BF02787825.  Google Scholar

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T. Downarowicz, Minimal models for noninvertible and not uniquely ergodic systems, Israel J. of Math., 156 (2006), 93-110.  doi: 10.1007/BF02773826.  Google Scholar

[22]

T. Downarowicz, Faces of simplexes of invariant measures, Israel J. of Math., 165 (2008), 189-210.  doi: 10.1007/s11856-008-1009-y.  Google Scholar

[23]

T. Downarowicz, Entropy in Dynamical Systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011.  Google Scholar

[24]

T. Downarowicz and D. Huczek, Faithful zero-dimensional principal extensions, Studia Math., 212 (2012), 1-19.  doi: 10.4064/sm212-1-1.  Google Scholar

[25]

T. Downarowicz and A. Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergod. Th. and Dynam. Sys., 28 (2008), 739-747.   Google Scholar

[26]

T. Downarowicz and J. Serafin, Possible entropy functions, Israel J. Math., 135 (2003), 221-250.  doi: 10.1007/BF02776059.  Google Scholar

[27]

F. Durand, Combinatorics on Bratteli diagrams and dynamical systems, Combinatorics, Automata and Number Theory, V. Berthé, M. Rigo (Eds). Encyclopedia of Mathematics and its Applications, Cambridge University Press, 135 (2010), 324-372.  Google Scholar

[28]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[29]

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

[30]

T. Giordano, H. Matui, I. Putnam and C. Skau, The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, 28 (2008), 1509-1531.   Google Scholar

[31]

T. Giordano, H. Matui, I. Putnam and C. Skau, Orbit equivalence for Cantor minimal $ \mathbb Z^d$-systems, Invent. Math., 179 (2010), 119-158.  doi: 10.1007/s00222-009-0213-7.  Google Scholar

[32]

T. Giordano, I. Putnam and C. Skau, Topological orbit equivalence and $ C^*$-crossed}products, J. Reine Angew. Math., 469 (1995), 51-111.   Google Scholar

[33]

T. Giordano, I. Putnam and C. Skau, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory and Dynam. Systems, 24 (2004), 441-475.  doi: 10.1017/S014338570300066X.  Google Scholar

[34]

E. Glasner and B. Weiss, Weak orbit equivalence of Cantor minimal systems, Internat. J. Math., 6 (1995), 559-579.  doi: 10.1142/S0129167X95000213.  Google Scholar

[35]

Y. Gutman, Mean dimension and Jaworski-type theorems, Proc. London Math. Soc., 111 (2015), 831-850.  doi: 10.1112/plms/pdv043.  Google Scholar

[36]

Y. Gutman, Embedding topological dynamical systems with periodic points in cubical shifts, Ergod. Th. and Dynam. Sys., 37 (2017), 512-538, https://doi.org/10.1017/etds.2015.40 doi: 10.1017/etds.2015.40.  Google Scholar

[37]

J. Hadamard, Les surfaces à courbures opposées et leur lignes geodesiques, Journal de Mathématiques Pures et Appliqués, 4 (1898), 27-73.   Google Scholar

[38]

T. Hamachi, M. Keane and H. Yuasa, Universally measure-preserving homeomorphisms of Cantor minimal systems, J. Anal. Math., 113 (2011), 1-51.  doi: 10.1007/s11854-011-0001-3.  Google Scholar

[39]

G. Hedlund, Endomorphisms and automorphisms of the shift dynamical systems, Math. Syst. Theory, 3 (1969), 320-375.  doi: 10.1007/BF01691062.  Google Scholar

[40]

R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Int. J. Math., 3 (1992), 827-864.  doi: 10.1142/S0129167X92000382.  Google Scholar

[41]

R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1970), 717-729.   Google Scholar

[42]

W. Krieger, On unique ergodicity, L. Le Cam, J. Neyman and E.L. Scott (eds), Proc. VIth Berkeley Symp. on Math. Statistics and Probability, 2 (1972), 327-346.   Google Scholar

[43]

W. Krieger, On the subsystems of topological Markov chains, Ergodic Theory Dynam. Sys., 2 (1982), 195-202.   Google Scholar

[44]

J. Kulesza, Zero-dimensional covers of finite-dimensional dynamical systems, Erg. Th. & Dyn. Syst., 15 (1995), 939-950.  doi: 10.1017/S014338570000969X.  Google Scholar

[45]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995.  Google Scholar

[46]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Publ. Math. I.H.E.S., 89 (1999), 227-262.   Google Scholar

[47]

E. Lindenstrauss and B. Weiss, Mean Topological Dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.  Google Scholar

[48]

K. Medynets, Cantor aperiodic systems and Bratteli diagrams, C. R., Math., Acad. Sci. Paris, 342 (2006), 43-46.  doi: 10.1016/j.crma.2005.10.024.  Google Scholar

[49]

J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems (College Park, MD, 1986-87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988.  Google Scholar

[50]

M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.  doi: 10.4064/sm-55-2-175-200.  Google Scholar

[51]

M. Morse and G. Hedlund, Symbolic dynamics, Amer. J. Math., 60 (1938), 815-866.  doi: 10.2307/2371264.  Google Scholar

[52]

I. Putnam, Orbit equivalence of Cantor minimal systems: A survey and a new proof, Expo. Math., 28 (2010), 101-131.  doi: 10.1016/j.exmath.2009.06.002.  Google Scholar

[53]

F. Ramsey, On a Problem of Formal Logic, Proc. London Math. Soc., 30 (1929), 264-286.   Google Scholar

[54]

A. Rosenthal, Strictly ergodic models for non-invertible transformations, Isr. J. Math., 64 (1988), 57-72.  doi: 10.1007/BF02767370.  Google Scholar

[55]

J. Serafin, A faithful symbolic extension, Communications on Pure and Applied Analysis, 11 (2012), 1051-1062.   Google Scholar

[56]

C. Skau, Ordered $K$-theory and minimal symbolic dynamical systems. Dedicated to the memory of Anzelm Iwanik, Colloq. Math., 84/85 (2000), 203-227.   Google Scholar

[57]

A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators, Dokl. Acad. Nauk SSSR, 259 (1981), 526-529.   Google Scholar

[58]

A. M. Vershik, A theorem on Markov periodic approximation in ergodic theory, Zap. Nauchn. Sem. LOMI, 115 (1982), 72-82.   Google Scholar

show all references

References:
[1]

M. Adamska, S. Bezuglyi, O. Karpel and J. Kwiatkowski, Subdiagrams and invariant measures on Bratteli diagrams, Ergodic Theory Dynam. Syst., 37 (2017), 2417-2452.  doi: 10.1017/etds.2016.8.  Google Scholar

[2]

V. Bergelson, Ergodic Ramsey theory - an update, Ergodic Theory of $ \mathbb{Z}^d$-actions, London Math. Soc. Lecture Note Series, 228 (1996), 1-61.  Google Scholar

[3]

V. Bergelson, Minimal idempotents and ergodic Ramsey theory, Topics in dynamics and ergodic theory, London Math. Soc. Lecture Note Series 310 (2003), Cambridge Univ. Press, Cambridge, 8-39  Google Scholar

[4]

S. Bezuglyi, A. H. Dooley and J. Kwiatkowski, Topologies on the group of Borel automorphisms of a standard Borel space, Topol. Methods Nonlinear Anal., 27 (2006), 333-385.   Google Scholar

[5]

S. Bezuglyi and O. Karpel, Bratteli diagrams: Structure, measures, dynamics, Contemp. Math., 669 (2016), 1-36.   Google Scholar

[6]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Syst., 30 (2010), 973-1007.  doi: 10.1017/S0143385709000443.  Google Scholar

[7]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Finite rank Bratteli diagrams: Structure of invariant measures, Trans. Amer. Math. Soc., 365 (2013), 2637-2679.   Google Scholar

[8]

S. Bezuglyi, J. Kwiatkowski and R. Yassawi, Perfect orderings on finite rank Bratteli diagrams, Canad. J. Math., 66 (2014), 57-101.  doi: 10.4153/CJM-2013-041-6.  Google Scholar

[9]

S. Bezuglyi and R. Yassawi, Orders that yield homeomorphisms on Bratteli diagrams, Dynamical Systems, 32 (2017), 249-282.  doi: 10.1080/14689367.2016.1197888.  Google Scholar

[10]

M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Sys., 3 (1983), 541-557.   Google Scholar

[11]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Inventiones Math., 156 (2004), 119-161.  doi: 10.1007/s00222-003-0335-2.  Google Scholar

[12]

M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Mathematicum, 14 (2002), 713-757.   Google Scholar

[13]

O. Bratteli, Inductive limits of finite-dimensional $ C^{*}$-algebras, Trans. Amer. Math. Soc., 171 (1972), 195-234.   Google Scholar

[14]

D. Burguet, Embedding asymptotically expansive systems, Monatsh Math., 184 (2017), 21-49.  doi: 10.1007/s00605-017-1079-1.  Google Scholar

[15]

D. Burguet and T. Downarowicz, Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits, preprint, arXiv:1705.08829. Google Scholar

[16]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.  doi: 10.1007/BF02773637.  Google Scholar

[17]

P. Collet and J. -P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Modern Birkhäuser Classics, Birkhäuser Basel, 2009.  Google Scholar

[18]

P. Domínguez, A. Hernández and G. Sienra, Totally disconnected Julia set for different classes of meromorphic functions, Conformal Geometry and Dynamics, An Electronic Journal of the Amer. Math. Soc. 18(2014), 1-7. doi: 10.1090/S1088-4173-2014-00258-6.  Google Scholar

[19]

T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Isr. J. Math., 74 (1991), 241-256.  doi: 10.1007/BF02775789.  Google Scholar

[20]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.  doi: 10.1007/BF02787825.  Google Scholar

[21]

T. Downarowicz, Minimal models for noninvertible and not uniquely ergodic systems, Israel J. of Math., 156 (2006), 93-110.  doi: 10.1007/BF02773826.  Google Scholar

[22]

T. Downarowicz, Faces of simplexes of invariant measures, Israel J. of Math., 165 (2008), 189-210.  doi: 10.1007/s11856-008-1009-y.  Google Scholar

[23]

T. Downarowicz, Entropy in Dynamical Systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011.  Google Scholar

[24]

T. Downarowicz and D. Huczek, Faithful zero-dimensional principal extensions, Studia Math., 212 (2012), 1-19.  doi: 10.4064/sm212-1-1.  Google Scholar

[25]

T. Downarowicz and A. Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergod. Th. and Dynam. Sys., 28 (2008), 739-747.   Google Scholar

[26]

T. Downarowicz and J. Serafin, Possible entropy functions, Israel J. Math., 135 (2003), 221-250.  doi: 10.1007/BF02776059.  Google Scholar

[27]

F. Durand, Combinatorics on Bratteli diagrams and dynamical systems, Combinatorics, Automata and Number Theory, V. Berthé, M. Rigo (Eds). Encyclopedia of Mathematics and its Applications, Cambridge University Press, 135 (2010), 324-372.  Google Scholar

[28]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[29]

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

[30]

T. Giordano, H. Matui, I. Putnam and C. Skau, The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, 28 (2008), 1509-1531.   Google Scholar

[31]

T. Giordano, H. Matui, I. Putnam and C. Skau, Orbit equivalence for Cantor minimal $ \mathbb Z^d$-systems, Invent. Math., 179 (2010), 119-158.  doi: 10.1007/s00222-009-0213-7.  Google Scholar

[32]

T. Giordano, I. Putnam and C. Skau, Topological orbit equivalence and $ C^*$-crossed}products, J. Reine Angew. Math., 469 (1995), 51-111.   Google Scholar

[33]

T. Giordano, I. Putnam and C. Skau, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory and Dynam. Systems, 24 (2004), 441-475.  doi: 10.1017/S014338570300066X.  Google Scholar

[34]

E. Glasner and B. Weiss, Weak orbit equivalence of Cantor minimal systems, Internat. J. Math., 6 (1995), 559-579.  doi: 10.1142/S0129167X95000213.  Google Scholar

[35]

Y. Gutman, Mean dimension and Jaworski-type theorems, Proc. London Math. Soc., 111 (2015), 831-850.  doi: 10.1112/plms/pdv043.  Google Scholar

[36]

Y. Gutman, Embedding topological dynamical systems with periodic points in cubical shifts, Ergod. Th. and Dynam. Sys., 37 (2017), 512-538, https://doi.org/10.1017/etds.2015.40 doi: 10.1017/etds.2015.40.  Google Scholar

[37]

J. Hadamard, Les surfaces à courbures opposées et leur lignes geodesiques, Journal de Mathématiques Pures et Appliqués, 4 (1898), 27-73.   Google Scholar

[38]

T. Hamachi, M. Keane and H. Yuasa, Universally measure-preserving homeomorphisms of Cantor minimal systems, J. Anal. Math., 113 (2011), 1-51.  doi: 10.1007/s11854-011-0001-3.  Google Scholar

[39]

G. Hedlund, Endomorphisms and automorphisms of the shift dynamical systems, Math. Syst. Theory, 3 (1969), 320-375.  doi: 10.1007/BF01691062.  Google Scholar

[40]

R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Int. J. Math., 3 (1992), 827-864.  doi: 10.1142/S0129167X92000382.  Google Scholar

[41]

R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1970), 717-729.   Google Scholar

[42]

W. Krieger, On unique ergodicity, L. Le Cam, J. Neyman and E.L. Scott (eds), Proc. VIth Berkeley Symp. on Math. Statistics and Probability, 2 (1972), 327-346.   Google Scholar

[43]

W. Krieger, On the subsystems of topological Markov chains, Ergodic Theory Dynam. Sys., 2 (1982), 195-202.   Google Scholar

[44]

J. Kulesza, Zero-dimensional covers of finite-dimensional dynamical systems, Erg. Th. & Dyn. Syst., 15 (1995), 939-950.  doi: 10.1017/S014338570000969X.  Google Scholar

[45]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995.  Google Scholar

[46]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Publ. Math. I.H.E.S., 89 (1999), 227-262.   Google Scholar

[47]

E. Lindenstrauss and B. Weiss, Mean Topological Dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.  Google Scholar

[48]

K. Medynets, Cantor aperiodic systems and Bratteli diagrams, C. R., Math., Acad. Sci. Paris, 342 (2006), 43-46.  doi: 10.1016/j.crma.2005.10.024.  Google Scholar

[49]

J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems (College Park, MD, 1986-87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988.  Google Scholar

[50]

M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.  doi: 10.4064/sm-55-2-175-200.  Google Scholar

[51]

M. Morse and G. Hedlund, Symbolic dynamics, Amer. J. Math., 60 (1938), 815-866.  doi: 10.2307/2371264.  Google Scholar

[52]

I. Putnam, Orbit equivalence of Cantor minimal systems: A survey and a new proof, Expo. Math., 28 (2010), 101-131.  doi: 10.1016/j.exmath.2009.06.002.  Google Scholar

[53]

F. Ramsey, On a Problem of Formal Logic, Proc. London Math. Soc., 30 (1929), 264-286.   Google Scholar

[54]

A. Rosenthal, Strictly ergodic models for non-invertible transformations, Isr. J. Math., 64 (1988), 57-72.  doi: 10.1007/BF02767370.  Google Scholar

[55]

J. Serafin, A faithful symbolic extension, Communications on Pure and Applied Analysis, 11 (2012), 1051-1062.   Google Scholar

[56]

C. Skau, Ordered $K$-theory and minimal symbolic dynamical systems. Dedicated to the memory of Anzelm Iwanik, Colloq. Math., 84/85 (2000), 203-227.   Google Scholar

[57]

A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators, Dokl. Acad. Nauk SSSR, 259 (1981), 526-529.   Google Scholar

[58]

A. M. Vershik, A theorem on Markov periodic approximation in ergodic theory, Zap. Nauchn. Sem. LOMI, 115 (1982), 72-82.   Google Scholar

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