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

Dynamics in dimension zero A survey

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.

Citation: Tomasz Downarowicz, Olena Karpel. Dynamics in dimension zero A survey. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1033-1062. doi: 10.3934/dcds.2018044
References:
[1]

M. AdamskaS. BezuglyiO. 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.

[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.

[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

[4]

S. BezuglyiA. 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. 

[5]

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

[6]

S. BezuglyiJ. KwiatkowskiK. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Syst., 30 (2010), 973-1007.  doi: 10.1017/S0143385709000443.

[7]

S. BezuglyiJ. KwiatkowskiK. Medynets and B. Solomyak, Finite rank Bratteli diagrams: Structure of invariant measures, Trans. Amer. Math. Soc., 365 (2013), 2637-2679. 

[8]

S. BezuglyiJ. 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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

T. GiordanoH. MatuiI. Putnam and C. Skau, The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, 28 (2008), 1509-1531. 

[31]

T. GiordanoH. MatuiI. 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.

[32]

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

[33]

T. GiordanoI. 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.

[34]

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

[35]

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

[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.

[37]

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

[38]

T. HamachiM. 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.

[39]

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

[40]

R. H. HermanI. 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.

[41]

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

[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. 

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[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.

[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.

[50]

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

[51]

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

[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.

[53]

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

[54]

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

[55]

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

[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. 

[57]

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

[58]

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

show all references

References:
[1]

M. AdamskaS. BezuglyiO. 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.

[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.

[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

[4]

S. BezuglyiA. 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. 

[5]

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

[6]

S. BezuglyiJ. KwiatkowskiK. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Syst., 30 (2010), 973-1007.  doi: 10.1017/S0143385709000443.

[7]

S. BezuglyiJ. KwiatkowskiK. Medynets and B. Solomyak, Finite rank Bratteli diagrams: Structure of invariant measures, Trans. Amer. Math. Soc., 365 (2013), 2637-2679. 

[8]

S. BezuglyiJ. 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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

T. GiordanoH. MatuiI. Putnam and C. Skau, The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, 28 (2008), 1509-1531. 

[31]

T. GiordanoH. MatuiI. 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.

[32]

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

[33]

T. GiordanoI. 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.

[34]

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

[35]

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

[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.

[37]

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

[38]

T. HamachiM. 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.

[39]

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

[40]

R. H. HermanI. 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.

[41]

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

[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. 

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[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.

[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.

[50]

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

[51]

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

[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.

[53]

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

[54]

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

[55]

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

[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. 

[57]

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

[58]

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

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