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What is topological about topological dynamics?
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 |
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.
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. |
[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. 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.
|
[5] |
S. Bezuglyi and O. Karpel,
Bratteli diagrams: Structure, measures, dynamics, Contemp. Math., 669 (2016), 1-36.
|
[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. |
[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.
|
[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. |
[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. Boyle, D. 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. Google Scholar |
[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. Giordano, H. Matui, I. Putnam and C. Skau,
The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, 28 (2008), 1509-1531.
|
[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. |
[32] |
T. Giordano, I. Putnam and C. Skau,
Topological orbit equivalence and $ C^*$-crossed}products, J. Reine Angew. Math., 469 (1995), 51-111.
|
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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.
|
[5] |
S. Bezuglyi and O. Karpel,
Bratteli diagrams: Structure, measures, dynamics, Contemp. Math., 669 (2016), 1-36.
|
[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. |
[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.
|
[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. |
[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. Boyle, D. 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. Google Scholar |
[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. Giordano, H. Matui, I. Putnam and C. Skau,
The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, 28 (2008), 1509-1531.
|
[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. |
[32] |
T. Giordano, I. Putnam and C. Skau,
Topological orbit equivalence and $ C^*$-crossed}products, J. Reine Angew. Math., 469 (1995), 51-111.
|
[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. |
[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. 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. |
[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. 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. |
[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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