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: |
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.![]() ![]() |
|
V. Bergelson, Ergodic Ramsey theory - an update, Ergodic Theory of $ \mathbb{Z}^d$-actions, London Math. Soc. Lecture Note Series, 228 (1996), 1-61.
![]() ![]() |
|
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
![]() ![]() |
|
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.
![]() ![]() |
|
S. Bezuglyi
and O. Karpel
, Bratteli diagrams: Structure, measures, dynamics, Contemp. Math., 669 (2016)
, 1-36.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
S. Bezuglyi
and R. Yassawi
, Orders that yield homeomorphisms on Bratteli diagrams, Dynamical Systems, 32 (2017)
, 249-282.
doi: 10.1080/14689367.2016.1197888.![]() ![]() ![]() |
|
M. Boyle
, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Sys., 3 (1983)
, 541-557.
![]() ![]() |
|
M. Boyle
and T. Downarowicz
, The entropy theory of symbolic extensions, Inventiones Math., 156 (2004)
, 119-161.
doi: 10.1007/s00222-003-0335-2.![]() ![]() ![]() |
|
M. Boyle
, D. Fiebig
and U. Fiebig
, Residual entropy, conditional entropy and subshift covers, Forum Mathematicum, 14 (2002)
, 713-757.
![]() ![]() |
|
O. Bratteli
, Inductive limits of finite-dimensional $ C^{*}$-algebras, Trans. Amer. Math. Soc., 171 (1972)
, 195-234.
![]() ![]() |
|
D. Burguet
, Embedding asymptotically expansive systems, Monatsh Math., 184 (2017)
, 21-49.
doi: 10.1007/s00605-017-1079-1.![]() ![]() ![]() |
|
D. Burguet and T. Downarowicz, Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits, preprint, arXiv:1705.08829.
![]() |
|
J. Buzzi
, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997)
, 125-161.
doi: 10.1007/BF02773637.![]() ![]() ![]() |
|
P. Collet and J. -P. Eckmann,
Iterated Maps on the Interval as Dynamical Systems, Modern Birkhäuser Classics, Birkhäuser Basel, 2009.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
T. Downarowicz
, The Choquet simplex of invariant measures for minimal flows, Isr. J. Math., 74 (1991)
, 241-256.
doi: 10.1007/BF02775789.![]() ![]() ![]() |
|
T. Downarowicz
, Entropy structure, J. Anal. Math., 96 (2005)
, 57-116.
doi: 10.1007/BF02787825.![]() ![]() ![]() |
|
T. Downarowicz
, Minimal models for noninvertible and not uniquely ergodic systems, Israel J. of Math., 156 (2006)
, 93-110.
doi: 10.1007/BF02773826.![]() ![]() ![]() |
|
T. Downarowicz
, Faces of simplexes of invariant measures, Israel J. of Math., 165 (2008)
, 189-210.
doi: 10.1007/s11856-008-1009-y.![]() ![]() ![]() |
|
T. Downarowicz,
Entropy in Dynamical Systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011.
![]() ![]() |
|
T. Downarowicz
and D. Huczek
, Faithful zero-dimensional principal extensions, Studia Math., 212 (2012)
, 1-19.
doi: 10.4064/sm212-1-1.![]() ![]() ![]() |
|
T. Downarowicz
and A. Maass
, Finite-rank Bratteli-Vershik diagrams are expansive, Ergod. Th. and Dynam. Sys., 28 (2008)
, 739-747.
![]() ![]() |
|
T. Downarowicz
and J. Serafin
, Possible entropy functions, Israel J. Math., 135 (2003)
, 221-250.
doi: 10.1007/BF02776059.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
H. Furstenberg,
Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N. J., 1981.
![]() ![]() |
|
T. Giordano
, H. Matui
, I. Putnam
and C. Skau
, The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, 28 (2008)
, 1509-1531.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
T. Giordano
, I. Putnam
and C. Skau
, Topological orbit equivalence and $ C^*$-crossed}products, J. Reine Angew. Math., 469 (1995)
, 51-111.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
E. Glasner
and B. Weiss
, Weak orbit equivalence of Cantor minimal systems, Internat. J. Math., 6 (1995)
, 559-579.
doi: 10.1142/S0129167X95000213.![]() ![]() ![]() |
|
Y. Gutman
, Mean dimension and Jaworski-type theorems, Proc. London Math. Soc., 111 (2015)
, 831-850.
doi: 10.1112/plms/pdv043.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. Hadamard
, Les surfaces à courbures opposées et leur lignes geodesiques, Journal de Mathématiques Pures et Appliqués, 4 (1898)
, 27-73.
![]() |
|
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.![]() ![]() ![]() |
|
G. Hedlund
, Endomorphisms and automorphisms of the shift dynamical systems, Math. Syst. Theory, 3 (1969)
, 320-375.
doi: 10.1007/BF01691062.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
R. I. Jewett
, The prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1970)
, 717-729.
![]() ![]() |
|
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.
![]() ![]() |
|
W. Krieger
, On the subsystems of topological Markov chains, Ergodic Theory Dynam. Sys., 2 (1982)
, 195-202.
![]() ![]() |
|
J. Kulesza
, Zero-dimensional covers of finite-dimensional dynamical systems, Erg. Th. & Dyn. Syst., 15 (1995)
, 939-950.
doi: 10.1017/S014338570000969X.![]() ![]() ![]() |
|
D. Lind and B. Marcus,
An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995.
![]() ![]() |
|
E. Lindenstrauss
, Mean dimension, small entropy factors and an embedding theorem, Publ. Math. I.H.E.S., 89 (1999)
, 227-262.
![]() ![]() |
|
E. Lindenstrauss
and B. Weiss
, Mean Topological Dimension, Israel J. Math., 115 (2000)
, 1-24.
doi: 10.1007/BF02810577.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
M. Misiurewicz
, Topological conditional entropy, Studia Math., 55 (1976)
, 175-200.
doi: 10.4064/sm-55-2-175-200.![]() ![]() ![]() |
|
M. Morse
and G. Hedlund
, Symbolic dynamics, Amer. J. Math., 60 (1938)
, 815-866.
doi: 10.2307/2371264.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
F. Ramsey
, On a Problem of Formal Logic, Proc. London Math. Soc., 30 (1929)
, 264-286.
![]() ![]() |
|
A. Rosenthal
, Strictly ergodic models for non-invertible transformations, Isr. J. Math., 64 (1988)
, 57-72.
doi: 10.1007/BF02767370.![]() ![]() ![]() |
|
J. Serafin
, A faithful symbolic extension, Communications on Pure and Applied Analysis, 11 (2012)
, 1051-1062.
![]() ![]() |
|
C. Skau
, Ordered $K$-theory and minimal symbolic dynamical systems. Dedicated to the memory of Anzelm Iwanik, Colloq. Math., 84/85 (2000)
, 203-227.
![]() ![]() |
|
A. M. Vershik
, Uniform algebraic approximation of shift and multiplication operators, Dokl. Acad. Nauk SSSR, 259 (1981)
, 526-529.
![]() ![]() |
|
A. M. Vershik
, A theorem on Markov periodic approximation in ergodic theory, Zap. Nauchn. Sem. LOMI, 115 (1982)
, 72-82.
![]() ![]() |