Ergodic universality of some topological dynamical systems

Quas, Anthony; Soo, Terry

doi:10.1090/tran/6489

Ergodic universality of some topological dynamical systems
HTML articles powered by AMS MathViewer

by Anthony Quas and Terry Soo PDF

Trans. Amer. Math. Soc. 368 (2016), 4137-4170 Request permission

Abstract:

The Krieger generator theorem says that every invertible ergodic measure-preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. We extend Krieger’s theorem to include toral automorphisms and, more generally, any topological dynamical system on a compact metric space that satisfies almost weak specification, asymptotic entropy expansiveness, and the small boundary property. As a corollary, one obtains a complete solution to a natural generalization of an open problem in Halmos’s 1956 book regarding an isomorphism invariant that he proposed.

References

Rufus Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc. 164 (1972), 323–331. MR 285689, DOI 10.1090/S0002-9947-1972-0285689-X
Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. MR 274707, DOI 10.1090/S0002-9947-1971-0274707-X
Rufus Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377–397. MR 282372, DOI 10.1090/S0002-9947-1971-0282372-0
Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. MR 339281, DOI 10.2307/2373793
Mike Boyle and Tomasz Downarowicz, The entropy theory of symbolic extensions, Invent. Math. 156 (2004), no. 1, 119–161. MR 2047659, DOI 10.1007/s00222-003-0335-2
M. Brin and A. Katok, On local entropy, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 30–38. MR 730261, DOI 10.1007/BFb0061408
David Burguet, A direct proof of the tail variational principle and its extension to maps, Ergodic Theory Dynam. Systems 29 (2009), no. 2, 357–369. MR 2486774, DOI 10.1017/S0143385708080425
R. M. Burton, M. S. Keane, and Jacek Serafin, Residuality of dynamical morphisms. part 2, Colloq. Math. 84/85 (2000), no. part 2, 307–317. Dedicated to the memory of Anzelm Iwanik. MR 1784199, DOI 10.4064/cm-84/85-2-307-317
R. Burton and A. Rothstein, Isomorphism theorems in ergodic theory, Tech. report, Oregon State University, 1977.
Jerome Buzzi, The almost Borel structure of diffeomorphisms with some hyperbolicity, 2014, arXiv:1403.2616.
Vaughn Climenhaga and Daniel J. Thompson, Intrinsic ergodicity beyond specification: $\beta$-shifts, $S$-gap shifts, and their factors, Israel J. Math. 192 (2012), no. 2, 785–817. MR 3009742, DOI 10.1007/s11856-012-0052-x
Masahito Dateyama, The almost weak specification property for ergodic group automorphisms of abelian groups, J. Math. Soc. Japan 42 (1990), no. 2, 341–351. MR 1041229, DOI 10.2969/jmsj/04220341
Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675, DOI 10.1007/BFb0082364
Tomasz Downarowicz, Minimal models for noninvertible and not uniquely ergodic systems, Israel J. Math. 156 (2006), 93–110. MR 2282370, DOI 10.1007/BF02773826
T. Downarowicz and J. Serafin, A short proof of the Ornstein theorem, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 587–597. MR 2901361, DOI 10.1017/S0143385711000265
Tomasz Downarowicz and Benjamin Weiss, Entropy theorems along times when $x$ visits a set, Illinois J. Math. 48 (2004), no. 1, 59–69. MR 2048214
Tomasz Downarowicz, Entropy structure, J. Anal. Math. 96 (2005), 57–116. MR 2177182, DOI 10.1007/BF02787825
Tomasz Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2809170, DOI 10.1017/CBO9780511976155
Richard M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. MR 982264
Ryszard Engelking, Dimension theory, North-Holland Mathematical Library, vol. 19, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author. MR 0482697
Eli Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR 1958753, DOI 10.1090/surv/101
P. Hall, On representatives of subsets, J. London Math. Soc.(1) 10 (1935), 26–30.
Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, vol. 3, Mathematical Society of Japan, Tokyo, 1956. MR 0097489
Paul R. Halmos, On automorphisms of compact groups, Bull. Amer. Math. Soc. 49 (1943), 619–624. MR 8647, DOI 10.1090/S0002-9904-1943-07995-5
Michael Hochman, Erratum to: Isomorphism and embedding of Borel systems on full sets [MR3077948], Acta Appl. Math. 128 (2013), 211. MR 3125641, DOI 10.1007/s10440-013-9846-z
Michael Hochman, Isomorphism and embedding of Borel systems on full sets, Acta Appl. Math. 126 (2013), 187–201. MR 3077948, DOI 10.1007/s10440-013-9813-8
Steven Kalikow and Randall McCutcheon, An outline of ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 122, Cambridge University Press, Cambridge, 2010. MR 2650005, DOI 10.1017/CBO9780511801600
Anatole Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 1245–1253. MR 804774
M. Keane and M. Smorodinsky, A class of finitary codes, Israel J. Math. 26 (1977), no. 3-4, 352–371. MR 450514, DOI 10.1007/BF03007652
Michael Keane and Meir Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math. (2) 109 (1979), no. 2, 397–406. MR 528969, DOI 10.2307/1971117
W. Krieger, On generators in ergodic theory, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 303–308. MR 0422576
Wolfgang Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc. 149 (1970), 453–464. MR 259068, DOI 10.1090/S0002-9947-1970-0259068-3
Wolfgang Krieger, Erratum to: “On entropy and generators of measure-preserving transformations”, Trans. Amer. Math. Soc. 168 (1972), 519. MR 294603, DOI 10.1090/S0002-9947-1972-0294603-2
Wolfgang Krieger, On unique ergodicity, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 327–346. MR 0393402
John Kulesza, Zero-dimensional covers of finite-dimensional dynamical systems, Ergodic Theory Dynam. Systems 15 (1995), no. 5, 939–950. MR 1356620, DOI 10.1017/S014338570000969X
D. A. Lind, Ergodic group automorphisms and specification, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 93–104. MR 550414
D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 49–68. MR 684244, DOI 10.1017/s0143385700009573
D. A. Lind and J.-P. Thouvenot, Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations, Math. Systems Theory 11 (1977/78), no. 3, 275–282. MR 584588, DOI 10.1007/BF01768481
Elon Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 227–262 (2000). MR 1793417, DOI 10.1007/BF02698858
Elon Lindenstrauss and Klaus Schmidt, Invariant sets and measures of nonexpansive group automorphisms, Israel J. Math. 144 (2004), 29–60. MR 2121533, DOI 10.1007/BF02984405
Elon Lindenstrauss and Klaus Schmidt, Symbolic representations of nonexpansive group automorphisms, Israel J. Math. 149 (2005), 227–266. Probability in mathematics. MR 2191216, DOI 10.1007/BF02772542
Brian Marcus, A note on periodic points for ergodic toral automorphisms, Monatsh. Math. 89 (1980), no. 2, 121–129. MR 572888, DOI 10.1007/BF01476590
Richard Miles, Periodic points of endomorphisms on solenoids and related groups, Bull. Lond. Math. Soc. 40 (2008), no. 4, 696–704. MR 2441142, DOI 10.1112/blms/bdn052
MichałMisiurewicz, Topological conditional entropy, Studia Math. 55 (1976), no. 2, 175–200. MR 415587, DOI 10.4064/sm-55-2-175-200
Donald Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337–352. MR 257322, DOI 10.1016/0001-8708(70)90029-0
Donald S. Ornstein, Ergodic theory, randomness, and dynamical systems, Yale Mathematical Monographs, No. 5, Yale University Press, New Haven, Conn.-London, 1974. James K. Whittemore Lectures in Mathematics given at Yale University. MR 0447525
Donald Samuel Ornstein and Benjamin Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory 39 (1993), no. 1, 78–83. MR 1211492, DOI 10.1109/18.179344
C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 929–956. MR 2322186, DOI 10.1017/S0143385706000824
Anthony Quas and Terry Soo, Weak mixing suspension flows over shifts of finite type are universal, J. Mod. Dyn. 6 (2012), no. 4, 427–449. MR 3008405, DOI 10.3934/jmd.2012.6.427
M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math. 15 (1973), 92–114. MR 339282, DOI 10.1007/BF02771776
E. Arthur Robinson Jr. and Ayşe A. Şahin, Modeling ergodic, measure preserving actions on $\Bbb Z^d$ shifts of finite type, Monatsh. Math. 132 (2001), no. 3, 237–253. MR 1844076, DOI 10.1007/s006050170043
Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631
Thierry de la Rue, An introduction to joinings in ergodic theory, Discrete Contin. Dyn. Syst. 15 (2006), no. 1, 121–142. MR 2191388, DOI 10.3934/dcds.2006.15.121
Karl Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc. 190 (1974), 285–299. MR 352411, DOI 10.1090/S0002-9947-1974-0352411-X
Ja. G. Sinaĭ, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.) 63 (105) (1964), 23–42 (Russian). MR 0161961
Yakov G. Sinai, Selecta. Volume I. Ergodic theory and dynamical systems, Springer, New York, 2010. MR 2766434, DOI 10.1007/978-0-387-87870-6
S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. MR 1619545, DOI 10.1007/978-3-642-85473-6
Peng Sun, Density of metric entropies for linear toral automorphisms, Dyn. Syst. 27 (2012), no. 2, 197–204. MR 2926697, DOI 10.1080/14689367.2011.649246
Benjamin Weiss, 2012, personal communication.
Benjamin Weiss, 2013, personal communication.
Kenichiro Yamamoto, On the weaker forms of the specification property and their applications, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3807–3814. MR 2529890, DOI 10.1090/S0002-9939-09-09937-7