SpringerLink
Log in

The Complexity and the Structure and Classification of Dynamical Systems

  • Reference work entry
  • First Online:
Ergodic Theory

Part of the book series: Encyclopedia of Complexity and Systems Science Series ((ECSSS))

  • 830 Accesses

  • Originally published in
  • R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, © This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2022

Glossary

Analytic set :

A subset A of a Polish space X is analytic if there is a Polish space Y and a Borel set B ⊆ X × Y such that A = {x: for some y ∈ Y, (x, y) ∈ B}. Equivalently A is the projection of B to the X-axis.

Anosov diffeomorphism :

Definition 13.

Benchmarking :

An informal term describing the location of a set or an equivalence relation in terms of reducibility to other sets or equivalence relations.

Bernoulli shift :

Definition 7.

Borel hierarchy :

Definition 70

Co-analytic set :

A subset C of a Polish space X is co-analytic if there is a Polish space Y and a Borel set B ⊆ X × Y such that C = {x for all y ∈ Y, (x, y) ∉ B. Equivalently, the complement of C is analytic.

Descriptive complexity :

This term is used for the placement of a structure or a classification problem among the benchmarks of reducibility.

Distal :

Topologically distal is definition 20, measure distal is definition 27.

=+ equivalence relation:

Introduced and discussed in section “=+and the...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 299.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  • Aaronson J (1997) An introduction to infinite ergodic theory, Mathematical surveys and monographs, vol 50. American Mathematical Society, Providence, RI

    Google Scholar 

  • Barreira L, Valls C (2013) Dynamical systems, an introduction. Universitext, Springer, London

    Google Scholar 

  • Becker H, Kechris AS (1996) The descriptive set theory of polish group actions. London Mathematical society lecture note series. Cambridge University Press, Cambridge

    Google Scholar 

  • Beleznay F, Foreman M (1995) The collection of distal flows is not Borel. Am J Math 117(1):203–239

    Article  MathSciNet  MATH  Google Scholar 

  • Beleznay F, Foreman M (1996) The complexity of the collection of measure-distal transformations. Ergodic Theory Dyn Syst 16(5):929–962

    Article  MathSciNet  MATH  Google Scholar 

  • Bonatti C, Grines V, Pochinka O (2019) Topological classification of Morse-Smale diffeomorphisms on 3-manifolds. Duke Math J 168(13):2507–2558

    Article  MathSciNet  MATH  Google Scholar 

  • Braverman M, Yampolsky M (2009) Computability of Julia sets, Algorithms and computation in mathematics, vol 23. Springer-Verlag, Berlin

    MATH  Google Scholar 

  • Camerlo R, Gao S (2001) The completeness of the isomorphism relation for countable Boolean algebras. Trans Am Math Soc 353(2):491–518

    Article  MathSciNet  MATH  Google Scholar 

  • Chacon RV (1969) Weakly mixing transformations which are not strongly mixing. Proc Am Math Soc 22:559–562

    Article  MathSciNet  MATH  Google Scholar 

  • Clemens JD (2009) Isomorphism of subshifts is a universal countable Borel equivalence relation. Israel J. Math. 170:113–123

    Article  MathSciNet  MATH  Google Scholar 

  • Ding L, Gao S (2014) Is there a spectral theory for all bounded linear operators? Not Am Math Soc 61(7):730–735

    Article  MathSciNet  MATH  Google Scholar 

  • Dougherty R, Jackson S, Kechris AS (1994) The structure of hyperfinite Borel equivalence relations. Trans Am Math Soc 341(1):193–225

    Article  MathSciNet  MATH  Google Scholar 

  • Effros EG (1965) Transformation groups and C*-algebras. Ann Math 81(2):38–55

    Article  MathSciNet  MATH  Google Scholar 

  • Feldman J (1974) Borel structures and invariants for measurable transformations. Proc Am Math Soc 46:383–394

    Article  MathSciNet  MATH  Google Scholar 

  • Feldman J, Moore CC (1977) Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans Am Math Soc 234(2):289–324

    Article  MathSciNet  MATH  Google Scholar 

  • Ferenczi V, Louveau A, Rosendal C (2009) The complexity of classifying separable Banach spaces up to isomorphism. J Lond Math Soc 79(2):323–345

    Article  MathSciNet  MATH  Google Scholar 

  • Foreman M (2000) A descriptive view of ergodic theory, Descriptive set theory and dynamical systems (Marseille-Luminy, 1996), London Mathematical Society lecture note series, vol 277. Cambridge Univ. Press, Cambridge, pp 87–171

    Google Scholar 

  • Foreman M (2010) Naive descriptive set theory, ar**v: 2110. 08881, 1–53.

    Google Scholar 

  • Foreman M (2018) What is a Borel reduction? Not Am Math Soc 65(10):1263–1268

    MathSciNet  MATH  Google Scholar 

  • Foreman M, Weiss B (2019a) From odometers to circular systems: a global structure theorem. J Mod Dyn 15:345–423

    Google Scholar 

  • Foreman M, Weiss B (2019b) A symbolic representation for Anosov-Katok systems. J Anal Math 137(2):603–661

    Google Scholar 

  • Foreman M, Weiss B (2022) Measure preserving diffeomorphisms of the torus are unclassifiable. J Eur Math Soc (JEMS), 1–80.

    Google Scholar 

  • Foreman M, Gorodetski A (2022) Anti-classification results for smooth dynamical systems, ar**v: 2206. 09322, 1–56.

    Google Scholar 

  • Foreman M, Weiss B (2004) An anti-classification theorem for ergodic measure preserving transformations. J Eur Math Soc (JEMS) 6(3):277–292

    Article  MathSciNet  MATH  Google Scholar 

  • Foreman M, Rudolph DJ, Weiss B (2011) The conjugacy problem in ergodic theory. Ann Math 173(3):1529–1586

    Article  MathSciNet  MATH  Google Scholar 

  • Friedman H, Stanley L (1989) A Borel reducibility theory for classes of countable structures. J Symb Log 54(3):894–914

    Google Scholar 

  • Furstenberg H (1963) The structure of distal flows. Am J Math 85:477–515

    Article  MathSciNet  MATH  Google Scholar 

  • Furstenberg H (1981) Recurrence in ergodic theory and combinatorial number theory, Porter lectures. Princeton University Press, Princeton

    Google Scholar 

  • Gao S (2009) Invariant descriptive set theory, A series of monographs and textbooks, vol 293. Taylor and Francis, Boca Raton

    Google Scholar 

  • Gao S (2022) Dynamical systems and countable structures, Lecture notes: https://www.birs.ca/workshops/2022/22w5134/files/Su%20Gao/banff%20lecture%20gao%20nopause.pdf

  • Gao S, Pestov V (2003) On a universality property of some abelian Polish groups. Fundam Math 179(1):1–15

    Article  MathSciNet  MATH  Google Scholar 

  • Gerber M, Kunde P (2021) Anti-classification results for the Kakutani equivalence relation, https://arxiv.org/abs/2109.06086v1, 1–72.

  • Glimm J (1961) Locally compact transformation groups. TAMS 101:124–138

    Article  MathSciNet  MATH  Google Scholar 

  • Graham L, Kantor J-M (2009) Naming infinity. Belknap Press, Cambridge

    Google Scholar 

  • Halmos PR (1944) In general a measure preserving transformation is mixing. Ann Math 45(2):786–792

    Article  MathSciNet  MATH  Google Scholar 

  • Halmos PR (1956) Lectures on ergodic theory. Chelsea Publishing Company, New York

    MATH  Google Scholar 

  • Halmos PR (1974) Naive set theory, Undergraduate texts in mathematics. Springer-Verlag, New York-Heidelberg

    Google Scholar 

  • Halmos PR and von Neumann J (1942) Operator methods in classical mechanics. II, Ann of Math. 43(2):332–350

    Google Scholar 

  • Harrington LA, Kechris AS, Louveau A (1990) A Glimm-Effros dichotomy for Borel equivalence relations. J Am Math Soc 3(4):903–928

    Article  MathSciNet  MATH  Google Scholar 

  • Hasselblatt B (2007) Problems in dynamical systems and related topics, Dynamics, ergodic theory, and geometry, Mathematical Sciences Research Institute Publications, vol 54. Cambridge University Press, Cambridge, pp 273–324

    MATH  Google Scholar 

  • Hjorth G (1999) Around nonclassifiability for countable torsion free abelian groups. In: Eklof PC, Göbel R (eds) Abelian Groups and Modules: International conference in Dublin, August 10–14, 1998. Birkhäuser, Basel, pp 269–292

    Chapter  Google Scholar 

  • Hjorth G (2000) Classification and orbit equivalence relations, Mathematical surveys and monographs, vol 75. American Mathematical Society, Providence, RI

    Google Scholar 

  • Hjorth G (2001) On invariants for measure preserving transformations. Fundam Math 169(1):51–84

    Article  MathSciNet  MATH  Google Scholar 

  • Jackson S, Kechris AS, Louveau A (2002) Countable Borel equivalence relations. J Math Log 2(1):1–80

    Article  MathSciNet  MATH  Google Scholar 

  • Jech TJ (2003) Set theory: The third millenium edition. Springer, Berlin

    Google Scholar 

  • Kanovei V (2008) Borel equivalence relations: structure and classification, University lecture series, vol 14. American Mathematical Society, Providence, RI

    Google Scholar 

  • Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its applications, vol 54. Cambridge University Press, Cambridge

    Google Scholar 

  • Kechris A (1995) Classical descriptive set theory, Graduate texts in mathematics, 1st edn. Springer, New York

    Google Scholar 

  • Kechris A (2010) Global aspects of Ergodic Group actions, Mathematical surveys and monographs. Am Math Soc 160

    Google Scholar 

  • Kechris AS, Louveau A (1997) The classification of Hypersmooth Borel equivalence relations. J Am Mathematical Soc 10(1):215–242

    Article  MathSciNet  MATH  Google Scholar 

  • Kechris A, Miller BD (2004) Topics in orbit equivalence, Lecture notes in mathematics. Springer, Berlin/Heidelberg

    Book  MATH  Google Scholar 

  • Kechris AS, Sofronidis NE (2001) A strong generic ergodicity property of unitary and self-adjoint operators. Ergodic Theory Dyn Syst 21(5):1459–1479

    Article  MathSciNet  MATH  Google Scholar 

  • Kechris AS, Tucker-Drob RD (2013) The complexity of classification problems in ergodic theory, Appalachian set theory 2006–2012, London Mathematical Society lecture note series, vol 406. Cambridge University Press, Cambridge, pp 265–299

    MATH  Google Scholar 

  • Khinchin AI (1949) Mathematical foundations of statistical mechanics. Dover Publications, Inc., New York. Translated by G. Gamow

    MATH  Google Scholar 

  • King J (1986) The commutant is the weak closure of the powers, for rank-1 transformations. Ergodic Theory Dyn Syst 6(3):363–384

    Article  MathSciNet  MATH  Google Scholar 

  • Krieger W (1970) On entropy and generators of measure-preserving transformations. Trans Am Math Soc 149:453–464

    Article  MathSciNet  MATH  Google Scholar 

  • Krieger W (1972a) Erratum to: “On entropy and generators of measure-preserving transformations”. Trans Am Math Soc 168:519

    MathSciNet  MATH  Google Scholar 

  • Krieger W (1972b) On unique ergodicity. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, University of California, Berkeley, 1970/1971, vol II: Probability theory, pp 327–346

    Google Scholar 

  • Levy A (2002) Basic set theory. Dover Publications, Inc., Mineola, Reprint of the 1979 original [Springer, Berlin; MR0533962 (80k:04001)].

    MATH  Google Scholar 

  • Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Lindenstrass J, Olsen G, Sternfeld Y (1978) The Poulsen simplex. Annales de l’institute Fourier 28(1):91–114

    Article  MathSciNet  MATH  Google Scholar 

  • Lusin N (1933) Sur les classes des constituantes des complémentaires analytiques. Ann Scuola Norm Super Pisa Cl Sci 2(3):269–282

    MathSciNet  MATH  Google Scholar 

  • Marker D (2002) Descriptive set theory, http://homepages.math.uic.edu/~marker/math512/dst.pdf, 1–56.

  • Maruyama G (1949) The harmonic analysis of stationary stochastic processes. Mem Fac Sci Kyūsyū Univ A 4:45–106

    MathSciNet  MATH  Google Scholar 

  • Mekler AH (1981) Stability of nilpotent groups of class 2 and prime exponent. J Symb Log 46(4):781–788

    Article  MathSciNet  MATH  Google Scholar 

  • Mirzakhani M, Feng T (2014) Introduction to ergodic theory, https://www.mit.edu/~fengt/ergodic_theory.pdf, 1–56.

  • Mitin AV (1998) Undecidability of the elementary theory of groups of measure-preserving transformations. Mat Zametki 63(3):414–420

    MathSciNet  MATH  Google Scholar 

  • Moschovakis YN (2009) Descriptive set theory: second edition, Mathematical surveys and monographs, no. 155, 2nd edn. American Mathematical Society, Providence, RI

    Google Scholar 

  • Newhouse SE (1970) Nondensity of axiom A(a) on S2, global analysis. In: Proceedings of the Symposium on Pure Mathematics, vol XIV, Berkeley, 1968, American Mathematical Society, Providence, pp 191–202.

    Google Scholar 

  • Newhouse SE (1974) Diffeomorphisms with infinitely many sinks. Topology 13:9–18

    Article  MathSciNet  MATH  Google Scholar 

  • Ornstein D (1970a) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4:337–352

    Article  MathSciNet  MATH  Google Scholar 

  • Ornstein D (1970b) The isomorphism problem for measure-preserving transformations. In: Proceedings of the symposium on functional analysis, Monterey, 1969. Academic Press, New York, pp 71–74. MR 0262463

    Google Scholar 

  • Ornstein D (1974) Ergodic theory, randomness, and dynamical systems, Yale mathematical monographs, No. 5. Yale University Press, New Haven/London, James K. Whittemore. Lectures in Mathematics given at Yale University

    Google Scholar 

  • Ornstein D (1975) Some open problems in Ergodic theory. Publications mathématiques et informatique de Rennes, no. S4

    Google Scholar 

  • Ornstein DS, Shields PC (1973) An uncountable family of K-automorphisms. Adv Math 10:63–88

    Article  MathSciNet  MATH  Google Scholar 

  • Ornstein DS, Weiss B (1974) Finitely determined implies very weak Bernoulli. Israel J Math 17:94–104

    Article  MathSciNet  MATH  Google Scholar 

  • Oshemkov AA, Sharko VV (1998) On the classification of Morse-Smale flows on two-dimensional manifolds. Mat Sb 189(8):93–140

    MathSciNet  MATH  Google Scholar 

  • Peixoto MM (1973) On the classification of flows on 2-manifolds. In: Proceedings of the Symposium dynamical systems, University of Bahia, Salvador, 1971, pp 389–419

    Google Scholar 

  • Petersen KE (1983) Ergodic theory, Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  • Rohlin V (1948) A “general” a measure-preserving transformation is not mixing. Doklady Akad Nauk SSSR 60:349–351

    MathSciNet  Google Scholar 

  • Rohlin V (1967) Lectures on the entropy theory of transformations with invariant measure. Uspehi Mat Nauk 22(5):3–56, 137.

    MathSciNet  Google Scholar 

  • Rosendal C (2005) Cofinal families of Borel equivalence relations and quasiorders. J Symb Log 70(4):1325–1340

    Article  MathSciNet  MATH  Google Scholar 

  • Ryzhikov VV (1985) Representation of transformations preserving the Lebesgue measure, in the form of a product of periodic transformations. Mat. Zametki 38(6):860–865, 957.

    MathSciNet  MATH  Google Scholar 

  • Sabok M (2016) Completeness of the isomorphism problem for separable C*-algebras. Invent Math 204(3):833–868

    Article  MathSciNet  MATH  Google Scholar 

  • Smale S (1963) Dynamical systems and the topological conjugacy problem for diffeomorphisms. In: Proceedings of the International Congress on Mathematicians, Stockholm, 1962. Institute Mittag-Leffler, Djursholm, pp 49–496

    Google Scholar 

  • Smale S (1967) Differentiable dynamical systems. Bull Am Math Soc 73:747–817

    Article  MathSciNet  MATH  Google Scholar 

  • Soare R (1987) Recursively enumerable sets and degrees, perspectives in mathematical logic. Springer, Berlin/Heidelberg

    Book  Google Scholar 

  • Thomas S (2003) The classification problem for torsion-free abelian groups of finite rank. J Am Math Soc 16(1):233–258

    Article  MathSciNet  MATH  Google Scholar 

  • Thomas S (2011) The classification problem for S-local torsion-free abelian groups of finite rank. Adv Math 226(4):3699–3723

    Article  MathSciNet  MATH  Google Scholar 

  • Thomas S, Velickovic B (1999) On the complexity of the isomorphism relation for finitely generated groups. J Algebra 217(1):352–373

    Article  MathSciNet  MATH  Google Scholar 

  • Thouvenot J-P (2002) Entropy, isomorphism and equivalence in ergodic theory, Handbook of dynamical systems, vol 1A. North-Holland, Amsterdam, pp 205–238

    MATH  Google Scholar 

  • Uspenskiĭ VV (1986) A universal topological group with a countable basis. Funktsional Anal i Prilozhen 20(2):86–87

    Article  MathSciNet  Google Scholar 

  • Walters P (1982) An introduction to ergodic theory, Graduate texts in mathematics. Springer, New York

    Book  Google Scholar 

  • Weihrauch K (2000) Computable analysis, Texts in theoretical computer science. An EATCS series. Springer, Berlin, An introduction

    Google Scholar 

  • Wikipedia. Hamiltonian system. https://en.wikipedia.org/wiki/Hamiltonian_system

  • Wikipedia. Rotation number. https://en.wikipedia.org/wiki/Rotation_number

  • Zimmer RJ (1976) Ergodic actions with generalized discrete spectrum. Ill J Math 20(4):555–588

    MathSciNet  MATH  Google Scholar 

  • Zippin L (2013) Transformation groups, Lectures in Topology; the University of Michigan Conference of 1940. University of Michigan Press, Ann Arbor

    Google Scholar 

Download references

Acknowledgments

The author received help from many people in writing and editing this entry. He particularly wants to acknowledge Filippo Calderoni, Marlies Gerber, Anton Gorodetski, Philipp Kunde, Andrew Marks, Ronnie Pavlov. Alexander Kechris made invaluable corrections on an early text and correspondence with Su Gao was essential in the completion of the text. As always, Benjamin Weiss provided important suggestions and comments and provided very helpful references.

Expository Books and Articles on Analytic Equivalence Relations

  1. 1.

    The book by Gao (2009),

  2. 2.

    The books by Kechris (1995), Kechris–Miller (2004) and Becker–Kechris (1996),

  3. 3.

    The paper by Friedman and Stanley (1989),

  4. 4.

    Dave Marker’s website (Marker 2002).

The author would like to acknowledge partial support from NSF grant DMS-2100367.

Author information

Authors and Affiliations

  1. University of California at Irvine, Irvine, CA, USA

    Matthew Foreman

Authors
  1. Matthew Foreman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew Foreman .

Editor information

Editors and Affiliations

  1. Mathematics, Williams College, Williamstown, MA, USA

    Cesar E. Silva

  2. B.Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv, Ukraine

    Alexandre I. Danilenko

Rights and permissions

Reprints and permissions

Copyright information

© 2023 Springer Science+Business Media, LLC, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Foreman, M. (2023). The Complexity and the Structure and Classification of Dynamical Systems. In: Silva, C.E., Danilenko, A.I. (eds) Ergodic Theory. Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-2388-6_726

Download citation

Publish with us

Policies and ethics

Search

Navigation