A measure-conjugacy invariant for free group actions

Pages 1387-1400 from Volume 171 (2010), Issue 2 by Lewis Phylip Bowen

Abstract

This paper introduces a new measure-conjugacy invariant for actions of free groups. Using this invariant, it is shown that two Bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy. This answers a question of Ornstein and Weiss.

  • [Gl03] E. Glasner, Ergodic Theory via Joinings, Providence, RI: Amer. Math. Soc., 2003, vol. 101.
    @book {Gl03, MRKEY = {MR1958753},
      AUTHOR = {Glasner, Eli},
      TITLE = {Ergodic Theory via Joinings},
      SERIES = {Mathematical Surveys and Monographs},
      VOLUME = {101},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2003},
      PAGES = {xii+384},
      ISBN = {0-8218-3372-3},
      MRCLASS = {37A15 (28Dxx 37A25 37A35 37A45 37B99 54H20)},
      MRNUMBER = {2004c:37011},
      MRREVIEWER = {Andr{é}s del Junco},
      ZBLNUMBER={1038.37002},
      }
  • [Ki75] J. C. Kieffer, "A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space," Ann. Probability, vol. 3, iss. 6, pp. 1031-1037, 1975.
    @article {Ki75, MRKEY = {MR0393422},
      AUTHOR = {Kieffer, J. C.},
      TITLE = {A generalized {S}hannon-{M}c{M}illan theorem for the action of an amenable group on a probability space},
      JOURNAL = {Ann. Probability},
      VOLUME = {3},
      YEAR = {1975},
      NUMBER = {6},
      PAGES = {1031--1037},
      MRCLASS = {28A65},
      MRNUMBER = {52 \#14232},
      MRREVIEWER = {Paul C. Shields},
      }
  • [Ko58] A. N. Kolmogorov, "A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces," Dokl. Akad. Nauk SSSR ( N.S.), vol. 119, pp. 861-864, 1958.
    @article {Ko58, MRKEY = {MR0103254},
      AUTHOR = {Kolmogorov, A. N.},
      TITLE = {A new metric invariant of transient dynamical systems and automorphisms in {L}ebesgue spaces},
      JOURNAL = {Dokl. Akad. Nauk SSSR ({\rm N.S.})},
      FJOURNAL = {Doklady Akademii Nauk SSSR},
      VOLUME = {119},
      YEAR = {1958},
      PAGES = {861--864},
      ISSN = {0002-3264},
      MRCLASS = {28.00 (94.00)},
      MRNUMBER = {21 \#2035a},
      ZBLNUMBER={0083.10602},
      }
  • [Ko59] A. N. Kolmogorov, "Entropy per unit time as a metric invariant of automorphisms," Dokl. Akad. Nauk SSSR, vol. 124, pp. 754-755, 1959.
    @article {Ko59, MRKEY = {MR0103255},
      AUTHOR = {Kolmogorov, A. N.},
      TITLE = {Entropy per unit time as a metric invariant of automorphisms},
      JOURNAL = {Dokl. Akad. Nauk SSSR},
      FJOURNAL = {Doklady Akademii Nauk SSSR},
      VOLUME = {124},
      YEAR = {1959},
      PAGES = {754--755},
      ISSN = {0002-3264},
      MRCLASS = {28.00 (94.00)},
      MRNUMBER = {21 \#2035b},
      MRREVIEWER = {Y. N. Dowker},
      ZBLNUMBER={0086.10101},
      }
  • [LM95] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge: Cambridge Univ. Press, 1995.
    @book {LM95, MRKEY = {MR1369092},
      AUTHOR = {Lind, Douglas and Marcus, Brian},
      TITLE = {An Introduction to Symbolic Dynamics and Coding},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1995},
      PAGES = {xvi+495},
      ISBN = {0-521-55124-2; 0-521-55900-6},
      MRCLASS = {58F03 (15A48 54H20 58F20 94A24 94B60)},
      MRNUMBER = {97a:58050},
      MRREVIEWER = {Petr K{\.u}rka},
      }
  • [Or70a] D. Ornstein, "Bernoulli shifts with the same entropy are isomorphic," Advances in Math., vol. 4, pp. 337-352 (1970), 1970.
    @article {Or70a, MRKEY = {MR0257322},
      AUTHOR = {Ornstein, Donald},
      TITLE = {Bernoulli shifts with the same entropy are isomorphic},
      JOURNAL = {Advances in Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {4},
      YEAR = {1970},
      PAGES = {337--352 (1970)},
      ISSN = {0001-8708},
      MRCLASS = {28.70},
      MRNUMBER = {41 \#1973},
      MRREVIEWER = {U. Krengel},
      }
  • [Or70b] D. Ornstein, "Two Bernoulli shifts with infinite entropy are isomorphic," Advances in Math., vol. 5, pp. 339-348 (1970), 1970.
    @article {Or70b, MRKEY = {MR0274716},
      AUTHOR = {Ornstein, Donald},
      TITLE = {Two {B}ernoulli shifts with infinite entropy are isomorphic},
      JOURNAL = {Advances in Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {5},
      YEAR = {1970},
      PAGES = {339--348 (1970)},
      ISSN = {0001-8708},
      MRCLASS = {28.70},
      MRNUMBER = {43 \#478a},
      }
  • [OW87] D. S. Ornstein and B. Weiss, "Entropy and isomorphism theorems for actions of amenable groups," J. Analyse Math., vol. 48, pp. 1-141, 1987.
    @article {OW87, MRKEY = {MR910005},
      AUTHOR = {Ornstein, Donald S. and Weiss, Benjamin},
      TITLE = {Entropy and isomorphism theorems for actions of amenable groups},
      JOURNAL = {J. Analyse Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {48},
      YEAR = {1987},
      PAGES = {1--141},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {28D20 (22D40)},
      MRNUMBER = {88j:28014},
      MRREVIEWER = {John C. Kieffer},
      }
  • [Pa69] W. Parry, Entropy and Generators in Ergodic Theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
    @book {Pa69, MRKEY = {MR0262464},
      AUTHOR = {Parry, William},
      TITLE = {Entropy and Generators in Ergodic Theory},
      PUBLISHER = {W. A. Benjamin, Inc., New York-Amsterdam},
      YEAR = {1969},
      PAGES = {xii+124},
      MRCLASS = {28.70},
      MRNUMBER = {41 \#7071},
      MRREVIEWER = {W. L. Reddy},
      }
  • [Pe83] K. Petersen, Ergodic Theory, Cambridge: Cambridge University Press, 1983, vol. 2.
    @book {Pe83, MRKEY = {MR833286},
      AUTHOR = {Petersen, Karl},
      TITLE = {Ergodic Theory},
      SERIES = {Cambridge Studies in Advanced Mathematics},
      VOLUME = {2},
      PUBLISHER = {Cambridge University Press},
      ADDRESS = {Cambridge},
      YEAR = {1983},
      PAGES = {xii+329},
      ISBN = {0-521-23632-0},
      MRCLASS = {28-02 (28D05 54H20 58F11)},
      MRNUMBER = {87i:28002},
      MRREVIEWER = {Nathaniel F. G. Martin},
      }
  • [Ro67] V. A. Rohlin, "Lectures on the entropy theory of transformations with invariant measure," Uspehi Mat. Nauk, vol. 22, iss. 5 (137), pp. 3-56, 1967.
    @article {Ro67, MRKEY = {MR0217258},
      AUTHOR = {Rohlin, V. A.},
      TITLE = {Lectures on the entropy theory of transformations with invariant measure},
      JOURNAL = {Uspehi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {22},
      YEAR = {1967},
      NUMBER = {5 (137)},
      PAGES = {3--56},
      ISSN = {0042-1316},
      MRCLASS = {28.70},
      MRNUMBER = {36 \#349},
      MRREVIEWER = {D. Newton},
      }
  • [Ru90] D. J. Rudolph, Fundamentals of Measurable Dynamics, New York: The Clarendon Press Oxford Univ. Press, 1990.
    @book {Ru90, MRKEY = {MR1086631},
      AUTHOR = {Rudolph, Daniel J.},
      TITLE = {Fundamentals of Measurable Dynamics},
      SERIES = {Oxford Science Publications},
      PUBLISHER = {The Clarendon Press Oxford Univ. Press},
      ADDRESS = {New York},
      YEAR = {1990},
      PAGES = {x+168},
      ISBN = {0-19-853572-4},
      MRCLASS = {28Dxx},
      MRNUMBER = {92e:28006},
      MRREVIEWER = {Andr{é}s del Junco},
      }
  • [Si59] J. Sinaui, "On the concept of entropy for a dynamic system," Dokl. Akad. Nauk SSSR, vol. 124, pp. 768-771, 1959.
    @article {Si59, MRKEY = {MR0103256},
      AUTHOR = {Sina{\u\i},
      Ja.},
      TITLE = {On the concept of entropy for a dynamic system},
      JOURNAL = {Dokl. Akad. Nauk SSSR},
      FJOURNAL = {Doklady Akademii Nauk SSSR},
      VOLUME = {124},
      YEAR = {1959},
      PAGES = {768--771},
      ISSN = {0002-3264},
      MRCLASS = {28.00 (94.00)},
      MRNUMBER = {21 \#2036a},
      }
  • [St75] A. M. Stepin, "Bernoulli shifts on groups," Dokl. Akad. Nauk SSSR, vol. 223, iss. 2, pp. 300-302, 1975.
    @article {St75, MRKEY = {MR0409769},
      AUTHOR = {Stepin, A. M.},
      TITLE = {Bernoulli shifts on groups},
      JOURNAL = {Dokl. Akad. Nauk SSSR},
      FJOURNAL = {Doklady Akademii Nauk SSSR},
      VOLUME = {223},
      YEAR = {1975},
      NUMBER = {2},
      PAGES = {300--302},
      ISSN = {0002-3264},
      MRCLASS = {28A65},
      MRNUMBER = {53 \#13521},
      MRREVIEWER = {R. K. Thomas},
      }
  • [Wi73] R. F. Williams, "Classification of subshifts of finite type," Ann. of Math., vol. 98, pp. 120-153; errata, ibid. 99 (1974), 380, 1973.
    @article {Wi73, MRKEY = {MR0331436},
      AUTHOR = {Williams, R. F.},
      TITLE = {Classification of subshifts of finite type},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {98},
      YEAR = {1973},
      PAGES = {120--153; errata, ibid. {\bf 99} (1974), 380--381},
      ISSN = {0003-486X},
      MRCLASS = {58F20 (28A65)},
      MRNUMBER = {48 \#9769},
      MRREVIEWER = {C. B. Thomas},
      }

Keywords
, , ,
Mathematical Subject Classification
Primary:
DOI
http://doi.org/10.4007/annals.2010.171.1387
MR
2630067
zbMATH
1201.37007
Milestones
Received: 14 February 2009
Accepted: 16 February 2009
Published online: 25 March 2010

Authors

Lewis Phylip Bowen

Mathematics Department
University of Hawaii at Manoa
2565 McCarthy Mall
Honolulu, HI 96822
United States
and
Mathematics Department
Mailstop 3368
Texas A&M University
College Station, TX 77843-3368
United States