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.
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@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},
} -
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@article {Ki75, MRKEY = {MR0393422},
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NUMBER = {6},
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MRNUMBER = {52 \#14232},
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} -
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@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},
} -
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@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},
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ISSN = {0002-3264},
MRCLASS = {28.00 (94.00)},
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ZBLNUMBER={0086.10101},
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@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},
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ISBN = {0-521-55124-2; 0-521-55900-6},
MRCLASS = {58F03 (15A48 54H20 58F20 94A24 94B60)},
MRNUMBER = {97a:58050},
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} -
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@article {Or70a, MRKEY = {MR0257322},
AUTHOR = {Ornstein, Donald},
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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},
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} -
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@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},
} -
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JOURNAL = {J. Analyse Math.},
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YEAR = {1987},
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ISSN = {0021-7670},
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@book {Pa69, MRKEY = {MR0262464},
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SERIES = {Cambridge Studies in Advanced Mathematics},
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JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
VOLUME = {22},
YEAR = {1967},
NUMBER = {5 (137)},
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} -
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@book {Ru90, MRKEY = {MR1086631},
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SERIES = {Oxford Science Publications},
PUBLISHER = {The Clarendon Press Oxford Univ. Press},
ADDRESS = {New York},
YEAR = {1990},
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ISBN = {0-19-853572-4},
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@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},
} -
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@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},
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ISSN = {0002-3264},
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} -
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@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},
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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},
}