Abstract
All common probability preserving transformations can be represented as elements of MPT, the group of measure preserving transformations of the unit interval with Lebesgue measure. This group has a natural Polish topology and the induced topology on the set of ergodic transformations is also Polish. Our main result is that the set of ergodic elements $T$ in MPT that are isomorphic to their inverse is a complete analytic set. This has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not Borel. This is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is Borel.
This result explains, perhaps, why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable. The construction that we use is general enough to show that the set of ergodic $T$’s with nontrivial centralizer is also complete analytic.
On the positive side we show that the isomorphism relation is Borel when restricted to the rank one transformations, which form a generic subset of MPT. It remains an open problem to find a good explicit method of checking when two rank one transformations are isomorphic.
-
[Orn-inv]
H. Anzai, "On an example of a measure preserving transformation which is not conjugate to its inverse," Proc. Japan Acad., vol. 27, pp. 517-522, 1951.
@article {Orn-inv, MRKEY = {0047742},
AUTHOR = {Anzai, Hirotada},
TITLE = {On an example of a measure preserving transformation which is not conjugate to its inverse},
JOURNAL = {Proc. Japan Acad.},
FJOURNAL = {Proceedings of the Japan Academy},
VOLUME = {27},
YEAR = {1951},
PAGES = {517--522},
ISSN = {0021-4280},
MRCLASS = {27.2X},
MRNUMBER = {0047742},
MRREVIEWER = {P. R. Halmos},
URL = {http://projecteuclid.org/euclid.pja/1195571227},
ZBLNUMBER = {0044.12502},
} -
[Ke-Beck]
H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Group Actions, Cambridge: Cambridge Univ. Press, 1996, vol. 232.
@book {Ke-Beck, MRKEY = {1425877},
AUTHOR = {Becker, Howard and Kechris, Alexander S.},
TITLE = {The Descriptive Set Theory of {P}olish Group Actions},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {232},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1996},
PAGES = {xii+136},
ISBN = {0-521-57605-9},
MRCLASS = {54H05 (04A15 04A20 28A05)},
MRNUMBER = {1425877},
MRREVIEWER = {Klaas Pieter Hart},
DOI = {10.1017/CBO9780511735264},
ZBLNUMBER = {0949.54052},
} -
[Foremancentral] M. Foreman, Examples of ergodic transformations with complicated centralizers.
@misc{Foremancentral,
author = {Foreman, Matthew},
TITLE = {Examples of ergodic transformations with complicated centralizers},
NOTE={in preparation},
sortyear = {2200},
} -
[Fo-survey] M. Foreman, "A descriptive view of ergodic theory," in Descriptive Set Theory and Dynamical Systems, Cambridge, 2000, pp. 87-171.
@inproceedings {Fo-survey, MRKEY = {1774425},
AUTHOR = {Foreman, Matthew},
TITLE = {A descriptive view of ergodic theory},
BOOKTITLE = {Descriptive Set Theory and Dynamical Systems},
VENUE={{M}arseille-{L}uminy, 1996},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {277},
PAGES = {87--171},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2000},
MRCLASS = {37A05 (03E15 28A05 37B05 54H05)},
MRNUMBER = {1774425},
MRREVIEWER = {A. I. Danilenko},
ZBLNUMBER = {0962.03043},
} -
[FRW] M. Foreman, D. J. Rudolph, and B. Weiss, Models for measure preserving transformations.
@misc{FRW,
author = {Foreman, Matthew and Rudolph, D. J. and Weiss, Benjamin},
TITLE={Models for measure preserving transformations},
NOTE={in preparation},
} -
[FoW]
M. Foreman and B. Weiss, "An anti-classification theorem for ergodic measure preserving transformations," J. Eur. Math. Soc. $($JEMS$)$, vol. 6, iss. 3, pp. 277-292, 2004.
@article {FoW, MRKEY = {2060477},
AUTHOR = {Foreman, Matthew and Weiss, Benjamin},
TITLE = {An anti-classification theorem for ergodic measure preserving transformations},
JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
VOLUME = {6},
YEAR = {2004},
NUMBER = {3},
PAGES = {277--292},
ISSN = {1435-9855},
MRCLASS = {37A35 (28D05 37A05 37A25)},
MRNUMBER = {2060477},
MRREVIEWER = {Nathaniel F. G. Martin},
URL = {http://link.springer.de/cgi/linkref?issn=1435-9855&year=04&volume=6&page=277},
ZBLNUMBER = {1063.37004},
} -
[furstbook] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton, N.J.: Princeton Univ. Press, 1981.
@book {furstbook, MRKEY = {0603625},
AUTHOR = {Furstenberg, H.},
TITLE = {Recurrence in Ergodic Theory and Combinatorial Number Theory},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1981},
PAGES = {xi+203},
ISBN = {0-691-08269-3},
MRCLASS = {28D05 (10K10 10L10 54H20)},
MRNUMBER = {0603625},
MRREVIEWER = {Michael Keane},
ZBLNUMBER = {0459.28023},
} -
[glasnerbook] E. Glasner, Ergodic Theory via Joinings, Providence, RI: Amer. Math. Soc., 2003, vol. 101.
@book {glasnerbook, MRKEY = {1958753},
AUTHOR = {Glasner, Eli},
TITLE = {Ergodic Theory via Joinings},
SERIES = {Math. Surveys Monogr.},
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 = {1958753},
MRREVIEWER = {Andr{é}s del Junco},
ZBLNUMBER = {1038.37002},
} -
[HvN]
P. R. Halmos and J. von Neumann, "Operator methods in classical mechanics. II," Ann. of Math., vol. 43, pp. 332-350, 1942.
@article {HvN, MRKEY = {0006617},
AUTHOR = {Halmos, Paul R. and von Neumann, John},
TITLE = {Operator methods in classical mechanics. {II}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {43},
YEAR = {1942},
PAGES = {332--350},
ISSN = {0003-486X},
MRCLASS = {46.3X},
MRNUMBER = {0006617},
MRREVIEWER = {J. L. Doob},
DOI = {10.2307/1968872},
ZBLNUMBER = {0063.01888},
} -
[Hj]
G. Hjorth, "On invariants for measure preserving transformations," Fund. Math., vol. 169, iss. 1, pp. 51-84, 2001.
@article {Hj, MRKEY = {1852353},
AUTHOR = {Hjorth, Greg},
TITLE = {On invariants for measure preserving transformations},
JOURNAL = {Fund. Math.},
FJOURNAL = {Fundamenta Mathematicae},
VOLUME = {169},
YEAR = {2001},
NUMBER = {1},
PAGES = {51--84},
ISSN = {0016-2736},
MRCLASS = {03E15 (28D05)},
MRNUMBER = {1852353},
MRREVIEWER = {Miroslav Repick{ý}},
DOI = {10.4064/fm169-1-2},
ZBLNUMBER = {0990.03036},
} -
[Hjbook] G. Hjorth, Classification and Orbit Equivalence Relations, Providence, RI: Amer. Math. Soc., 2000, vol. 75.
@book {Hjbook, MRKEY = {1725642},
AUTHOR = {Hjorth, Greg},
TITLE = {Classification and Orbit Equivalence Relations},
SERIES = {Math. Surveys Monogr.},
VOLUME = {75},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {xviii+195},
ISBN = {0-8218-2002-8},
MRCLASS = {03E15 (22A05 37A20 37B99 54H05 54H20)},
MRNUMBER = {1725642},
MRREVIEWER = {Miroslav Repick{ý}},
ZBLNUMBER = {0942.03056},
} -
[Kechbook] A. S. Kechris, Classical Descriptive Set Theory, New York: Springer-Verlag, 1995, vol. 156.
@book {Kechbook, MRKEY = {1321597},
AUTHOR = {Kechris, Alexander S.},
TITLE = {Classical Descriptive Set Theory},
SERIES = {Grad. Texts in Math.},
VOLUME = {156},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1995},
PAGES = {xviii+402},
ISBN = {0-387-94374-9},
MRCLASS = {03E15 (03-01 03-02 04A15 28A05 54H05 90D44)},
MRNUMBER = {1321597},
MRREVIEWER = {Jakub Jasi{ń}ski},
ZBLNUMBER = {0819.04002},
} -
[kerud]
A. S. Kechris, "On the concept of ${\Pi}^1_1$-completeness," Proc. Amer. Math. Soc., vol. 125, iss. 6, pp. 1811-1814, 1997.
@article {kerud, MRKEY = {1372034},
AUTHOR = {Kechris, Alexander S.},
TITLE = {On the concept of {${\Pi}^1_1$}-completeness},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {125},
YEAR = {1997},
NUMBER = {6},
PAGES = {1811--1814},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {03E15 (28A05 54H05)},
MRNUMBER = {1372034},
MRREVIEWER = {Howard Becker},
DOI = {10.1090/S0002-9939-97-03770-2},
ZBLNUMBER = {0864.03034},
} -
[Kingcent]
J. King, "The commutant is the weak closure of the powers, for rank-$1$ transformations," Ergodic Theory Dynam. Systems, vol. 6, iss. 3, pp. 363-384, 1986.
@article {Kingcent, MRKEY = {0863200},
AUTHOR = {King, Jonathan},
TITLE = {The commutant is the weak closure of the powers, for rank-{$1$} transformations},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {6},
YEAR = {1986},
NUMBER = {3},
PAGES = {363--384},
ISSN = {0143-3857},
MRCLASS = {28D05},
MRNUMBER = {0863200},
MRREVIEWER = {Nathaniel F. G. Martin},
DOI = {10.1017/S0143385700003552},
} -
[loustr]
A. Louveau and J. Saint-Raymond, "Borel classes and closed games: Wadge-type and Hurewicz-type results," Trans. Amer. Math. Soc., vol. 304, iss. 2, pp. 431-467, 1987.
@article {loustr, MRKEY = {0911079},
AUTHOR = {Louveau, A. and Saint-Raymond, J.},
TITLE = {Borel classes and closed games: {W}adge-type and {H}urewicz-type results},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {304},
YEAR = {1987},
NUMBER = {2},
PAGES = {431--467},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {03E15 (04A15 28A05 54H05)},
MRNUMBER = {0911079},
MRREVIEWER = {D. W. Bressler},
DOI = {10.2307/2000725},
ZBLNUMBER = {0655.04001},
} -
[or-cent] D. S. Ornstein, "On the root problem in ergodic theory," in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory, Berkeley, CA, 1972, pp. 347-356.
@inproceedings {or-cent, MRKEY = {0399415},
AUTHOR = {Ornstein, Donald S.},
TITLE = {On the root problem in ergodic theory},
BOOKTITLE = {Proceedings of the {S}ixth {B}erkeley {S}ymposium on {M}athematical {S}tatistics and {P}robability, {V}ol. {\rm II}: {P}robability Theory},
VENUE={{U}niv. {C}alifornia, {B}erkeley, {C}alif., 1970/1971},
PAGES = {347--356},
PUBLISHER = {Univ. California Press},
ADDRESS = {Berkeley, CA},
YEAR = {1972},
MRCLASS = {28A65},
MRNUMBER = {0399415},
MRREVIEWER = {Nathaniel Freidman},
ZBLNUMBER = {0262.28009},
} -
[orn] D. S. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, New Haven, Conn.: Yale Univ. Press, 1974, vol. 5.
@book {orn, MRKEY = {0447525},
AUTHOR = {Ornstein, Donald S.},
TITLE = {Ergodic Theory, Randomness, and Dynamical Systems},
NOTE = {James K. Whittemore Lectures in Mathematics given at Yale University},
SERIES={ Yale Math. Monogr.},
VOLUME={5},
PUBLISHER = {Yale Univ. Press},
ADDRESS = {New Haven, Conn.},
YEAR = {1974},
PAGES = {vii+141},
MRCLASS = {28A65},
MRNUMBER = {0447525},
MRREVIEWER = {Paul C. Shields},
ZBLNUMBER = {0296.28016},
} -
[dansbook] D. J. Rudolph, Fundamentals of Measurable Dynamics, New York: The Clarendon Press Oxford University Press, 1990.
@book {dansbook, MRKEY = {1086631},
AUTHOR = {Rudolph, Daniel J.},
TITLE = {Fundamentals of Measurable Dynamics},
SERIES = {Oxford Sci. Publ.},
NOTE = {Ergodic theory on Lebesgue spaces},
PUBLISHER = {The Clarendon Press Oxford University Press},
ADDRESS = {New York},
YEAR = {1990},
PAGES = {x+168},
ISBN = {0-19-853572-4},
MRCLASS = {28Dxx},
MRNUMBER = {1086631},
MRREVIEWER = {Andr{é}s del Junco},
ZBLNUMBER = {0718.28008},
} -
[Sarton] G. Sarton, A History of Science. Hellenistic Science and Culture in the Last Three Centuries B.C., Cambridge, MA: Harvard Univ. Press, 1959.
@book{Sarton,
author={Sarton, G.},
TITLE={A History of Science. Hellenistic Science and Culture in the Last Three Centuries B.C.},
PUBLISHER={Harvard Univ. Press},
ADDRESS={Cambridge, MA},
YEAR={1959},
ZBLNUMBER = {0092.24201},
} -
[Suslin] M. Suslin, "Sur une définition des ensembles B sans nombres transfinis," Compte Rendu Acad Science, vol. 164, pp. 88-91, 1917.
@article{Suslin,
author={Suslin, M.},
TITLE={Sur une définition des ensembles {B} sans nombres transfinis},
JOURNAL={Compte Rendu Acad Science},
VOLUME={164},
PAGES={88--91},
YEAR={1917},
} -
[1932]
J. von Neumann, "Zur Operatorenmethode in der klassischen Mechanik," Ann. of Math., vol. 33, iss. 3, pp. 587-642, 1932.
@article {1932, MRKEY = {1503078},
AUTHOR = {von Neumann, J.},
TITLE = {Zur {O}peratorenmethode in der klassischen {M}echanik},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {33},
YEAR = {1932},
NUMBER = {3},
PAGES = {587--642},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {Contributed Item},
MRNUMBER = {1503078},
DOI = {10.2307/1968537},
ZBLNUMBER = {0005.12203},
} -
[Wagon] S. Wagon, The Banach-Tarski Paradox, Cambridge: Cambridge Univ. Press, 1985, vol. 24.
@book {Wagon, MRKEY = {0803509},
AUTHOR = {Wagon, Stan},
TITLE = {The {B}anach-{T}arski Paradox},
SERIES = {Encyclopedia Math. Appl.},
VOLUME = {24},
NOTE = {with a foreword by Jan Mycielski},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1985},
PAGES = {xvi+251},
ISBN = {0-521-30244-7},
MRCLASS = {04A25 (03E25 04-02 28-02 43A05)},
MRNUMBER = {0803509},
MRREVIEWER = {W{\l}odzimierz Bzyl},
ZBLNUMBER = {0569.43001},
} -
[walters] P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982, vol. 79.
@book {walters, MRKEY = {0648108},
AUTHOR = {Walters, Peter},
TITLE = {An Introduction to Ergodic Theory},
SERIES = {Grad. Texts in Math.},
VOLUME = {79},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1982},
PAGES = {ix+250},
ISBN = {0-387-90599-5},
MRCLASS = {28Dxx (54H20 58F11)},
MRNUMBER = {0648108},
MRREVIEWER = {M. A. Akcoglu},
ZBLNUMBER = {0475.28009},
}