2019, 15: 345-423. doi: 10.3934/jmd.2019024

From odometers to circular systems: A global structure theorem

1. 

Department of Mathematics, University of California, Irvine, CA 92697, USA

2. 

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received  November 27, 2017 Revised  May 2019 Published  December 2019

Fund Project: Supported in part by NSF grant DMS-1700143.

The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings that preserve underlying timing factors. The classes are canonically isomorphic by a continuous map that takes synchronous and anti-synchronous factor maps to synchronous and anti-synchronous factor maps, synchronous and anti-synchronous measure-isomorphisms to synchronous and anti-synchronous measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. The first class includes all finite entropy ergodic transformations that have an odometer factor. By results in [6], the second class contains all transformations realizable as diffeomorphisms using the untwisted Anosov–Katok method. An application of the main result will appear in a forthcoming paper [7] that shows that the diffeomorphisms of the torus are inherently unclassifiable up to measure-isomorphism. Other consequences include the existence of measure distal diffeomorphisms of arbitrary countable distal height.

Citation: Matthew Foreman, Benjamin Weiss. From odometers to circular systems: A global structure theorem. Journal of Modern Dynamics, 2019, 15: 345-423. doi: 10.3934/jmd.2019024
References:
[1]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3-36. 

[2]

F. Beleznay and M. Foreman, The complexity of the collection of measure-distal transformations, Ergodic Theory Dynam. Systems, 16 (1996), 929-962.  doi: 10.1017/S0143385700010129.

[3]

T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math., 74 (1991), 241-256.  doi: 10.1007/BF02775789.

[4]

J. Feldman, Borel structures and invariants for measurable transformations, Proc. Amer. Math. Soc., 46 (1974), 383-394.  doi: 10.1090/S0002-9939-1974-0355002-5.

[5]

M. ForemanD. J. Rudolph and B. Weiss, The conjugacy problem in ergodic theory, Ann. of Math. (2), 173 (2011), 1529-1586.  doi: 10.4007/annals.2011.173.3.7.

[6]

M. Foreman and B. Weiss, A symbolic representation of Anosov-Katok systems, J. Anal. Math., 137 (2019), 603-661.  doi: 10.1007/s11854-019-0010-1.

[7]

M. Foreman and B. Weiss, Measure preserving diffeomorphisms of the torus are unclassifiable, arXiv: 1705.04414, 2017.

[8]

M. Foreman and B. Weiss, Odometer and circular systems: measure invariant under diffeomorphisms, in preparation.

[9]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.

[10]

H. Furstenberg and B. Weiss, A mean ergodic theorem for $(1/N)\sum^N_{n = 1}f(T^nx)g(T^{n^2}x)$, in Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996,193–227.

[11]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[12]

P. R. Halmos, Lectures on Ergodic Theory, Chelsea Publishing Co., New York, 1960.

[13]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. of Math. (2), 43 (1942), 332-350.  doi: 10.2307/1968872.

[14]

G. Hjorth, On invariants for measure preserving transformations, Fund. Math., 169 (2001), 51-84.  doi: 10.4064/fm169-1-2.

[15]

A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, 30, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/030.

[16]

K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1989.

[17]

D. J. Rudolph, Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990.

[18]

P. C. Shields, The Ergodic Theory of Discrete Sample Paths, Graduate Studies in Mathematics, 13, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/gsm/013.

[19]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.  doi: 10.1007/BF01667386.

[20]

J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. (2), 33 (1932), 587-642.  doi: 10.2307/1968537.

[21]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[22]

B. Weiss, Single Orbit Dynamics, CBMS Regional Conference Series in Mathematics, 95, American Mathematical Society, Providence, RI, 2000.

[23]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.

show all references

References:
[1]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3-36. 

[2]

F. Beleznay and M. Foreman, The complexity of the collection of measure-distal transformations, Ergodic Theory Dynam. Systems, 16 (1996), 929-962.  doi: 10.1017/S0143385700010129.

[3]

T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math., 74 (1991), 241-256.  doi: 10.1007/BF02775789.

[4]

J. Feldman, Borel structures and invariants for measurable transformations, Proc. Amer. Math. Soc., 46 (1974), 383-394.  doi: 10.1090/S0002-9939-1974-0355002-5.

[5]

M. ForemanD. J. Rudolph and B. Weiss, The conjugacy problem in ergodic theory, Ann. of Math. (2), 173 (2011), 1529-1586.  doi: 10.4007/annals.2011.173.3.7.

[6]

M. Foreman and B. Weiss, A symbolic representation of Anosov-Katok systems, J. Anal. Math., 137 (2019), 603-661.  doi: 10.1007/s11854-019-0010-1.

[7]

M. Foreman and B. Weiss, Measure preserving diffeomorphisms of the torus are unclassifiable, arXiv: 1705.04414, 2017.

[8]

M. Foreman and B. Weiss, Odometer and circular systems: measure invariant under diffeomorphisms, in preparation.

[9]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.

[10]

H. Furstenberg and B. Weiss, A mean ergodic theorem for $(1/N)\sum^N_{n = 1}f(T^nx)g(T^{n^2}x)$, in Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996,193–227.

[11]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[12]

P. R. Halmos, Lectures on Ergodic Theory, Chelsea Publishing Co., New York, 1960.

[13]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. Ⅱ, Ann. of Math. (2), 43 (1942), 332-350.  doi: 10.2307/1968872.

[14]

G. Hjorth, On invariants for measure preserving transformations, Fund. Math., 169 (2001), 51-84.  doi: 10.4064/fm169-1-2.

[15]

A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, 30, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/030.

[16]

K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1989.

[17]

D. J. Rudolph, Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990.

[18]

P. C. Shields, The Ergodic Theory of Discrete Sample Paths, Graduate Studies in Mathematics, 13, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/gsm/013.

[19]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.  doi: 10.1007/BF01667386.

[20]

J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. (2), 33 (1932), 587-642.  doi: 10.2307/1968537.

[21]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[22]

B. Weiss, Single Orbit Dynamics, CBMS Regional Conference Series in Mathematics, 95, American Mathematical Society, Providence, RI, 2000.

[23]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.

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