Amenable relations for endomorphisms

Hawkins, J. M.

doi:10.1090/S0002-9947-1994-1179396-3

Amenable relations for endomorphisms
HTML articles powered by AMS MathViewer

by J. M. Hawkins PDF

Trans. Amer. Math. Soc. 343 (1994), 169-191 Request permission

Abstract:

We give necessary and sufficient conditions for an endomorphism to admit an equivalent invariant $\sigma$-finite measure in terms of a generalized Perron-Frobenius operator. The assumptions are that the endomorphism is nonsingular (preserves sets of measure zero), conservative, and finite-to-1. We study two orbit equivalence relations associated to an endomorphism, and their connections to nonsingularity, ergodicity, and exactness. We also discuss Radon-Nikodym derivative cocycles for the relations and the endomorphism, and relate these to the Jacobian of the endomorphism.

References

Jonathan Aaronson, Michael Lin, and Benjamin Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products, Israel J. Math. 33 (1979), no. 3-4, 198–224 (1980). A collection of invited papers on ergodic theory. MR 571530, DOI 10.1007/BF02762161
Gavin Brown and A. H. Dooley, Ergodic measures are of weak product type, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 1, 129–145. MR 789727, DOI 10.1017/S0305004100063325
Roger Butler and Klaus Schmidt, An information cocycle for groups of nonsingular transformations, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 3, 347–360. MR 787603, DOI 10.1007/BF00532739
A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI 10.1017/s014338570000136x
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
Karma Dajani and Jane Hawkins, Rohlin factors, product factors, and joinings for $n$-to-one maps, Indiana Univ. Math. J. 42 (1993), no. 1, 237–258. MR 1218714, DOI 10.1512/iumj.1993.42.42012
Stanley J. Eigen and Cesar E. Silva, A structure theorem for $n$-to-$1$ endomorphisms and existence of nonrecurrent measures, J. London Math. Soc. (2) 40 (1989), no. 3, 441–451. MR 1053613, DOI 10.1112/jlms/s2-40.3.441
Toshihiro Hamachi, On a Bernoulli shift with nonidentical factor measures, Ergodic Theory Dynam. Systems 1 (1981), no. 3, 273–283 (1982). MR 662470, DOI 10.1017/s0143385700001255
Toshihiro Hamachi and Motosige Osikawa, Ergodic groups of automorphisms and Krieger’s theorems, Seminar on Mathematical Sciences, vol. 3, Keio University, Department of Mathematics, Yokohama, 1981. MR 617740
Jane M. Hawkins, Ratio sets of endomorphisms which preserve a probability measure, Measure and measurable dynamics (Rochester, NY, 1987) Contemp. Math., vol. 94, Amer. Math. Soc., Providence, RI, 1989, pp. 159–169. MR 1012986, DOI 10.1090/conm/094/1012986
Jane M. Hawkins and Cesar E. Silva, Remarks on recurrence and orbit equivalence of nonsingular endomorphisms, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 281–290. MR 970561, DOI 10.1007/BFb0082837

Noninvertible transformations admitting no absolutely continuous

finite invariant measure

111

Shizuo Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. MR 23331, DOI 10.2307/1969123
Wolfgang Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), no. 1, 19–70. MR 415341, DOI 10.1007/BF01360278
Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464
V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 55. MR 0047744

Exact endomorphisms of a Lebesgue space

39

Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Co. of India, Ltd., Delhi, 1977. MR 0578731
Klaus Schmidt, Strong ergodicity and quotients of equivalence relations, Miniconferences on harmonic analysis and operator algebras (Canberra, 1987) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 16, Austral. Nat. Univ., Canberra, 1988, pp. 300–311. MR 954006
Klaus Schmidt, Algebraic ideas in ergodic theory, CBMS Regional Conference Series in Mathematics, vol. 76, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1074576, DOI 10.1090/cbms/076
F. Schweiger, Numbertheoretical endomorphisms with $\sigma$-finite invariant measure, Israel J. Math. 21 (1975), no. 4, 308–318. MR 384735, DOI 10.1007/BF02757992
Cesar E. Silva, On $\mu$-recurrent nonsingular endomorphisms, Israel J. Math. 61 (1988), no. 1, 1–13. MR 937578, DOI 10.1007/BF02776298
Cesar E. Silva and Philippe Thieullen, The subadditive ergodic theorem and recurrence properties of Markovian transformations, J. Math. Anal. Appl. 154 (1991), no. 1, 83–99. MR 1087960, DOI 10.1016/0022-247X(91)90072-8
Peter Walters, Some results on the classification of non-invertible measure preserving transformations, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., Vol. 318, Springer, Berlin, 1973, pp. 266–276. MR 0393424
Ulrich Krengel, Transformations without finite invariant measure have finite strong generators, Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970) Springer, Berlin, 1970, pp. 133–157. MR 0269808