Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Amenable relations for endomorphisms


Author: J. M. Hawkins
Journal: Trans. Amer. Math. Soc. 343 (1994), 169-191
MSC: Primary 28D99
MathSciNet review: 1179396
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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, 10.1007/BF02762161
  • [2] 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, 10.1017/S0305004100063325
  • [3] 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, 10.1007/BF00532739
  • [4] A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems 1 (1981), no. 4, 431–450 (1982). MR 662736
  • [5] 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
  • [6] 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, 10.1512/iumj.1993.42.42012
  • [7] 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, 10.1112/jlms/s2-40.3.441
  • [8] Toshihiro Hamachi, On a Bernoulli shift with nonidentical factor measures, Ergodic Theory Dynamical Systems 1 (1981), no. 3, 273–283 (1982). MR 662470
  • [9] 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
  • [10] 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, 10.1090/conm/094/1012986
  • [11] 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, 10.1007/BFb0082837
  • [12] -, Noninvertible transformations admitting no absolutely continuous $ \sigma $-finite invariant measure, Proc. Amer. Math. Soc. 111 (1989), 455-463.
  • [13] Shizuo Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. MR 0023331
  • [14] Wolfgang Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), no. 1, 19–70. MR 0415341
  • [15] 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
  • [16] William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464
  • [17] V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 55. MR 0047744
  • [18] -, Exact endomorphisms of a Lebesgue space, Amer. Math. Soc. Transl. Ser. 2 39 (1964), 1-36.
  • [19] Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Company of India, Ltd., Delhi, 1977. Macmillan Lectures in Mathematics, Vol. 1. MR 0578731
  • [20] 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
  • [21] 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
  • [22] F. Schweiger, Numbertheoretical endomorphisms with 𝜎-finite invariant measure, Israel J. Math. 21 (1975), no. 4, 308–318. MR 0384735
  • [23] Cesar E. Silva, On 𝜇-recurrent nonsingular endomorphisms, Israel J. Math. 61 (1988), no. 1, 1–13. MR 937578, 10.1007/BF02776298
  • [24] 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, 10.1016/0022-247X(91)90072-8
  • [25] 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), Springer, Berlin, 1973, pp. 266–276. Lecture Notes in Math., Vol. 318. MR 0393424
  • [26] 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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28D99

Retrieve articles in all journals with MSC: 28D99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1179396-3
Article copyright: © Copyright 1994 American Mathematical Society