Examples of natural extensions of nonsingular endomorphisms
HTML articles powered by AMS MathViewer
- by K. G. Dajani and J. M. Hawkins PDF
- Proc. Amer. Math. Soc. 120 (1994), 1211-1217 Request permission
Abstract:
We extend a nonsingular noninvertible map $T$ on $(X,\mathcal {B},\mu )$ to a minimal nonsingular automorphism containing $T$ as a factor and preserving some measure theoretic properties of $T$. We call the automorphism the natural extension of $T$. We compute some examples of natural extensions: first when $T$ is a shift with $\mu$ a product measure, and then generalize the result to some nonexact examples as well.References
-
K. Dajani and J. Hawkins, Quotient relations and automorphic factors of countable-to-one maps, preprint, 1991.
- 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
- 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
- J. M. Hawkins, Amenable relations for endomorphisms, Trans. Amer. Math. Soc. 343 (1994), no. 1, 169–191. MR 1179396, DOI 10.1090/S0002-9947-1994-1179396-3
- Jane M. Hawkins and Cesar E. Silva, Noninvertible transformations admitting no absolutely continuous $\sigma$-finite invariant measure, Proc. Amer. Math. Soc. 111 (1991), no. 2, 455–463. MR 1045139, DOI 10.1090/S0002-9939-1991-1045139-X
- Shizuo Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224. MR 23331, DOI 10.2307/1969123
- 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 —, A skew product entropy for nonsingular endomorphisms, preprint, 1991.
- 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
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 1211-1217
- MSC: Primary 28D99
- DOI: https://doi.org/10.1090/S0002-9939-1994-1174489-4
- MathSciNet review: 1174489