Noninvertible transformations admitting no absolutely continuous 𝜎-finite invariant measure

Hawkins, Jane M.; Silva, Cesar E.

doi:10.1090/S0002-9939-1991-1045139-X

Noninvertible transformations admitting no absolutely continuous $\sigma$-finite invariant measure
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by Jane M. Hawkins and Cesar E. Silva PDF

Proc. Amer. Math. Soc. 111 (1991), 455-463 Request permission

Abstract:

We study a family of $n$-to-1 conservative ergodic endomorphisms which we will show to admit no $\sigma$-finite absolutely continuous invariant measure. We exhibit recurrent measures for these transformations and study their ratio sets; the examples can be realized as ${C^\infty }$ endomorphisms of the $2$-torus.

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
J. R. Choksi, J. M. Hawkins, and V. S. Prasad, Abelian cocycles for nonsingular ergodic transformations and the genericity of type $\textrm {III}_1$ transformations, Monatsh. Math. 103 (1987), no. 3, 187–205. MR 894170, DOI 10.1007/BF01364339
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
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
P. Góra and B. Schmitt, Un exemple de transformation dilatante et $C^1$ par morceaux de l’intervalle, sans probabilité absolument continue invariante, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 101–113 (French, with English summary). MR 991491, DOI 10.1017/S0143385700004831
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
Y. Katznelson, Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse Math. 31 (1977), 1–18. MR 486415, DOI 10.1007/BF02813295
Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
Wolfgang Krieger, On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space, Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970) Lecture Notes in Math., Vol. 160, Springer, Berlin, 1970, pp. 158–177. MR 0414823
A. Lasota and James A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488 (1974). MR 335758, DOI 10.1090/S0002-9947-1973-0335758-1
Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Co. of India, Ltd., Delhi, 1977. MR 0578731
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, Roots of $n\,:\,1$ measure-preserving transformations, J. London Math. Soc. 44 (1969), 7–14. MR 232913, DOI 10.1112/jlms/s1-44.1.7