Invariant measures for general(ized) induced transformations
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- by Roland Zweimüller PDF
- Proc. Amer. Math. Soc. 133 (2005), 2283-2295 Request permission
Abstract:
We show that the general(ized) induced transformation $T^{\tau }$ derived from an ergodic measure preserving transformation $T$ by means of an inducing time $\tau$ has an invariant measure canonically related to that of the original system iff a suitable induced version of $\tau$ is integrable. Moreover, we prove an Abramov-type entropy formula.References
- Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400, DOI 10.1090/surv/050
- Jon Aaronson, Rational ergodicity and a metric invariant for Markov shifts, Israel J. Math. 27 (1977), no. 2, 93–123. MR 584018, DOI 10.1007/BF02761661
- L.M.Abramov: Entropy of a derived automorphism. Amer. Math. Soc. Transl. Ser. II, 49 (1960), 162-176.
- H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys. 168 (1995), no. 3, 571–580. MR 1328254
- Jérôme Buzzi, Markov extensions for multi-dimensional dynamical systems, Israel J. Math. 112 (1999), 357–380. MR 1714974, DOI 10.1007/BF02773488
- N.Dunford, J.T.Schwartz: Linear operators I. Wiley 1957.
- Gilbert Helmberg, Über konservative Transformationen, Math. Ann. 165 (1966), 44–61 (German). MR 197679, DOI 10.1007/BF01351666
- Shizuo Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Tokyo 19 (1943), 635–641. MR 14222
- Gerhard Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989), no. 2-3, 183–200. MR 1026617, DOI 10.1007/BF01308670
- Ulrich Krengel, Entropy of conservative transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 (1967), 161–181. MR 218522, DOI 10.1007/BF00532635
- Y.Pesin, S.Senti: Equilibrium states for unimodal maps. Preprint 2003.
- Fritz Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1419320
- Maximilian Thaler, Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math. 46 (1983), no. 1-2, 67–96. MR 727023, DOI 10.1007/BF02760623
Additional Information
- Roland Zweimüller
- Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
- Email: r.zweimueller@imperial.ac.uk
- Received by editor(s): July 23, 2003
- Published electronically: March 14, 2005
- Additional Notes: This research was partially supported by the Austrian Science Foundation FWF, project P14734-MAT, and by an APART [Austrian programme for advanced research and technology] fellowship of the Austrian Academy of Sciences.
- Communicated by: Michael Handel
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2283-2295
- MSC (2000): Primary 28D05, 28D20, 37A05, 60G10, 60G40
- DOI: https://doi.org/10.1090/S0002-9939-05-07772-5
- MathSciNet review: 2138871