Article Contents
Article Contents

# Invariant measures for general induced maps and towers

• Absolutely continuous invariant measures (acims) for general induced transformations are shown to be related, in a natural way, to popular tower constructions regardless of any particulars of the latter. When combined with (an appropriate generalization of) the known integrability criterion for the existence of such acims, this leads to necessary and sufficient conditions under which acims can be lifted to, or projected from, nonsingular extensions.
Mathematics Subject Classification: Primary: 28D05, 37A40, 37C40.

 Citation:

•  [1] J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50, AMS, Providence, RI, 1997. [2] J. Aaronson and T. Meyerovitch, Absolutely continuous, invariant measures for dissipative, ergodic transformations, Colloq. Math., 110 (2008), 193-199.doi: 10.4064/cm110-1-7. [3] H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Commun. Math. Phys., 168 (1995), 571-580. [4] H. Bruin, M. Nicol and D. Terhesiu, On Young towers associated with infinite measure preserving transformations, Stoch. Dyn., 9 (2009), 635-655.doi: 10.1142/S0219493709002816. [5] A. O. Gel'fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809-814. [6] G. Helmberg, Über konservative Transformationen, Math. Annalen, 165 (1966), 44-61. [7] F. Hofbauer, $\beta$-shifts have unique maximal measure, Mh. Math., 85 (1978), 189-198. [8] F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Isr. J. Math., 34 (1979), 213-237.doi: 10.1007/BF02760884. [9] S. Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Sci. Tokyo, 19 (1943), 635-641. [10] G. Keller, Markov extensions, zeta-functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc., 314 (1989), 433-497.doi: 10.2307/2001395. [11] G. Keller, Lifting measures to Markov extensions, Mh. Math., 108 (1989), 183-200.doi: 10.1007/BF01308670. [12] W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. [13] K. Petersen, "Ergodic Theory," Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1983. [14] M. Thaler, Transformations on [0,1] with infinite invariant measures, Isr. J. Math., 46 (1983), 67-96.doi: 10.1007/BF02760623. [15] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. (2), 147 (1998), 585-650.doi: 10.2307/120960. [16] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.doi: 10.1007/BF02808180. [17] R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergod. Th. & Dynam. Sys., 20 (2000), 1519-1549.doi: 10.1017/S0143385700000821. [18] R. Zweimüller, Invariant measures for general(ized) induced transformations, Proc. Amer. Math. Soc., 133 (2005), 2283-2295.doi: 10.1090/S0002-9939-05-07772-5. [19] R. Zweimüller, Measure preserving extensions and minimal wandering rates, Israel J. Math., 181 (2011), 295-303.doi: 10.1007/s11856-011-0009-5.