Invariant measures for general induced maps and towers

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September 2013, 33(9): 3885-3901. doi: 10.3934/dcds.2013.33.3885

Invariant measures for general induced maps and towers

Arno Berger ^1, and Roland Zweimüller ^2,

1.	Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1, Canada
2.	Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria

Received August 2012 Revised February 2013 Published March 2013

Abstract
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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.

Keywords: jump transformation, Hofbauer tower, Young tower, infinite ergodic theory., Markov extension, Induced transformation.

Mathematics Subject Classification: Primary: 28D05, 37A40, 37C4.

Citation: Arno Berger, Roland Zweimüller. Invariant measures for general induced maps and towers. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3885-3901. doi: 10.3934/dcds.2013.33.3885

References:

[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.

show all references

References:

[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.

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