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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Commun. Math. Phys., 168 (1995), 571-580.  Google Scholar

[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.  Google Scholar

[5]

A. O. Gel'fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809-814.  Google Scholar

[6]

G. Helmberg, Über konservative Transformationen, Math. Annalen, 165 (1966), 44-61.  Google Scholar

[7]

F. Hofbauer, $\beta $-shifts have unique maximal measure, Mh. Math., 85 (1978), 189-198.  Google Scholar

[8]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Isr. J. Math., 34 (1979), 213-237. doi: 10.1007/BF02760884.  Google Scholar

[9]

S. Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Sci. Tokyo, 19 (1943), 635-641.  Google Scholar

[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.  Google Scholar

[11]

G. Keller, Lifting measures to Markov extensions, Mh. Math., 108 (1989), 183-200. doi: 10.1007/BF01308670.  Google Scholar

[12]

W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  Google Scholar

[13]

K. Petersen, "Ergodic Theory," Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1983.  Google Scholar

[14]

M. Thaler, Transformations on [0,1] with infinite invariant measures, Isr. J. Math., 46 (1983), 67-96. doi: 10.1007/BF02760623.  Google Scholar

[15]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. (2), 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar

[16]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Zweimüller, Measure preserving extensions and minimal wandering rates, Israel J. Math., 181 (2011), 295-303. doi: 10.1007/s11856-011-0009-5.  Google Scholar

show all references

References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50, AMS, Providence, RI, 1997.  Google Scholar

[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.  Google Scholar

[3]

H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Commun. Math. Phys., 168 (1995), 571-580.  Google Scholar

[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.  Google Scholar

[5]

A. O. Gel'fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809-814.  Google Scholar

[6]

G. Helmberg, Über konservative Transformationen, Math. Annalen, 165 (1966), 44-61.  Google Scholar

[7]

F. Hofbauer, $\beta $-shifts have unique maximal measure, Mh. Math., 85 (1978), 189-198.  Google Scholar

[8]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Isr. J. Math., 34 (1979), 213-237. doi: 10.1007/BF02760884.  Google Scholar

[9]

S. Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Sci. Tokyo, 19 (1943), 635-641.  Google Scholar

[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.  Google Scholar

[11]

G. Keller, Lifting measures to Markov extensions, Mh. Math., 108 (1989), 183-200. doi: 10.1007/BF01308670.  Google Scholar

[12]

W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  Google Scholar

[13]

K. Petersen, "Ergodic Theory," Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1983.  Google Scholar

[14]

M. Thaler, Transformations on [0,1] with infinite invariant measures, Isr. J. Math., 46 (1983), 67-96. doi: 10.1007/BF02760623.  Google Scholar

[15]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. (2), 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar

[16]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Zweimüller, Measure preserving extensions and minimal wandering rates, Israel J. Math., 181 (2011), 295-303. doi: 10.1007/s11856-011-0009-5.  Google Scholar

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