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Invariant measures for general induced maps and towers
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 |
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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