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Non-uniformly expanding dynamical systems: Multi-dimension
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
Dynamical systems on the interval $ [0, 1] $, satisfying the Thaler's condition, have been extensively studied. In this paper we consider invariant density and statistical properties of non-uniformly expanding dynamical systems on $ {\Bbb{R}}^d $ ($ d \geq 1 $). We present a critical regular condition that is a supplement and a development of the Thaler's condition, and it is very closely related to Lamperti's criterion. Under this new condition, we offer a method for studying the dynamical systems. A continuity description of the invariant density is presented; and a convergence theorem for iterations of Perron-Frobenius operator is set up. Furthermore, we establish a more exact result for one-dimensional systems.
References:
[1] |
J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical surveys and monographs, Vol. 50, Amer. Math. Soc. 1997.
doi: 10.1090/surv/050. |
[2] |
V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, Vol. 16, World Scientific, Singapore, 2000.
doi: 10.1142/9789812813633. |
[3] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer, Berlin, 1975. |
[4] |
R. Bowen,
Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17.
doi: 10.1007/BF01941319. |
[5] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I, Wily-Interscience, New York, 1958. Google Scholar |
[6] |
S. Gouëzel,
Decay of correlations for nonuniformly expanding systems, Bull. Soc. math. France, 134 (2006), 1-31.
doi: 10.24033/bsmf.2500. |
[7] |
S. Gouézel,
Correlation asymptotics from large deviations in dynamical systems with infinite measure, Colloq. Math., 125 (2011), 193-212.
doi: 10.4064/cm125-2-5. |
[8] |
M. Holland,
Slowly mixing systems and intermittency maps, Ergod. Th. and Dynam. Sys., 25 (2005), 133-159.
doi: 10.1017/S0143385704000343. |
[9] |
H. Y. Hu,
Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergod. Th. and Dynam. Sys., 24 (2004), 495-524.
doi: 10.1017/S0143385703000671. |
[10] |
H. Y. Hu, Conditions for the existence of SBR measures of "almost Anosov'' diffeomorphisms, Trans. Amer. Math. Soc., 352 (2000), 2331-2367. Google Scholar |
[11] |
H. Y. Hu and S. Vaienti,
Absolutely continuous invariant measures for some non-uniformly expanding maps, Ergod. Th. and Dynam. Sys., 29 (2009), 1185-1215.
doi: 10.1017/S0143385708000576. |
[12] |
H. Y. Hu and L. S. Young,
Nonexistence of SBR measures for some diffeomorphisms that are "almost Anosov", Ergod. Th. and Dynam. Sys., 15 (1995), 67-76.
doi: 10.1017/S0143385700008245. |
[13] |
S. Ito and M. Yuri,
Number theoretical transformations with finite range structure and their ergodic properties, Tokyo J. Math., 10 (1987), 1-32.
doi: 10.3836/tjm/1270141789. |
[14] |
J. Lamperti,
An occupation time theorem for a class of stochastic processes, Trans. Amer. Math. Soc., 88 (1958), 380-387.
doi: 10.1090/S0002-9947-1958-0094863-X. |
[15] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic aspects of dynamics, Applied Math. Sci., 97 (2nd ed.), Springer, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[16] |
A. Lasota and J. A. Yorke,
On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.
doi: 10.1090/S0002-9947-1973-0335758-1. |
[17] |
K. S. Lau and Y. L. Ye,
Ruelle operator with nonexpansive IFS, Studia Math., 148 (2001), 143-169.
doi: 10.4064/sm148-2-4. |
[18] |
C. Liverani, B. Saussol and S. Vaienti,
A probabilistic approach to intermittency, Ergod. Th. and Dynam. Sys., 19 (1999), 671-685.
doi: 10.1017/S0143385799133856. |
[19] |
I. Melbourne and D. Terhesiu,
Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math., 189 (2012), 61-110.
doi: 10.1007/s00222-011-0361-4. |
[20] |
I. Melbourne and D. Terhesiu,
Decay of correlations for non-uniformly expanding systems with general return times, Ergod. Th. and Dynam. Sys., 34 (2014), 893-918.
doi: 10.1017/etds.2012.158. |
[21] |
M. Pollicott and M. Yuri,
Statistical properties of maps with indifferent periodic points, Comm. Math. Phys., 217 (2001), 503-520.
doi: 10.1007/s002200100368. |
[22] |
T. Prellberg and J. Slawny,
Maps of intervals with indifferent fixed points: Thermodynamic formalism and phase transition, J. Statist. Phys., 66 (1992), 503-514.
doi: 10.1007/BF01060077. |
[23] |
O. Sarig,
Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653.
doi: 10.1007/s00222-002-0248-5. |
[24] |
F. Schweiger,
Some remarks on ergodicity and invariant measures, Michigan Math. J., 22 (1975), 308-318.
doi: 10.1307/mmj/1029001477. |
[25] |
M. Thaler,
Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.
doi: 10.1007/BF02788928. |
[26] |
M. Thaler,
Transformations on $[0, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.
doi: 10.1007/BF02760623. |
[27] |
M. Thaler,
A limit theorem for the Perron-Frobenius operator of transformations on $[0, 1]$ with indifferent fixed points, Israel J. Math., 91 (1995), 111-127.
doi: 10.1007/BF02761642. |
[28] |
M. Thaler,
Asymptotic distributions and large deviations for iterated maps with an indifferent fixed point, Stochastics and Dynamics, 5 (2005), 425-440.
doi: 10.1142/S0219493705001535. |
[29] |
Y. L. Ye,
Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733.
doi: 10.1088/1361-6544/aa639e. |
[30] |
Y. L. Ye,
Ruelle operator with weakly contractive iterated function systems, Ergod. Th. and Dynam. Sys., 33 (2013), 1265-1290.
doi: 10.1017/S0143385712000211. |
[31] |
L. S. Young,
Recurence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
[32] |
M. Yuri,
Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indag. Math. (N. S.), 6 (1995), 355-383.
doi: 10.1016/0019-3577(95)93202-L. |
[33] |
M. Yuri,
On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points, Nonlinearity, 15 (2002), 429-445.
doi: 10.1088/0951-7715/15/2/311. |
[34] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications. Ⅰ. Fixed-Point Theorems, Translated from the German by Peter R. Wadsack. Springer-Verlag, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[35] |
R. Zweimüller,
Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergod. Th. and Dynam. Sys., 20 (2000), 1519-1549.
doi: 10.1017/S0143385700000821. |
show all references
References:
[1] |
J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical surveys and monographs, Vol. 50, Amer. Math. Soc. 1997.
doi: 10.1090/surv/050. |
[2] |
V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, Vol. 16, World Scientific, Singapore, 2000.
doi: 10.1142/9789812813633. |
[3] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer, Berlin, 1975. |
[4] |
R. Bowen,
Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17.
doi: 10.1007/BF01941319. |
[5] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I, Wily-Interscience, New York, 1958. Google Scholar |
[6] |
S. Gouëzel,
Decay of correlations for nonuniformly expanding systems, Bull. Soc. math. France, 134 (2006), 1-31.
doi: 10.24033/bsmf.2500. |
[7] |
S. Gouézel,
Correlation asymptotics from large deviations in dynamical systems with infinite measure, Colloq. Math., 125 (2011), 193-212.
doi: 10.4064/cm125-2-5. |
[8] |
M. Holland,
Slowly mixing systems and intermittency maps, Ergod. Th. and Dynam. Sys., 25 (2005), 133-159.
doi: 10.1017/S0143385704000343. |
[9] |
H. Y. Hu,
Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergod. Th. and Dynam. Sys., 24 (2004), 495-524.
doi: 10.1017/S0143385703000671. |
[10] |
H. Y. Hu, Conditions for the existence of SBR measures of "almost Anosov'' diffeomorphisms, Trans. Amer. Math. Soc., 352 (2000), 2331-2367. Google Scholar |
[11] |
H. Y. Hu and S. Vaienti,
Absolutely continuous invariant measures for some non-uniformly expanding maps, Ergod. Th. and Dynam. Sys., 29 (2009), 1185-1215.
doi: 10.1017/S0143385708000576. |
[12] |
H. Y. Hu and L. S. Young,
Nonexistence of SBR measures for some diffeomorphisms that are "almost Anosov", Ergod. Th. and Dynam. Sys., 15 (1995), 67-76.
doi: 10.1017/S0143385700008245. |
[13] |
S. Ito and M. Yuri,
Number theoretical transformations with finite range structure and their ergodic properties, Tokyo J. Math., 10 (1987), 1-32.
doi: 10.3836/tjm/1270141789. |
[14] |
J. Lamperti,
An occupation time theorem for a class of stochastic processes, Trans. Amer. Math. Soc., 88 (1958), 380-387.
doi: 10.1090/S0002-9947-1958-0094863-X. |
[15] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic aspects of dynamics, Applied Math. Sci., 97 (2nd ed.), Springer, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[16] |
A. Lasota and J. A. Yorke,
On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.
doi: 10.1090/S0002-9947-1973-0335758-1. |
[17] |
K. S. Lau and Y. L. Ye,
Ruelle operator with nonexpansive IFS, Studia Math., 148 (2001), 143-169.
doi: 10.4064/sm148-2-4. |
[18] |
C. Liverani, B. Saussol and S. Vaienti,
A probabilistic approach to intermittency, Ergod. Th. and Dynam. Sys., 19 (1999), 671-685.
doi: 10.1017/S0143385799133856. |
[19] |
I. Melbourne and D. Terhesiu,
Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math., 189 (2012), 61-110.
doi: 10.1007/s00222-011-0361-4. |
[20] |
I. Melbourne and D. Terhesiu,
Decay of correlations for non-uniformly expanding systems with general return times, Ergod. Th. and Dynam. Sys., 34 (2014), 893-918.
doi: 10.1017/etds.2012.158. |
[21] |
M. Pollicott and M. Yuri,
Statistical properties of maps with indifferent periodic points, Comm. Math. Phys., 217 (2001), 503-520.
doi: 10.1007/s002200100368. |
[22] |
T. Prellberg and J. Slawny,
Maps of intervals with indifferent fixed points: Thermodynamic formalism and phase transition, J. Statist. Phys., 66 (1992), 503-514.
doi: 10.1007/BF01060077. |
[23] |
O. Sarig,
Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653.
doi: 10.1007/s00222-002-0248-5. |
[24] |
F. Schweiger,
Some remarks on ergodicity and invariant measures, Michigan Math. J., 22 (1975), 308-318.
doi: 10.1307/mmj/1029001477. |
[25] |
M. Thaler,
Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.
doi: 10.1007/BF02788928. |
[26] |
M. Thaler,
Transformations on $[0, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.
doi: 10.1007/BF02760623. |
[27] |
M. Thaler,
A limit theorem for the Perron-Frobenius operator of transformations on $[0, 1]$ with indifferent fixed points, Israel J. Math., 91 (1995), 111-127.
doi: 10.1007/BF02761642. |
[28] |
M. Thaler,
Asymptotic distributions and large deviations for iterated maps with an indifferent fixed point, Stochastics and Dynamics, 5 (2005), 425-440.
doi: 10.1142/S0219493705001535. |
[29] |
Y. L. Ye,
Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733.
doi: 10.1088/1361-6544/aa639e. |
[30] |
Y. L. Ye,
Ruelle operator with weakly contractive iterated function systems, Ergod. Th. and Dynam. Sys., 33 (2013), 1265-1290.
doi: 10.1017/S0143385712000211. |
[31] |
L. S. Young,
Recurence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
[32] |
M. Yuri,
Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indag. Math. (N. S.), 6 (1995), 355-383.
doi: 10.1016/0019-3577(95)93202-L. |
[33] |
M. Yuri,
On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points, Nonlinearity, 15 (2002), 429-445.
doi: 10.1088/0951-7715/15/2/311. |
[34] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications. Ⅰ. Fixed-Point Theorems, Translated from the German by Peter R. Wadsack. Springer-Verlag, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[35] |
R. Zweimüller,
Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergod. Th. and Dynam. Sys., 20 (2000), 1519-1549.
doi: 10.1017/S0143385700000821. |
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