# American Institute of Mathematical Sciences

May 2019, 39(5): 2511-2553. doi: 10.3934/dcds.2019106

## Non-uniformly expanding dynamical systems: Multi-dimension

 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received  February 2018 Revised  August 2018 Published  January 2019

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.

Citation: Yuan-Ling Ye. Non-uniformly expanding dynamical systems: Multi-dimension. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2511-2553. doi: 10.3934/dcds.2019106
##### 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. [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. [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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##### 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. [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. [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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