# 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.  Google Scholar [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.  Google Scholar [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer, Berlin, 1975.  Google Scholar [4] R. Bowen, Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17.  doi: 10.1007/BF01941319.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. Holland, Slowly mixing systems and intermittency maps, Ergod. Th. and Dynam. Sys., 25 (2005), 133-159.  doi: 10.1017/S0143385704000343.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] O. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653.  doi: 10.1007/s00222-002-0248-5.  Google Scholar [24] F. Schweiger, Some remarks on ergodicity and invariant measures, Michigan Math. J., 22 (1975), 308-318.  doi: 10.1307/mmj/1029001477.  Google Scholar [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.  Google Scholar [26] M. Thaler, Transformations on $[0, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] Y. L. Ye, Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733.  doi: 10.1088/1361-6544/aa639e.  Google Scholar [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.  Google Scholar [31] L. S. Young, Recurence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer, Berlin, 1975.  Google Scholar [4] R. Bowen, Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17.  doi: 10.1007/BF01941319.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. Holland, Slowly mixing systems and intermittency maps, Ergod. Th. and Dynam. Sys., 25 (2005), 133-159.  doi: 10.1017/S0143385704000343.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] O. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653.  doi: 10.1007/s00222-002-0248-5.  Google Scholar [24] F. Schweiger, Some remarks on ergodicity and invariant measures, Michigan Math. J., 22 (1975), 308-318.  doi: 10.1307/mmj/1029001477.  Google Scholar [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.  Google Scholar [26] M. Thaler, Transformations on $[0, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] Y. L. Ye, Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733.  doi: 10.1088/1361-6544/aa639e.  Google Scholar [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.  Google Scholar [31] L. S. Young, Recurence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002 [2] Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409 [3] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [4] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [5] Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404 [6] Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 [7] Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040 [8] Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003 [9] Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377 [10] Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325 [11] Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097 [12] Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 [13] Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $BV$ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405 [14] Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400 [15] Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399 [16] Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 [17] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [18] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [19] Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006 [20] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

2019 Impact Factor: 1.338

## Metrics

• HTML views (144)
• Cited by (0)

• on AIMS