# American Institute of Mathematical Sciences

December  2021, 41(12): 5887-5914. doi: 10.3934/dcds.2021100

## On the number of invariant measures for random expanding maps in higher dimensions

 School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

* Corresponding author: Fawwaz Batayneh

Received  June 2020 Revised  April 2021 Published  December 2021 Early access  June 2021

In [22], Jabłoński proved that a piecewise expanding $C^{2}$ multidimensional Jabłoński map admits an absolutely continuous invariant probability measure (ACIP). In [6], Boyarsky and Lou extended this result to the case of i.i.d. compositions of the above maps, with an on average expanding condition. We generalize these results to the (quenched) setting of random Jabłoński maps, where the randomness is governed by an ergodic, invertible and measure preserving transformation. We prove that the skew product associated to this random dynamical system admits a finite number of ergodic ACIPs. Furthermore, we provide two different upper bounds on the number of mutually singular ergodic ACIPs, motivated by the works of Buzzi [9] in one dimension and Góra, Boyarsky and Proppe [19] in higher dimensions.

Citation: Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
##### References:
 [1] K. Adl-Zarabi, Absolutely continuous invariant measures for piecewise expanding $C^{2}$ transformations in $\mathbb{R} ^{N}$ on domains with cusps on the boundaries, Ergodic Theory and Dynamical Systems, 16 (1996), 1-18.  doi: 10.1017/S0143385700008683.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems, Courier Corporation, 2000.  Google Scholar [3] V. Araujo and J. Solano, Absolutely continuous invariant measures for random non-uniformly expanding maps, Mathematische Zeitschrift, 3-4 (2014), 1199-1235.  doi: 10.1007/s00209-014-1300-z.  Google Scholar [4] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [5] C. Bose, On the existence and approximation of invariant densities for nonsingular transformations on $\mathbb{R} ^{d}$, Journal of Approximation Theory, 79 (1994), 260-270.  doi: 10.1006/jath.1994.1128.  Google Scholar [6] A. Boyarsky and Y. Lou, Existence of absolutely continuous invariant measures for higher-dimensional random maps, Dynamics and Stability of Systems, 7 (1992), 233-244.  doi: 10.1080/02681119208806141.  Google Scholar [7] A. Boyarsky and Y. Lou, Approximating measures invariant under higher-dimensional chaotic transformations, Journal of Approximation Theory, 65 (1991), 231-244.  doi: 10.1016/0021-9045(91)90105-J.  Google Scholar [8] A. Boyarsky, P. Góra and Y. S. Lou, Constructive approximations to the invariant densities of higher-dimensional transformations, Constructive Approximation, 10 (1994), 1-13.  doi: 10.1007/BF01205163.  Google Scholar [9] J. Buzzi, Absolutely continuous SRB measures for random Lasota-Yorke maps, Transactions of the American Mathematical Society, 352 (2000), 3289-3303.  doi: 10.1090/S0002-9947-00-02607-6.  Google Scholar [10] J. Buzzi, No or infinitely many ACIP for piecewise expanding $C^{r}$ maps in higher dimensions, Communications in Mathematical Physics, 3 (2001), 495-501.  doi: 10.1007/s002200100509.  Google Scholar [11] W. J. Cowieson, Stochastic stability for piecewise expanding maps in $\mathbb{R} ^{d}$, Nonlinearity, 13 (2000), 1745-1760.  doi: 10.1088/0951-7715/13/5/316.  Google Scholar [12] G. Froyland, S. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operators cocycles, Discrete and Continuous Dynamical Systems, 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.  Google Scholar [13] I. Ghenciu, Weakly precompact subsets of $L^{1}(\mu, X)$, Colloquium Mathematicum, 129 (2012), 133-143.  doi: 10.4064/cm129-1-10.  Google Scholar [14] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar [15] C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory and Dynamical Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.  Google Scholar [16] C. González-Tokman and A. Quas, Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles, Journal of the European Mathematical Society, (to appear), arXiv: 1806.08873. Google Scholar [17] P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^{2}$ transformations in $\mathbb{R} ^{N}$, Israel Journal of Mathematics, 67 (1989), 272-286.  doi: 10.1007/BF02764946.  Google Scholar [18] P. Góra and A. Boyarsky, Higher-dimensional point transformations and asymptotic measures for cellular automata, Computers and Mathematics with Applications, 19 (1990), 13-31.  doi: 10.1016/0898-1221(90)90247-H.  Google Scholar [19] P. Góra, A. Boyarsky and H. Proppe, On the number of invariant measures for higher-dimensional chaotic transformations, Journal of Statistical Physics, 62 (1991), 709-728.  doi: 10.1007/BF01017979.  Google Scholar [20] L. Hsieh, Ergodic Theory of Multidimensional Random Dynamical Systems, Master thesis, University of Victoria, 2008. Google Scholar [21] C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d'opérations non complètement continues, Annals of Mathematics, (1950), 140–147. doi: 10.2307/1969514.  Google Scholar [22] M. Jabłoński, On invariant measures for piecewise $C^{2}$-transformations of the $n$- dimensional cube, Annales Polonici Mathematici, 2 (1983), 185-195.  doi: 10.4064/ap-43-2-185-195.  Google Scholar [23] G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory and Dynamical Systems, 10 (1990), 717-744.  doi: 10.1017/S0143385700005861.  Google Scholar [24] G. Keller, Propriétés Ergordiques Des Endomorphismes Dilatants, $C^{2}$ Par Morceaux, Des Régions Bornées Du Plan, Thèse, Université de Rennes, 1979. Google Scholar [25] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society, (1973), 481–488. doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar [26] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182.  doi: 10.1017/S0143385711000939.  Google Scholar [27] T. Morita, Random iteration of one-dimensional transformations, Osaka Journal of Mathematics, 22 (1985), 489-518.   Google Scholar [28] F. Nakamura and H. Toyokawa, Random invariant densities for Markov operator cocycles and random mean ergodic theorem, preprint, arXiv: 2101.04878. Google Scholar [29] V. I. Oseledets, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 19 (1968), 179-210.   Google Scholar [30] S. Pelikan, Invariant densities for random maps of the interval, Transactions of the American Mathematical Society, 281 (1984), 813-825.  doi: 10.1090/S0002-9947-1984-0722776-1.  Google Scholar [31] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal of Mathematics, 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar [32] P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987) 49–97. doi: 10.1016/S0294-1449(16)30373-0.  Google Scholar [33] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete and Continuous Dynamical Systems, 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917.  Google Scholar [34] S. Ulam and J. von Neumann, Random ergodic theorem, Bulletin of the American Mathematical Society, 51 (1947), 660. doi: 10.1090/S0002-9904-1958-10189-5.  Google Scholar

show all references

##### References:
 [1] K. Adl-Zarabi, Absolutely continuous invariant measures for piecewise expanding $C^{2}$ transformations in $\mathbb{R} ^{N}$ on domains with cusps on the boundaries, Ergodic Theory and Dynamical Systems, 16 (1996), 1-18.  doi: 10.1017/S0143385700008683.  Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems, Courier Corporation, 2000.  Google Scholar [3] V. Araujo and J. Solano, Absolutely continuous invariant measures for random non-uniformly expanding maps, Mathematische Zeitschrift, 3-4 (2014), 1199-1235.  doi: 10.1007/s00209-014-1300-z.  Google Scholar [4] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [5] C. Bose, On the existence and approximation of invariant densities for nonsingular transformations on $\mathbb{R} ^{d}$, Journal of Approximation Theory, 79 (1994), 260-270.  doi: 10.1006/jath.1994.1128.  Google Scholar [6] A. Boyarsky and Y. Lou, Existence of absolutely continuous invariant measures for higher-dimensional random maps, Dynamics and Stability of Systems, 7 (1992), 233-244.  doi: 10.1080/02681119208806141.  Google Scholar [7] A. Boyarsky and Y. Lou, Approximating measures invariant under higher-dimensional chaotic transformations, Journal of Approximation Theory, 65 (1991), 231-244.  doi: 10.1016/0021-9045(91)90105-J.  Google Scholar [8] A. Boyarsky, P. Góra and Y. S. Lou, Constructive approximations to the invariant densities of higher-dimensional transformations, Constructive Approximation, 10 (1994), 1-13.  doi: 10.1007/BF01205163.  Google Scholar [9] J. Buzzi, Absolutely continuous SRB measures for random Lasota-Yorke maps, Transactions of the American Mathematical Society, 352 (2000), 3289-3303.  doi: 10.1090/S0002-9947-00-02607-6.  Google Scholar [10] J. Buzzi, No or infinitely many ACIP for piecewise expanding $C^{r}$ maps in higher dimensions, Communications in Mathematical Physics, 3 (2001), 495-501.  doi: 10.1007/s002200100509.  Google Scholar [11] W. J. Cowieson, Stochastic stability for piecewise expanding maps in $\mathbb{R} ^{d}$, Nonlinearity, 13 (2000), 1745-1760.  doi: 10.1088/0951-7715/13/5/316.  Google Scholar [12] G. Froyland, S. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operators cocycles, Discrete and Continuous Dynamical Systems, 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.  Google Scholar [13] I. Ghenciu, Weakly precompact subsets of $L^{1}(\mu, X)$, Colloquium Mathematicum, 129 (2012), 133-143.  doi: 10.4064/cm129-1-10.  Google Scholar [14] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar [15] C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory and Dynamical Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.  Google Scholar [16] C. González-Tokman and A. Quas, Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles, Journal of the European Mathematical Society, (to appear), arXiv: 1806.08873. Google Scholar [17] P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^{2}$ transformations in $\mathbb{R} ^{N}$, Israel Journal of Mathematics, 67 (1989), 272-286.  doi: 10.1007/BF02764946.  Google Scholar [18] P. Góra and A. Boyarsky, Higher-dimensional point transformations and asymptotic measures for cellular automata, Computers and Mathematics with Applications, 19 (1990), 13-31.  doi: 10.1016/0898-1221(90)90247-H.  Google Scholar [19] P. Góra, A. Boyarsky and H. Proppe, On the number of invariant measures for higher-dimensional chaotic transformations, Journal of Statistical Physics, 62 (1991), 709-728.  doi: 10.1007/BF01017979.  Google Scholar [20] L. Hsieh, Ergodic Theory of Multidimensional Random Dynamical Systems, Master thesis, University of Victoria, 2008. Google Scholar [21] C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d'opérations non complètement continues, Annals of Mathematics, (1950), 140–147. doi: 10.2307/1969514.  Google Scholar [22] M. Jabłoński, On invariant measures for piecewise $C^{2}$-transformations of the $n$- dimensional cube, Annales Polonici Mathematici, 2 (1983), 185-195.  doi: 10.4064/ap-43-2-185-195.  Google Scholar [23] G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory and Dynamical Systems, 10 (1990), 717-744.  doi: 10.1017/S0143385700005861.  Google Scholar [24] G. Keller, Propriétés Ergordiques Des Endomorphismes Dilatants, $C^{2}$ Par Morceaux, Des Régions Bornées Du Plan, Thèse, Université de Rennes, 1979. Google Scholar [25] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society, (1973), 481–488. doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar [26] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182.  doi: 10.1017/S0143385711000939.  Google Scholar [27] T. Morita, Random iteration of one-dimensional transformations, Osaka Journal of Mathematics, 22 (1985), 489-518.   Google Scholar [28] F. Nakamura and H. Toyokawa, Random invariant densities for Markov operator cocycles and random mean ergodic theorem, preprint, arXiv: 2101.04878. Google Scholar [29] V. I. Oseledets, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 19 (1968), 179-210.   Google Scholar [30] S. Pelikan, Invariant densities for random maps of the interval, Transactions of the American Mathematical Society, 281 (1984), 813-825.  doi: 10.1090/S0002-9947-1984-0722776-1.  Google Scholar [31] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal of Mathematics, 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar [32] P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987) 49–97. doi: 10.1016/S0294-1449(16)30373-0.  Google Scholar [33] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete and Continuous Dynamical Systems, 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917.  Google Scholar [34] S. Ulam and J. von Neumann, Random ergodic theorem, Bulletin of the American Mathematical Society, 51 (1947), 660. doi: 10.1090/S0002-9904-1958-10189-5.  Google Scholar
$I^{2}$ partitioned into $25$ equal squares
Bounds in (45) (solid) and (46) (dashed)
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