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 and 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.

[2]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems, Courier Corporation, 2000.

[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.

[4]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.

[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.

[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.

[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.

[8]

A. BoyarskyP. 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.

[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.

[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.

[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.

[12]

G. FroylandS. 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.

[13]

I. Ghenciu, Weakly precompact subsets of $L^{1}(\mu, X)$, Colloquium Mathematicum, 129 (2012), 133-143.  doi: 10.4064/cm129-1-10.

[14]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. doi: 10.1007/978-1-4684-9486-0.

[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.

[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.

[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.

[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.

[19]

P. GóraA. 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.

[20]

L. Hsieh, Ergodic Theory of Multidimensional Random Dynamical Systems, Master thesis, University of Victoria, 2008.

[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.

[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.

[23]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory and Dynamical Systems, 10 (1990), 717-744.  doi: 10.1017/S0143385700005861.

[24]

G. Keller, Propriétés Ergordiques Des Endomorphismes Dilatants, $C^{2}$ Par Morceaux, Des Régions Bornées Du Plan, Thèse, Université de Rennes, 1979.

[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.

[26]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182.  doi: 10.1017/S0143385711000939.

[27]

T. Morita, Random iteration of one-dimensional transformations, Osaka Journal of Mathematics, 22 (1985), 489-518. 

[28]

F. Nakamura and H. Toyokawa, Random invariant densities for Markov operator cocycles and random mean ergodic theorem, preprint, arXiv: 2101.04878.

[29]

V. I. Oseledets, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 19 (1968), 179-210. 

[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.

[31]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal of Mathematics, 116 (2000), 223-248.  doi: 10.1007/BF02773219.

[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.

[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.

[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.

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.

[2]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variations and Free Discontinuity Problems, Courier Corporation, 2000.

[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.

[4]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.

[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.

[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.

[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.

[8]

A. BoyarskyP. 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.

[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.

[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.

[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.

[12]

G. FroylandS. 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.

[13]

I. Ghenciu, Weakly precompact subsets of $L^{1}(\mu, X)$, Colloquium Mathematicum, 129 (2012), 133-143.  doi: 10.4064/cm129-1-10.

[14]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. doi: 10.1007/978-1-4684-9486-0.

[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.

[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.

[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.

[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.

[19]

P. GóraA. 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.

[20]

L. Hsieh, Ergodic Theory of Multidimensional Random Dynamical Systems, Master thesis, University of Victoria, 2008.

[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.

[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.

[23]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory and Dynamical Systems, 10 (1990), 717-744.  doi: 10.1017/S0143385700005861.

[24]

G. Keller, Propriétés Ergordiques Des Endomorphismes Dilatants, $C^{2}$ Par Morceaux, Des Régions Bornées Du Plan, Thèse, Université de Rennes, 1979.

[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.

[26]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182.  doi: 10.1017/S0143385711000939.

[27]

T. Morita, Random iteration of one-dimensional transformations, Osaka Journal of Mathematics, 22 (1985), 489-518. 

[28]

F. Nakamura and H. Toyokawa, Random invariant densities for Markov operator cocycles and random mean ergodic theorem, preprint, arXiv: 2101.04878.

[29]

V. I. Oseledets, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 19 (1968), 179-210. 

[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.

[31]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal of Mathematics, 116 (2000), 223-248.  doi: 10.1007/BF02773219.

[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.

[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.

[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.

Figure 1.  $ I^{2} $ partitioned into $ 25 $ equal squares
Figure 2.  Bounds in (45) (solid) and (46) (dashed)
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