doi: 10.3934/dcdsb.2022058
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Lyapunov exponents for random maps

1. 

Faculty of Engineering, Kitami Institute of Technology, Hokkaido, 090-8507, Japan

2. 

Department of Mathematics, Tokai University, Kanagawa 259-1292, Japan

*Corresponding author: Hisayoshi Toyokawa

Received  April 2021 Revised  February 2022 Early access March 2022

Fund Project: This work is partially supported by JSPS KAKENHI Grant Numbers 19K14575, 19K21834 and 21K20330

It has been recently realized that for abundant dynamical systems on a compact manifold, the set of points for which Lyapunov exponents fail to exist, called the Lyapunov irregular set, has positive Lebesgue measure. In the present paper, we show that under any physical noise, the Lyapunov irregular set has zero Lebesgue measure and the number of such Lyapunov exponents is finite. This result is a Lyapunov exponent version of Araújo's theorem on the existence and finitude of time averages. Furthermore, we numerically compute the Lyapunov exponents for a surface flow with an attracting heteroclinic connection, which enjoys the Lyapunov irregular set of positive Lebesgue measure, under a physical noise. This paper also contains the proof of the disappearance of Lyapunov irregular behavior on a positive Lebesgue measure set for a surface flow with an attracting homoclinic/heteroclinic connection under a non-physical noise.

Citation: Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa. Lyapunov exponents for random maps. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022058
References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Nonuniform hyperbolicity for C1-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.

[2]

K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, 1997. doi: 10.1007/b97589.

[3]

V. Araújo, Attractors and time averages for random maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 307-369.  doi: 10.1016/s0294-1449(00)00112-8.

[4]

V. Araújo, Infinitely many stochastically stable attractors, Nonlinearity, 14 (2001), 583-596.  doi: 10.1088/0951-7715/14/3/308.

[5]

V. Araújo, Random perturbations of codimension one homoclinic tangencies in dimension 3, Dyn. Syst., 18 (2003), 35-55.  doi: 10.1080/1468936031000080803.

[6]

V. Araújo and H. Aytaç, Decay of correlations and laws of rare events for transitive random maps, Nonlinearity, 30 (2017), 1834-1852.  doi: 10.1088/1361-6544/aa64e8.

[7]

V. Araújo and V. Pinheiro, Abundance of wild historic behavior, Bull. Braz. Math. Soc. (N.S.), 52 (2021), 41-76.  doi: 10.1007/s00574-019-00191-8.

[8]

V. Araújo and A. Tahzibi, Stochastic stability at the boundary of expanding maps, Nonlinearity, 18 (2005), 939-958.  doi: 10.1088/0951-7715/18/3/001.

[9]

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

[10]

L. BarreiraJ. Li and C. Valls, Irregular sets are residual, Tohoku Math. J., 66 (2014), 471-489.  doi: 10.2748/tmj/1432229192.

[11]

L. BarreiraJ. Li and C. Valls, Topological entropy of irregular sets, Rev. Mat. Iberoam., 34 (2018), 853-878.  doi: 10.4171/RMI/1006.

[12]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.  doi: 10.1007/BF02773211.

[13]

L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions, Ergodic Theory Dynam. Systems, 26 (2006), 653-671.  doi: 10.1017/S0143385705000672.

[14]

P. G. BarrientosS. KirikiY. NakanoA. Raibekas and T. Soma, Historic behavior in nonhyperbolic homoclinic classes, Proc. Amer. Math. Soc., 148 (2020), 1195-1206.  doi: 10.1090/proc/14809.

[15]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Springer, 2005. doi: doi.org/10.1007/b138174.

[16]

E. Catsigeras, Empiric stochastic stability of physical and pseudo-physical measures, Springer Proc. Math. Stat., 285 (2019), 113-136.  doi: 10.1007/978-3-030-16833-9_7.

[17]

E. ChenT. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.  doi: 10.1017/S0143385704000872.

[18]

E. Colli and E. Vargas, Non-trivial wandering domains and homoclinic bifurcations, Ergodic Theory Dynam. Systems, 21 (2001), 1657-1681.  doi: 10.1017/S0143385701001791.

[19]

S. CrovisierD. Yang and J. Zhang, Empirical measures of partially hyperbolic attractors, Comm. Math. Phys, 375 (2020), 725-764.  doi: 10.1007/s00220-019-03668-1.

[20]

M. F. DemersF. Pène and H.-K. Zhang, Local limit theorem for randomly deforming billiards, Comm. Math. Phys., 375 (2020), 2281-2334.  doi: 10.1007/s00220-019-03670-7.

[21]

S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. doi: 10.1007/BF02760066.

[22]

A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.

[23]

M. Gianfelice and S. Vaienti, Stochastic stability of the classical Lorenz flow under impulsive type forcing, J. Stat. Phys., 181 (2020), 163-211.  doi: 10.1007/s10955-020-02572-6.

[24]

P. GuarinoP.-A. Guihéneuf and B. Santiago, Dirac physical measures on saddle-type fixed points, J. Dynam. Differ. Equat., 34 (2020), 1-61.  doi: 10.1007/s10884-020-09911-x.

[25]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys., 127 (1990), 319-337.  doi: 10.1007/BF02096761.

[26]

J. JostM. Kell and C. S. Rodrigues, Representation of Markov chains by random maps: Existence and regularity conditions, Calc. Var. Partial Differential Equations, 54 (2015), 2637-2655.  doi: 10.1007/s00526-015-0878-2.

[27] A. Katok and B. Hasselblatt, Introduction To the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. 
[28]

S. KirikiX. LiY. Nakano and T. Soma, Abundance of observable Lyapunov irregular sets, Comm. Math. Phys., 33 (2022), 1-29.  doi: 10.1007/s00220-022-04337-6.

[29]

S. KirikiY. Nakano and T. Soma, Historic behaviour for nonautonomous contraction mappings, Nonlinearity, 32 (2019), 1111-1124.  doi: 10.1088/1361-6544/aaf253.

[30]

S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains, Adv. Math., 306 (2017), 524-588.  doi: 10.1016/j.aim.2016.10.019.

[31]

I. S. Labouriau and A. A. Rodrigues, On Takens' last problem: Tangencies and time averages near heteroclinic networks, Nonlinearity, 30 (2017), 1876-1910.  doi: 10.1088/1361-6544/aa64e9.

[32]

Y. Nakano, Historic behaviour for random expanding maps on the circle, Tokyo J. Math., 40 (2017), 165-184.  doi: 10.3836/tjm/1502179221.

[33]

V. I. Oseledec, A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. 

[34]

W. Ott and J. A. Yorke, When Lyapunov exponents fail to exist, Phys. Rev. E, 78 (2008), 056203, 6 pp. doi: 10.1103/PhysRevE.78.056203.

[35]

Y. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. I Prilozhen, 18 (1984), 50-63.  doi: 10.1007/BF01083692.

[36]

D. Ruelle, Historical behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, Inst. Phys., Bristol, (2001), 63–66.

[37]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 107-120.  doi: 10.1007/BF01232938.

[38]

X. Tian, Nonexistence of Lyapunov exponents for matrix cocycles, Ann. Inst. Henri Poincaré Probab. Stat., 53 (2017), 493-502.  doi: 10.1214/15-AIHP733.

[39] M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781139976602.

show all references

References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Nonuniform hyperbolicity for C1-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.

[2]

K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, 1997. doi: 10.1007/b97589.

[3]

V. Araújo, Attractors and time averages for random maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 307-369.  doi: 10.1016/s0294-1449(00)00112-8.

[4]

V. Araújo, Infinitely many stochastically stable attractors, Nonlinearity, 14 (2001), 583-596.  doi: 10.1088/0951-7715/14/3/308.

[5]

V. Araújo, Random perturbations of codimension one homoclinic tangencies in dimension 3, Dyn. Syst., 18 (2003), 35-55.  doi: 10.1080/1468936031000080803.

[6]

V. Araújo and H. Aytaç, Decay of correlations and laws of rare events for transitive random maps, Nonlinearity, 30 (2017), 1834-1852.  doi: 10.1088/1361-6544/aa64e8.

[7]

V. Araújo and V. Pinheiro, Abundance of wild historic behavior, Bull. Braz. Math. Soc. (N.S.), 52 (2021), 41-76.  doi: 10.1007/s00574-019-00191-8.

[8]

V. Araújo and A. Tahzibi, Stochastic stability at the boundary of expanding maps, Nonlinearity, 18 (2005), 939-958.  doi: 10.1088/0951-7715/18/3/001.

[9]

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

[10]

L. BarreiraJ. Li and C. Valls, Irregular sets are residual, Tohoku Math. J., 66 (2014), 471-489.  doi: 10.2748/tmj/1432229192.

[11]

L. BarreiraJ. Li and C. Valls, Topological entropy of irregular sets, Rev. Mat. Iberoam., 34 (2018), 853-878.  doi: 10.4171/RMI/1006.

[12]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.  doi: 10.1007/BF02773211.

[13]

L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions, Ergodic Theory Dynam. Systems, 26 (2006), 653-671.  doi: 10.1017/S0143385705000672.

[14]

P. G. BarrientosS. KirikiY. NakanoA. Raibekas and T. Soma, Historic behavior in nonhyperbolic homoclinic classes, Proc. Amer. Math. Soc., 148 (2020), 1195-1206.  doi: 10.1090/proc/14809.

[15]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Springer, 2005. doi: doi.org/10.1007/b138174.

[16]

E. Catsigeras, Empiric stochastic stability of physical and pseudo-physical measures, Springer Proc. Math. Stat., 285 (2019), 113-136.  doi: 10.1007/978-3-030-16833-9_7.

[17]

E. ChenT. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.  doi: 10.1017/S0143385704000872.

[18]

E. Colli and E. Vargas, Non-trivial wandering domains and homoclinic bifurcations, Ergodic Theory Dynam. Systems, 21 (2001), 1657-1681.  doi: 10.1017/S0143385701001791.

[19]

S. CrovisierD. Yang and J. Zhang, Empirical measures of partially hyperbolic attractors, Comm. Math. Phys, 375 (2020), 725-764.  doi: 10.1007/s00220-019-03668-1.

[20]

M. F. DemersF. Pène and H.-K. Zhang, Local limit theorem for randomly deforming billiards, Comm. Math. Phys., 375 (2020), 2281-2334.  doi: 10.1007/s00220-019-03670-7.

[21]

S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. doi: 10.1007/BF02760066.

[22]

A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815.  doi: 10.1016/S0246-0203(97)80113-6.

[23]

M. Gianfelice and S. Vaienti, Stochastic stability of the classical Lorenz flow under impulsive type forcing, J. Stat. Phys., 181 (2020), 163-211.  doi: 10.1007/s10955-020-02572-6.

[24]

P. GuarinoP.-A. Guihéneuf and B. Santiago, Dirac physical measures on saddle-type fixed points, J. Dynam. Differ. Equat., 34 (2020), 1-61.  doi: 10.1007/s10884-020-09911-x.

[25]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys., 127 (1990), 319-337.  doi: 10.1007/BF02096761.

[26]

J. JostM. Kell and C. S. Rodrigues, Representation of Markov chains by random maps: Existence and regularity conditions, Calc. Var. Partial Differential Equations, 54 (2015), 2637-2655.  doi: 10.1007/s00526-015-0878-2.

[27] A. Katok and B. Hasselblatt, Introduction To the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. 
[28]

S. KirikiX. LiY. Nakano and T. Soma, Abundance of observable Lyapunov irregular sets, Comm. Math. Phys., 33 (2022), 1-29.  doi: 10.1007/s00220-022-04337-6.

[29]

S. KirikiY. Nakano and T. Soma, Historic behaviour for nonautonomous contraction mappings, Nonlinearity, 32 (2019), 1111-1124.  doi: 10.1088/1361-6544/aaf253.

[30]

S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains, Adv. Math., 306 (2017), 524-588.  doi: 10.1016/j.aim.2016.10.019.

[31]

I. S. Labouriau and A. A. Rodrigues, On Takens' last problem: Tangencies and time averages near heteroclinic networks, Nonlinearity, 30 (2017), 1876-1910.  doi: 10.1088/1361-6544/aa64e9.

[32]

Y. Nakano, Historic behaviour for random expanding maps on the circle, Tokyo J. Math., 40 (2017), 165-184.  doi: 10.3836/tjm/1502179221.

[33]

V. I. Oseledec, A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. 

[34]

W. Ott and J. A. Yorke, When Lyapunov exponents fail to exist, Phys. Rev. E, 78 (2008), 056203, 6 pp. doi: 10.1103/PhysRevE.78.056203.

[35]

Y. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. I Prilozhen, 18 (1984), 50-63.  doi: 10.1007/BF01083692.

[36]

D. Ruelle, Historical behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, Inst. Phys., Bristol, (2001), 63–66.

[37]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 107-120.  doi: 10.1007/BF01232938.

[38]

X. Tian, Nonexistence of Lyapunov exponents for matrix cocycles, Ann. Inst. Henri Poincaré Probab. Stat., 53 (2017), 493-502.  doi: 10.1214/15-AIHP733.

[39] M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781139976602.
Figure 1.  The finite time flow Lyapunov exponent for the system (7) are illustrated without noise (left), with additive noise (right). They are plotted by $ \log $ time scale
Figure 2.  The image of the model with perturbation of impulsive type
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