Multiplicative ergodic theorem of semi-discrete dynamic system

Feng, Jiahui; Yang, Xue

doi:10.1090/proc/15999

Multiplicative ergodic theorem of semi-discrete dynamic system
HTML articles powered by AMS MathViewer

by Jiahui Feng and Xue Yang PDF

Proc. Amer. Math. Soc. 150 (2022), 4393-4405 Request permission

Abstract:

In this paper, Multiplicative Ergodic Theorem (MET) on manifolds with semi-discrete time variable is proved. Considering that there is no cocycle property with any semi-discrete time variable $t\in \mathbb {T}$, we define the quasi-cocycle property on forward and backward time scales. We obtain the skew-product quasi-flow with semi-discrete time variable $t\in \mathbb {T}$. For dynamic equations with $\Delta$-derivative and $\nabla$-derivative on $\mathbb {T}$, we present a more generalized version about MET of semi-discrete system. The result is more suitable for treating models with nonuniform time difference and studying the stability of systems induced by both differential and difference operators.

References

Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Martin Bohner and Allan Peterson, Dynamic equations on time scales, Birkhäuser Boston, Inc., Boston, MA, 2001. An introduction with applications. MR 1843232, DOI 10.1007/978-1-4612-0201-1
Stefan Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18–56. MR 1066641, DOI 10.1007/BF03323153
Wen Huang and Pengfei Zhang, Pointwise dimension, entropy and Lyapunov exponents for $C^1$ maps, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6355–6370. MR 2958939, DOI 10.1090/S0002-9947-2012-05527-9
Zeng Lian and Kening Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc. 206 (2010), no. 967, vi+106. MR 2674952, DOI 10.1090/S0065-9266-10-00574-0
Zeng Lian and Yi Wang, On random linear dynamical systems in a Banach space. I. Multiplicative ergodic theorem and Krein-Rutman type theorems, Adv. Math. 312 (2017), 374–424. MR 3635815, DOI 10.1016/j.aim.2017.03.024
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
Mark Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds, London Mathematical Society Lecture Note Series, vol. 180, Cambridge University Press, Cambridge, 1993. MR 1215938, DOI 10.1017/CBO9780511752537
Ja. B. Pesin, Characteristic Ljapunov exponents, and ergodic properties of smooth dynamical systems with invariant measure, Dokl. Akad. Nauk SSSR 226 (1976), no. 4, 774–777 (Russian). MR 0410804
J. Michael Steele, Kingman’s subadditive ergodic theorem, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 1, 93–98 (English, with French summary). MR 995293