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Entropy formula for random $ \mathbb{Z}^k$-actions


Author: Yujun Zhu
Journal: Trans. Amer. Math. Soc. 369 (2017), 4517-4544
MSC (2010): Primary 37A35, 37C85, 37H99
DOI: https://doi.org/10.1090/tran/6798
Published electronically: December 22, 2016
MathSciNet review: 3632542
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Abstract: In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $ \mathbb{Z}^k$-actions which are generated by random compositions of the generators of $ \mathbb{Z}^k$-actions. Applying Pesin's theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $ C^{2}$ random $ \mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $ \mathbb{Z}^k$(or $ \mathbb{Z}_+^k)$-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and $ C^{1}$ expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedland's entropy for certain $ C^{2}$ $ \mathbb{Z}^k$-actions.


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Additional Information

Yujun Zhu
Affiliation: College of Mathematics and Information Science and Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, Hebei, 050024, People’s Republic of China

DOI: https://doi.org/10.1090/tran/6798
Keywords: Entropy formula, random $\mathbb{Z}^k$-action, Lyapunov exponent, Friedland's entropy
Received by editor(s): December 17, 2014
Received by editor(s) in revised form: June 11, 2015
Published electronically: December 22, 2016
Additional Notes: The author was supported by NSFC (No. 11371120), NSFHB (No. A2014205154), BRHB (No. BR2-219) and GCCHB (No. GCC2014052)
Article copyright: © Copyright 2016 American Mathematical Society