Preimage entropy for random dynamical systems
Yujun Zhu
In [6], Cheng and Newhouse introduced and studied the new invariants - preimage entropies for deterministic dynamical systems. In this paper, the analogous notions, measure-theoretic preimage entropy and topological preimage entropy, are formulated for random dynamical systems. Analogues of many known results for entropies, such as the Shannon-McMillan-Breiman Theorem, the Kolmogorov-Sinai Theorem, the Abromov-Rokhlin formula and the power rule, are obtained for preimage entropies. In particular, a variational principle is given.
keywords: random dynamical system; preimage entropy.
Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations
Yujun Zhu
In this paper, $C^0$ random perturbations of a partially hyperbolic diffeomorphism are considered. It is shown that a partially hyperbolic diffeomorphism is quasi-stable under such perturbations.
keywords: Partial hyperbolicity topological quasi-stability random perturbation.
Center specification property and entropy for partially hyperbolic diffeomorphisms
Lin Wang Yujun Zhu
Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $\mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, \mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $\Lambda$ is center topologically mixing then $f|_{\Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $ h(f)\leq h(f,\mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $\mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.
keywords: center specification property Partial hyperbolicity uniformly compact center foliation topological entropy.
Shadowing in random dynamical systems
Lianfa He Hongwen Zheng Yujun Zhu
In this paper we consider the shadowing property for $C^1$ random dynamical systems. We first define a type of hyperbolicity on the full measure invariant set which is given by Oseledec's multiplicative ergodic theorem, and then prove that the system has the "Lipschitz" shadowing property on it.
keywords: multiplicative ergodic theorem shadowing property. Random dynamical system
Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting
Xinsheng Wang Lin Wang Yujun Zhu

Metric entropies along a hierarchy of unstable foliations are investigated for $C^1 $ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.

keywords: Metric entropy along unstable foliations dominated splitting Lyapunov exponents Ruelle inequality Pesin's entropy formula

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