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### Open Access Journals

DCDS

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.

DCDS

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.

DCDS

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 $h(f)= p^{c}(f)$.

DCDS

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.

DCDS

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.

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