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### Open Access Journals

DCDS

Let $f$ be a $C^1$ diffeomorphism on a compact Riemannian manifold without boundary and $\mu$ an ergodic $f$-invariant measure whose Oseledets splitting admits domination. We give a hybrid estimate from above for the metric entropy of $\mu$ in terms of Lyapunov exponents and volume growth. Furthermore, for any $C^1$ diffeomorphism away from tangencies, its topological entropy is bounded by the volume growth.

DCDS

In this paper we establish a closing
property and a hyperbolic closing property for thin
trapped chain hyperbolic homoclinic classes with one dimensional
center in partial hyperbolicity setting. Taking advantage of theses
properties, we prove that the growth rate of the number of
hyperbolic periodic points is equal to the topological entropy. We also
obtain that the hyperbolic periodic measures are dense in the space of invariant measures.

DCDS

We construct a pair of equivalent flows with fixed
points, such that one has infinite topological entropy and the other
has zero topological entropy.

DCDS

Let $M$ be a compact manifold and $f:\,M\rightarrow M$ be a $C^1$
diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability
measure which is absolutely continuous relative to Lebesgue measure
and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting
$T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$, then we give an
estimation through Lyapunov characteristic exponents from below in
Pesin's entropy formula, i.e., the metric entropy $h_\mu(f)$
satisfies
$$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ where
$\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and
$\lambda_1(x)\geq\lambda_2(x)\geq\cdots\geq\lambda_{dim\,M}(x)$ are
the Lyapunov exponents at $x$ with respect to $\mu.$

Consequently, we obtain that Pesin's entropy formula always holds for (1) volume-preserving

Consequently, we obtain that Pesin's entropy formula always holds for (1) volume-preserving

*Anosov*diffeomorphisms, (2) volume-preserving*partially hyperbolic*diffeomorphisms with*one-dimensional*center bundle, (3) volume-preserving diffeomorphisms*far away from homoclinic tangency*, and (4)*generic*volume-preserving diffeomorphisms.
DCDS

In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero
Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we
can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic
diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.

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