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### Open Access Journals

DCDS

In this paper, we study the subharmonic bifurcations in the restricted
three-body problem. By study the Melnikov integrals for the subharmonic
solutions, we obtain the precise bifurcation scenario nearby the circular
solutions when one of the two primaries is small.

DCDS

We consider the monotone twist map $\bar f$ on $(\mathbb R/\mathbb Z)\times R$, itslift $f$ on $R^2$ and its associated variational principle $h:\mathbb R^2\to\mathbb R$ through its generating function. By working with the variationalprinciple $h$, we first show that for an adjacent minimal chain$\{(u^k, v^k)\}_{k=s}^t$ of fixed points of $f$, if there exists abarrier $B_k$ for each adjacent minimal pair $u^k < u^{k+1}$, $ s \le k \le {t-1} $, then there exists a heteroclinic orbit between $(u^s, v^s)$ and$(u^t, v^t)$, then by assuming that there is a barrier for any twoneighboring globally minimal critical points and $m$ is sufficientlylarge, we construct an invariant set $\Lambda^m\subset (\mathbb R/\mathbb Z)\times\mathbb R$ such that the shift map of the $n$-symbol space is a factor of$\bar f^m|_{\Lambda^m}$, where $n$ is the total number of the globallyminimal fixed points of $\bar f$.

DCDS

In this short paper we prove some results concerning volume-preserving
Anosov diffeomorphisms on compact manifolds. The first theorem is that
if a $C^{1 + \alpha}$, $\alpha >0$, volume-preserving diffeomorphism
on a compact connected manifold has a hyperbolic invariant set with
positive volume, then the map is Anosov. The same result had been
obtained by Bochi and Viana [2]. This result is not
necessarily true for $C^1$ maps. The proof uses a Pugh-Shub type of
dynamically defined measure density points, which are different from
the standard Lebesgue density points. We then give a

*direct*proof of the ergodicity of $C^{1+\alpha}$ volume preserving Anosov diffeomorphisms, without using the usual Hopf arguments or the Birkhoff ergodic theorem. The method we introduced also has interesting applications to partially hyperbolic and non-uniformly hyperbolic systems.
DCDS

In this paper, we prove certain persistence properties of
the homoclinic
points in Hamiltonian systems and symplectic
diffeomorphisms. We show that, under
some general conditions, stable and unstable manifolds
of hyperbolic periodic points
intersect in a very persistent way and we also give
some simple criteria for positive topological entropy.
The method used is the intersection theory of Lagrangian
submanifolds of symplectic manifolds.

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