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We study the orbit behavior of a four dimensional smooth symplecticdiffeomorphism $f$ near a homoclinic orbit $\Gamma$ to an 1-elliptic fixedpoint under some natural genericity assumptions. 1-elliptic fixed point has tworeal eigenvalues out of unit circle and two others on the unit circle. Thusthere is a smooth 2-dimensional center manifold $W^c$ where the restriction ofthe diffeomorphism has the elliptic fixed point supposed to be generic (nostrong resonances and first Birkhoff coefficient is nonzero). Moser's theoremguarantees the existence of a positive measure set of KAM invariant curves.$W^c$ itself is a normally hyperbolic manifold in the whole phase space and dueto Fenichel results every point on $W^c$ has 1-dimensional stable and unstablesmooth invariant curves forming two smooth foliations. In particular, each KAMinvariant curve has stable and unstable smooth 2-dimensional invariantmanifolds being Lagrangian. The related stable and unstable manifolds of $W^c$are 3-dimensional smooth manifolds which are supposed to be transverse alonghomoclinic orbit $\Gamma$. One of our theorems presents conditions under whicheach KAM invariant curve on $W^c$ in a sufficiently small neighborhood of$\Gamma$ has four transverse homoclinic orbits. Another result ensures thatunder some Birkhoff genericity assumption for the restriction of $f$ on $W^c$saddle periodic orbits in resonance zones also have homoclinic orbits thoughits transversality or tangency cannot be verified directly.

On symplectic dynamics near a homoclinic orbit to 1-elliptic fixed point