Topologically transversal reversible homoclinic sets

An $R$-reversible diffeomorphism on $\mathbb{R}^{2N}$ is studied possessing a hyperbolic fixed point. If the stable manifold of the hyperbolic fixed point and the fixed point set ${Fix}\, R$ of $R$ have a nontrivial local topological crossing, then an infinite number of $R$-symmetric periodic orbits of the diffeomorphism is shown. A perturbed problem is also studied by showing the relationship between a corresponding Melnikov function and the nontriviality of a local topological crossing of the set ${Fix}\, R$ and the stable manifold for the perturbed diffeomorphism.