Homoclinic tangencies of arbitrarily high order in conservative two-dimensional maps

Homoclinit tangencies of arbitrarily high order in conservative two-dimensional maps are studied. It is found that small smooth perturbations of a two-dimensional diffeomorphism with nontransversal Poincaré homoclinic orbit may lead to homoclinic tangencies of arbitrary high order and to arbitrary degenerate periodic orbits. It is possible to obtain a complete description of the dynamics and bifurcations of systems with homoclinic tangencies. The results on the density in the open regions of the space of dynamical systems with infinitely degenerate periodic and homoclinic orbits give evidence that the behavior of the trajectories of systems from Newhouse regions is extremely complicated. This make it possible to establish the genericity of global properties of the dynamics. It is proved for general smooth maps that systems with homoclinic tangencies of arbitrarily high orders are dense in Newhouse regions.