Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

We show that maps with homoclinic tangencies of arbitrarily high orders and, as a consequence, with arbitrarily degenerate periodic orbits are dense in the Newhouse regions in spaces of real-analytic area-preserving two-dimensional maps and general real-analytic two-dimensional maps (the result was earlier known only for the space of smooth non-conservative maps). Based on this, we show that a generic area-preserving map from the Newhouse region is 'universal' in the sense that its iterations approximate the dynamics of any other area-preserving map with arbitrarily good accuracy. In fact, we show that every dynamical phenomenon which occurs generically in any open set of symplectic diffeomorphisms of a two-dimensional disc, or in any open set of finite-parameter families of such diffeomorphisms, can be encountered at a perturbation of any area-preserving two-dimensional map with a homoclinic tangency.