Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders

Consider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold W^s(O), one-dimensional unstable manifold W^u(O), and with the saddle value σ different from 1. We assume that W^s(O) and W^u(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when σ > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.