Perturbations of Nonhyperbolic Algebraic Automorphisms of the 2-Torus

All nonhyperbolic automorphisms of the 2-torus are not structurally stable, and it is generally impossible to predict the dynamics of their arbitrarily small perturbations. In this paper, given a representative of each algebraic conjugacy class of nonperiodic nonhyperbolic maps, a one-parameter family of diffeomorphisms is constructed, in which the zero value of the parameter corresponds to the given map and the nonzero values, to Morse–Smale diffeomorphisms. According to results of V. Z. Grines and A. N. Bezdenezhnykh, a Morse–Smale diffeomorphism of a closed orientable surface which induces a nonperiodic action on the fundamental group has nonempty heteroclinic set. It is proved that, in all of the constructed families, the diffeomorphisms corresponding to nonzero parameter values have nonempty orientable heteroclinic sets in which the number of orbits is determined by the automorphism being perturbed.