Weakly homoclinic groups of ergodic actions

The homoclinic group of an ergodic action was introduced by M. I. Gordin. The present paper establishes a connection between homoclinic groups and the factors of an action and the K-property. We introduce the concept of a weakly homoclinic group and demonstrate the completeness of its trajectory. We prove the ergodicity of weakly homoclinic groups of Gaussian and Poisson actions. We establish the triviality of homoclinic groups for the classes of rank-one actions and the connection between weakly homoclinic groups and such asymptotic invariants as rigidity of action, local rank, and weak multiple mixing. We consider other analogues of homoclinic groups and discuss unsolved problems.