On interrelations between trivial and nontrivial basic sets of structurally stable diffeomorphisms of surfaces

Abstract

The article is devoted to interrelations between an existence of trivial and nontrivial basic sets of A-diffeomorphisms of surfaces. We prove that if all trivial basic sets of a structurally stable diffeomorphism of surface M2 are source periodic points α1,…αk, then the non-wandering set of this diffeomorphism consists of points α1,…,αk and exactly one one-dimensional attractor Λ. We give some sufficient conditions for attractor Λ to be widely situated. Also, we prove that if a non-wandering set of a structurally stable diffeomorphism contains a nontrivial zero-dimensional basic set, then it also contains source and sink periodic points.