On hyperbolic attractors and repellers of endomorphisms
It is well known that the topological classiﬁcation of dynamical systems with hyperbolic dynamics is signiﬁcantly deﬁned by dynamics on a nonwandering set. F.Przytycki generalized axiom A for smooth endomorphisms that was previously introduced by S.Smale for diﬀeomorphisms, and proved the spectral decomposition theorem which claims that the nonwandering set of an A-endomorphism is a union of a ﬁnite number of basic sets. In the present paper the criterion for a basic set of an A-endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown that if an attractor is a topological submanifold of codimension one of type (n - 1, 1), then it is smoothly embedded in the ambient manifold, and the restriction of the endomorphism to this basic set is an expanding endomorphism. If a basic set of type (n, 0) is a topological submanifold of codimension one, then it is a repeller, and the restriction of the endomorphism to this basic set is also an expanding endomorphism.