2017/11/09 by В. З. Гринес, Grines, Viacheslav Z., Evgeniy D. Kurenkov +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1711.03338
openalex publication_date 2017/11/09 · openalex created_date 2017/11/10 · openalex updated_date 2026/07/28
It is well known that topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on nonwandering set. F. Przytycki generalized axiom A for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms and proved spectral decomposition theorem which claims that nonwandering set of an A-endomorphism is a union of a finite number basic sets. In present paper the criterion for a basic sets 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 ambient manifold and 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 restriction of the endomorphism to this basic set is also an expanding endomorphism.