Equivalent conditions for \(\Omega\)-inverse limit stability (Q1300111)

The paper deals with an extension of results on \(\Omega\)-stability for diffeomorphisms to an analogy for endomorphisms. The basic problem is an absence of inverse mapping. Before the concept of \(\Omega\)-stability was generalized to the concept of \(\Omega\)-inverse limit stability that means the existence of conjugacy between spaces of bi-infinite orbits for any two endomorphisms near the original one. The hyperbolicity has been transformed to prehyperbolicity with an additional condition on identified points. The main results are two theorems. Theorem 1. For an endomorphism \(f\) the following are equivalent: (a) \(f\) satisfies weak Axiom A and the no-cycles condition; (b) the chain recurrent set is prehyperbolic; (c) \(\omega\)-limit points are prehyperbolic with no cycles. Theorem 2. Let \(f\) be an endomorphism. (a) If \(f\) satisfies weak Axiom A and the no-cycles condition, then \(f\) is \(\Omega\)-inverse limit stable. (b) If the chain recurrent set \(CR\) is prehyperbolic , then \(f\) is \(CR\)-inverse limit stable. (c) If \(\omega\)-limit points \(L\) are prehyperbolic with no cycles, then \(f\) is \(L\)-inverse limit stabe.