Stability of dynamical structures under perturbation of the generating function

Abstract

We consider the set valued functions C, NW and ℒ taking f in 𝒞(I, I) to its centre C( f ), its set of nonwandering points NW( f ) and its collection of ω-limit sets ℒ( f ) = {ω (x, f ) : x ∈ I}, and consider how these sets are affected by pertubations of f. Our main results characterize those functions g in 𝒞(I, I) at which C, NW and ℒ are continuous. In particular, we show that either of the maps C and NW is continuous at g if and only if one of the following conditions is satisfied: (i) The map ω which takes a function f to its set ω ( f ) of ω-limit points is continuous at g; (ii) the periodic orbits of g which are p-stable, i.e. stable with respect to small perturbations of g, are dense in the set CR(g) of chain recurrent points of g; (iii) CR(g) = ω (g) and the p-stable periodic orbits of g are dense in the set of periodic points of g.