Let M be the Cantor space or an n-manifold with C (M, M) the set of continuous self-maps of M. We prove the following:(1) There is a residual set of points (x, f ) in M × C(M, M) all of which generate as their ω-limit set a particular, unique adding machine.(2) Moreover, if M has the fixed point property, then a generic f ∈ C(M, M) generates uncountably many distinct copies of every possible adding machine.
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 v-limit sets L( f ) ¼ {v (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 v which takes a function f to its set v ( f ) of v-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) ¼ v (g) and the p-stable periodic orbits of g are dense in the set of periodic points of g.
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