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.
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