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Let X be a metric space. A function f: X → X is said to be non-sensitive at a point a E X if for every e > 0 there is a δ > 0 such that for any choice of points a 0 E B(a;δ), a 1 E B(f(a 0 );δ), a 2 ∈ B(f(a 1 );δ),..., we have that d(a m , f m (a)) 0. Let H(X) be the set of all homeomorphisms from X onto X endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces X, most functions in H(X) are non-sensitive at most points of X.

On the predictability of discrete dynamical systems