On the predictability of discrete dynamical systems

Let $X$ be a metric space. A function $f: X \to X$ is said to be non-sensitive at a point $a \in X$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that for any choice of points $a_0 \in B(a;\delta)$, $a_1 \in B(f(a_0);\delta)$, $a_2 \in B(f(a_1);\delta),\ldots$, we have that $d(a_m,f^m(a)) < \epsilon$ for every $m \geq 0$. 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 $X$.