ions there is complete uncertainty of the outcome of an experiment conducted at either of these two categories of initial points, whereas on $\mathcal{D}_{+}$ this is due to a random choice from a multifunctional set of equally probable outputs as dictated by the specific conditions under which the experiment was conducted at that instant, on its complement the uncertainty is due to the chaotic behaviour of the functional iterates themselves. Nevertheless it must be clearly understood *that this later behaviour is* *entirely due to the multifunctional limits at the $\mathcal{D}_{+}$ points which completely determine the behaviour of the system on its complement.* As an explicit illustration of this situation, recall that for the shift map $2x\textrm{ mod}(1)$ the $\mathcal{D}_{+}$ points are the rationals on $[0,1]$, and any irrational is represented by a non-terminating and non-repeating decimal so that almost all decimals in $[0,1]$ in any base contain all possible sequences of any number of digits. For the logistic map, the situation is more complex, however. Here the onset of chaos marking the end of the period doubling sequence at $\lambda_{*}=3.5699456$ is signaled by the disappearance of all stable fixed points, Fig. \[Fig: logcob357\](c), with Fig. \[Fig: logcob357\](a) being a demonstration of the stable limits for $\lambda=3.569$ that show up as convergence of the iterates to constant valued functions (rather than as constant valued inverse functions) at stable fixed points, shown more emphatically in Fig\[Fig: log357\](a). What actually happens at $\lambda_{*}$ is shown in Fig. \[Fig: attractor\](a) in the next subsection: the almost vertical lines produced at a large, but finite, iterations $i$ (the multifunctions are generated only in the limiting sense of $i\rightarrow\infty$ and represent a boundary between functional and non-functional relations on a set), decrease in magnitude with increasing iterations until they reduce to points. This gives rise to a (totally disconnected) Cantor set on the $y$