  In work started in [17] and continued in this paper our objective is to study
selectors of multivalued functions which have interesting dynamical properties,
such as possessing absolutely continuous invariant measures. We specify the
graph of a multivalued function by means of lower and upper boundary maps
$\tau_{1}$ and $\tau_{2}.$ On these boundary maps we define a position
dependent random map $R_{p}=\{\tau_{1},\tau_{2};p,1-p\},$ which, at each time
step, moves the point $x$ to $\tau_{1}(x)$ with probability $p(x)$ and to
$\tau_{2}(x)$ with probability $1-p(x)$. Under general conditions, for each
choice of $p$, $R_{p}$ possesses an absolutely continuous invariant measure
with invariant density $f_{p}.$ Let $\boldsymbol\tau$ be a selector which has
invariant density function $f.$ One of our objectives is to study conditions
under which $p(x)$ exists such that $R_{p}$ has $f$ as its invariant density
function. When this is the case, the long term statistical dynamical behavior
of a selector can be represented by the long term statistical behavior of a
random map on the boundaries of $G.$ We refer to such a result as a
mathematical holographic principle. We present examples and study the
relationship between the invariant densities attainable by classes of selectors
and the random maps based on the boundaries and show that, under certain
conditions, the extreme points of the invariant densities for selectors are
achieved by bang-bang random maps, that is, random maps for which $p(x)\in
\{0,1\}.$
