Toward a Mathematical Holographic Principle

Abstract

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 $\varvec{\tau }$ 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 $p(x)\in \{0,1\}.$