## On the Entropy Geometry of Cellular Automata

**John Milnor***Institute for Advanced Study, Princeton University,**Princeton, NJ 08540, USA*

#### Abstract

We consider *configurations* which assign some elements of a fixed finite alphabet to each point of an -dimensional lattice. An -dimensional *cellular automaton map* assigns a new configuration to each such configuration , in a translation invariant manner, and in such a way that the values of throughout any finite subset of the latice depend only on the values of throughout some larger finite subset. If we iterate such a map over and over, then the complete history of the resulting configurations throughout time can be described as a new configuration over an -dimensional "space-time" lattice. This note will describe the distribution and flow of information throughout this -dimensional lattice by introducing an -*dimensional entropy function* which measures the density of information in very large finite sets.