The Topological Pressure of Linear Cellular Automata

Jung-Chao Ban,Chih-Hung Chang

doi:10.3390/e11020271

Abstract

This elucidation studies ergodicity and equilibrium measures for additive cellular automata with prime states. Additive cellular automata are ergodic with respect to Bernoulli measure unless it is either an identity map or constant. The formulae of measure-theoretic and topological entropies can be expressed in closed forms and the topological pressure is demonstrated explicitly for potential functions that depend on finitely many coordinates. According to these results, Parry measure is inferred to be an equilibrium measure.

Highlights

Cellular automaton (CA) is a particular class of dynamical systems introduced by Ulam [1] and von Neumann [2] as a model for self-production and is widely studied in a variety of contexts in physics, biology and computer science [3,4,5,6,7,8,9,10,11]
One-dimensional CA consists of infinite lattice with finite states and an associated mapping, say local rule
[13, 14] makes a decisive impulse to the mathematical study; he proposes a classification of CA by means of asymptotical dynamics

Summary

Introduction

Cellular automaton (CA) is a particular class of dynamical systems introduced by Ulam [1] and von Neumann [2] as a model for self-production and is widely studied in a variety of contexts in physics, biology and computer science [3,4,5,6,7,8,9,10,11]. Local rules in same class admit the same dynamics and preserve invariants such as topological entropy, ergodicity, mixing, and so on. Among these 256 rules, there are eight of them being additive. When periodic boundary condition is considered, Chua et al [17] investigate the dynamical behavior of these rules such as Isles of Eden, period of attractors, and so on [17, 18] Shirvani and Rogers [19] demonstrate that a one-dimensional two states CA is ergodic provided its local rule is either rightmost or leftmost permutive. Section 5. extends the results to additive CA with prime symbols

Notation and Definition

Equivalent rules

Measures with Maximal Entropy

Topological entropy

Ergodicity and Topological Pressure

Topological pressure

Potential functions depend on one coordinate

The variational principle for topological pressure says that

Potential functions depend on finitely many coordinates

It is easily seen that and

Multiple Symbols and Larger Neighborhood

Ergodicity and topological pressure