The Topological Pressure of Linear Cellular Automata

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.

Some Properties of topological pressure on cellular automata

“…As an application, the topological pressure of linear CA can be formulated rigorously (cf. [3]), which extends our previous result [4].…”

The Measure-Theoretic Entropy and Topological Entropy of Actions overℤm

“…Recently, Ban et al [15] studied the complexity of permutative CA (defined later) in thermodynamics and topological aspects, and they also gave the formulae to compute measure-theoretic and topological entropies. In this paper, for both measure-theoretic and topological entropies, we extend results obtained in [8,14,15,20] to the case that is a weakly permutive CA over the ring Z with respect to any invariant measure. We remark that LCA is a special case of weakly permutive CA.…”

“…It is important to know how these notions are related with each other. In the last years, a lot of works are devoted to this subject (see, e.g., [5][6][7][8][9][10][11][12][13]). Recall that by the Variational Principle the topological entropy is the supremum of the entropies of invariant measures.…”

“…Recall that by the Variational Principle the topological entropy is the supremum of the entropies of invariant measures. In [8,14,15], the authors showed that the uniform Bernoulli measure is a measure of maximal entropy for some one-dimensional CA.…”