On the Quantitative Behavior of the Linear Cellular Automata

In this paper, we study the quantitative behavior of one-dimensional linear cellular automata Tf [−r,r], defined by local rule f(x−r, . . . , xr) = r ∑ i=−r λixi (mod m), acting on the space of all doubly infinite sequences with values in a finite ring Zm, m ≥ 2. Once generalize the formulas given by Ban et al. [J. Cellular Automata 6 (2011) 385-397] for measure-theoretic entropy and topological pressure of one-dimensional cellular automata, we calculate the measure entropy and the topological pressure of the linear cellular automata with respect to the Bernoulli measure on the set Zm. Also, it is shown that the uniform Bernoulli measure is the unique equilibrium measure for linear cellular automata. We compare values of topological entropy and topological directional entropy by using the formula obtained by Akın [J. Computation and Appl. Math. 225 (2) (2009) 459-466]. The topological directional entropy is interpreted by means of figures. As an application, we demonstrate that the Hausdorff of the limit set of a linear cellular automaton is the unique root of Bowen’s equation. Some open problems remain to be of interest. Mathematics Subject Classification: Primary 28D20; Secondary 37A35, 37B40.