On the Ergodicity of One-Dimensional Linear Cellular Automata with Additive Error

We consider one-dimensional two-state elementary linear cellular automata, perturbed by an additive noise: Each cell is updated by a linear function of its finite neighborhood, and the value is erroneously reversed with a probability which may depend on the time step and the cell, independently of each other. The state given by fair coin tossing is invariant for this system. The aim of this paper is to give a necessary and sufficient condition for ergodicity, which means that irrespective of choice of the initial state, the system approaches to the state given by fair coin tossing as time goes on. When error probabilities are space-homogeneous but time-inhomogeneous, the condition for ergodicity is the same provided that the linear cellular automata is 'non-trivial' i.e. it is not an identity map or a shift map. On the other hand, when error probabilities are time-homogeneous but space-inhomogeneous, the condition for ergodicity heavily depends on choice of a rule of linear cellular automata.