TRANSITIVITY IN TWO-DIMENSIONAL LOCAL LANGUAGES DEFINED BY DOT SYSTEMS

Abstract

The paper investigates two-dimensional recognizable languages that are defined by the so-called "dot systems" that are special subgroups of (ℤ/2ℤ)ℤ2. The dot shapes that provide directional transitivity or mixing for the related language are investigated. It is shown that languages defined by parallelogram shapes fail to be transitive in the direction of a defining vector and hence fail to be mixing, while certain triangular shapes guarantee that the factor language of the associated dot system will be mixing. Dot systems belong to a class of two-dimensional shift spaces that have a factor language such that every admissable block can be extended to a configuration of the entire plane. For this class of shift spaces we introduce a finite graph (i.e., a finite state automaton) that recognizes two-dimensional local languages, then show that certain transitivity properties may be observed from the structure of the finite graph.