Interval maps associated to the cellular automaton rule 184

We associate to the cellular automaton elementary rule 184 an interval map defined in [0,1]. We show that this interval map is characterized by a functional equation which depends directly on the local rule and also depends on the choice to represent numbers in base 2. The functional equation is the analytical expression of the interval map self-similarity. We also compute a family of transition matrices which characterizes the effect of the interval map on a family of partitions of the interval [0,1]. We show how the family of matrices can be built with a recursive algorithm which depends on the local rule.