The common dynamics of the Tribonacci substitutions (Q1840637)

The author considers dynamical systems induced by the Tribonacci substitutions: \(1\to 12\), \(2\to 13\), \(3\to 1\) and \(1\to 12\), \(2\to 31\), \(3\to 1\). These substitutions have the same incidence matrix therefore the same recurrence relation, the Tribonacci recurrence relation. The fixed points of each substitution have the same digit frequencies. But they have very different dynamical and geometrical properties. The author discusses and describes the topological properties of the geometrical realization in the plane of the symbolic dynamics obtained by the product of the prefix automata of these two substitutions. Generalization concerning the \(k\)-bonacci substitutions are also discussed.