Orbit representations from matrices

Abstract

Each Markov interval map f naturally produces a transition 0–1 matrix of interval type (in every row, the entries equal to 1 should be consecutive). We show that any 0–1 matrix A can be transformed into an interval type matrix AI, by a careful use of the state splitting. We then prove that AI can be realized as a transition matrix of an interval map fAI ,λAI arising from the Perron–Frobenius eigenvalue λAI and eigenvector of AI . Finally, we construct orbit representations associated with A from those of AI arising from the dynamical system ([0, 1], fAI ,λAI ).