Orbit representations from matrices

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 AIAI, by a careful use of the state splitting. We then prove that AIAI can be realized as a transition matrix of an interval map fAI,λAIfAI,λAI arising from the Perron–Frobenius eigenvalue λAIλAI and eigenvector of AIAI. Finally, we construct orbit representations associated with A   from those of AIAI arising from the dynamical system ([0,1],fAI,λAI)([0,1],fAI,λAI).