We consider a class of Markov interval maps \(f\in \mathcal {M}(I)\) with I an interval, whose associated transition matrix \(A_f\) is necessarily primitive. Then we search for subdynamics, i.e., a subset \(J\subset I\) and \(g=f\vert _{J} \in \mathcal { M}([J])\) , with [J] the minimal closed interval containing J. The transition matrix \(A_g\) of g is obtained through successive state splittings of \(A_f\) , followed by the removal of appropriate row(s) and column(s). We prove the existence of such J ensuring \(g\in \mathcal {M}([J])\) . We also consider the Cuntz–Krieger algebra \(\mathcal {O}_{A_{f}}\) representation \(\pi _{f,x}\) on the Hilbert space associated to the f-orbit of each point \(x\in J\) . We similarly obtain a representation \(\pi _{g,x}\) of \(\mathcal {O}_{A_{g}}\) . We prove that \(\pi _{g,x}(\mathcal {O}_{A_{g}})\) is a subalgebra of \(\pi _{f,x}(\mathcal {O}_{A_{f}})\) . By exploring this further, we show that in fact \(\mathcal {O}_{A_{g}}\) is a corner algebra of \(\mathcal {O}_{A_{f}}\) by finding a projection \(p_J\) such that \(\mathcal {O}_{A_{g}}=p_{J}\,\mathcal {O}_{A_{f}}p_{J}\) . We explicitly enumerate such corner algebras for each splitting state. This method provides a systematic way to construct concrete examples of specific Cuntz–Krieger corner algebras.
