Escape rates and Perron-Frobenius operators: Open and closed dynamical systems

We study the Perron-Frobenius operator $\mathcal{P}$ of closed dynamical systems and certain open dynamical systems. We prove that the presence of a large positive eigenvalue $\rho$ of $\mathcal{P}$ guarantees the existence of a 2-partition of the phase space for which the escape rates of the open systems defined on the two partition sets are both slower than $-\log\rho$. The open systems with slow escape rates are easily identified from the Perron-Frobenius operators of the closed systems. Numerical results are presented for expanding maps of the unit interval. We also apply our technique to shifts of finite type to show that if the adjacency matrix for the shift has a large positive second eigenvalue, then the shift may be decomposed into two disjoint subshifts, both of which have high topological entropies.