Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty}$-norm

In this article we study a piecewise linear discretization schemes for transfer operators (Perron-Frobenius operators) associated with interval maps. We show how these can be used to provide rigorous {\bf pointwise} approximations for invariant densities of Markov interval maps. We also derive the order of convergence of the approximate invariant density to the real one in the $L^{\infty}$-norm. The outcome of this paper complements rigorous results on $L^1$ approximations of invariant densities \cite{KMY} and recent results on the formulae of escape rates of open dynamical systems \cite{KL2}. We implement our computations on two examples (one rigorous and one non-rigorous) to illustrate the feasibility and efficiency of our schemes.