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In this article we study a piecewise linear discretization schemes fortransfer 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 derivethe order of convergence of the approximate invariant density to the real onein the $L^{\infty}$-norm. The outcome of this paper complements rigorousresults on $L^1$ approximations of invariant densities \cite{KMY} and recentresults on the formulae of escape rates of open dynamical systems \cite{KL2}.We implement our computations on two examples (one rigorous and onenon-rigorous) to illustrate the feasibility and efficiency of our schemes.

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