Invariant densities and escape rates: Rigorous and computable approximations in the L∞-norm

In this article, we study piecewise linear discretization schemes for transfer operators (Perron-Frobenius operators) associated with interval maps. We show how these can be used to provide rigorous 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∞-norm. The outcome of this paper complements recent results on the formulae of escape rates of open dynamical systems, (Keller and Liverani, 2009) [7]. In particular, the novelty of our work over previous results on BV and L∞ approximations is that it provides a method for explicit computation of the approximation error.