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We study the relation between escape rates and pressure in general dynamicalsystems with holes, where pressure is defined to be the difference betweenentropy and the sum of positive Lyapunov exponents. Central to the discussionis the formulation of a class of invariant measures supported on the survivorset over which we take the supremum to measure the pressure. Upper bounds forescape rates are proved for general diffeomorphisms of manifolds, possibly withsingularities, for arbitrary holes and natural initial distributions includingLebesgue and SRB measures. Lower bounds do not hold in such generality, but forsystems admitting Markov tower extensions with spectral gaps, we prove theequality of the escape rate with the absolute value of the pressure and theexistence of an invariant measure realizing the escape rate, i.e. we prove afull variational principle. As an application of our results, we prove avariational principle for the billiard map associated with a planar Lorentz gasof finite horizon with holes.

Entropy, Lyapunov Exponents and Escape Rates in Open Systems