Distances in inhomogeneous quintessence cosmology

We investigate the properties of cosmological distances in locally inhomoge neous universes with pressureless matter and dark energy (quintessence), wi th constant equation of state, p(x) = w(x)rho (x), - 1 less than or equal t o w(x) < 0. We give exact solutions for angular diameter distances in the e mpty beam approximation. In this hypothesis, the distance - redshift equati on is derived from the multiple-lens-plane theory. The case of a flat unive rse is considered with particular attention. We show how this general schem e makes distances degenerate with respect to w(x) and the smoothness parame ters <alpha>(i), accounting for the homogeneously distributed fraction of e nergy of the i-components. We analyse how this degeneracy influences the cr itical redshift where the angular diameter distance takes its maximum, and discuss future prospects for measuring the smoothness parameter of the pres sureless matter, alpha (M).