  A central question in the field of inhomogeneous attractors has been to
relate the dimension of an inhomogeneous attractor to the condensation set and
associated homogeneous attractor. This has been achieved only in specific
settings, with notable results by Olsen, Snigireva, Fraser and K\"aenm\"aki on
inhomogeneous self-similar sets, and by Burrell and Fraser on inhomogeneous
self-affine sets. This paper is devoted to filling a significant gap in the
dimension theory of inhomogeneous attractors, by studying those formed from
arbitrary bi-Lipschitz contractions. We show that the maximum of the dimension
of the condensation set and a quantity related to pressure, which we term upper
Lipschitz dimension, forms a natural and general upper bound on the dimension.
Additionally, we begin a new line of enquiry; the methods developed are used to
investigate the Hausdorff measure of inhomogeneous attractors. Our results have
applications for affine systems with affinity dimension less than or equal to
one and systems satisfying bounded distortion, such as conformal systems in
dimensions greater than one.
