  Given a dynamical system with a uniformly hyperbolic (`chaotic') attractor,
the physically relevant Sinai-Ruelle-Bowen (SRB) measure can be obtained as the
limit of the dynamical evolution of the leaf volume along local unstable
manifolds. We extend this geometric construction to the substantially broader
class of equilibrium states corresponding to H\"older continuous potentials;
these states arise naturally in statistical physics and play a crucial role in
studying stochastic behavior of dynamical systems. The key step in our
construction is to replace leaf volume with a reference measure that is
obtained from a Carath\'eodory dimension structure via an analogue of the
construction of Hausdorff measure. In particular, we give a new proof of
existence and uniqueness of equilibrium states that does not use standard
techniques based on Markov partitions or the specification property; our
approach can be applied to systems that do not have Markov partitions and do
not satisfy the specification property.
