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

Equilibrium states in dynamical systems via geometric measure theory