  The ECH capacities are a sequence of real numbers associated to any
symplectic four-manifold, which are monotone with respect to symplectic
embeddings. It is known that for a compact star-shaped domain in R^4, the ECH
capacities asymptotically recover the volume of the domain. We conjecture, with
a heuristic argument, that generically the error term in this asymptotic
formula converges to a constant determined by a "Ruelle invariant" which
measures the average rotation of the Reeb flow on the boundary. Our main result
is a proof of this conjecture for a large class of toric domains. As a
corollary, we obtain a general obstruction to symplectic embeddings of open
toric domains with the same volume. For more general domains in R^4, we bound
the error term with an improvement on the previously known exponent from 2/5 to
1/4.
