  We study the asymptotic behavior of solutions to the steady Navier-Stokes
equations outside of an infinite cylinder in $\mathbb{R}^3$. We assume that the
flow is periodic in $x_3$-direction and has no swirl. This problem is closely
related with two-dimensional exterior problem. Under a condition on the
generalized finite Dirichlet integral, we give a pointwise decay estimate of
the vorticity at the spatial infinity. Moreover, we prove a Liouville-type
theorem only from the condition of the generalized finite Dirichlet integral.
