  We classify the locally finite ergodic invariant measures of certain infinite
interval exchange transformations (IETs). These transformations naturally arise
from return maps of the straight-line flow on certain translation surfaces, and
the study of the invariant measures for these IETs is equivalent to the study
of invariant measures for the straight-line flow in some direction on these
translation surfaces. For the surfaces and directions for which our methods
apply, we can characterize the locally finite ergodic invariant measures of the
straight-line flow in a set of directions of Hausdorff dimension larger than
1/2. We promote this characterization to a classification in some cases. For
instance, when the surfaces admit a cocompact action by a nilpotent group, we
prove each ergodic invariant measure for the straight-line flow is a Maharam
measure, and we describe precisely which Maharam measures arise. When the
surfaces under consideration are finite area, the straight-line flows in the
directions we understand are uniquely ergodic. Our methods apply to translation
surfaces admitting multi-twists in a pair of cylinder decompositions in
non-parallel directions.
