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 to 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
. 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 of finite area, the straight-line flows in the directions we
understand are uniquely ergodic. Our methods apply to translation surfaces
admitting multitwists in a pair of cylinder decompositions in nonparallel
directions.
