  We introduce a new method for studying length spectrum rigidity problems
based on a combination of ideas from dynamical systems and geometric group
theory. This allows us to compare the marked length spectrum of metrics and
distance-like functions coming from various geometric origins. Using our new
perspective, we provide concise proofs of well-known length spectrum rigidity
results and are able to extend classical results to a variety of new settings.
Our methods rely on studying Manhattan curves and a coarse geometric analogue
of Teichm\"uller space equipped with a symmetrized version of the Thurston
metric.
