  We explore the impact of obliquity variations on planetary habitability in
hypothetical systems with high mutual inclination. We show that large
amplitude, high frequency obliquity oscillations on Earth-like exoplanets can
suppress the ice-albedo feedback, increasing the outer edge of the habitable
zone. We restrict our exploration to hypothetical systems consisting of a
solar-mass star, an Earth-mass planet at 1 AU, and 1 or 2 larger planets. We
verify that these systems are stable for $10^8$ years with N-body simulations,
and calculate the obliquity variations induced by the orbital evolution of the
Earth-mass planet and a torque from the host star. We run a simplified energy
balance model on the terrestrial planet to assess surface temperature and ice
coverage on the planet's surface, and we calculate differences in the outer
edge of the habitable zone for planets with rapid obliquity variations. For
each hypothetical system, we calculate the outer edge of habitability for two
conditions: 1) the full evolution of the planetary spin and orbit, and 2) the
eccentricity and obliquity fixed at their average values. We recover previous
results that higher values of fixed obliquity and eccentricity expand the
habitable zone, but also find that obliquity oscillations further expand
habitable orbits in all cases. Terrestrial planets near the outer edge of the
habitable zone may be more likely to support life in systems that induce rapid
obliquity oscillations as opposed to fixed-spin planets. Such planets may be
the easiest to directly characterize with space-borne telescopes.
