  The Unique Games Conjecture (UGC) constitutes a highly dynamic subarea within
computational complexity theory, intricately linked to the outstanding P versus
NP problem. Despite multiple insightful results in the past few years, a proof
for the conjecture remains elusive. In this work, we construct a novel
dynamical systems-based approach for studying unique games and, more generally,
the field of computational complexity. We propose a family of dynamical systems
whose equilibria correspond to solutions of unique games and prove that
unsatisfiable instances lead to ergodic dynamics. Moreover, as the instance
hardness increases, the weight of the invariant measure in the vicinity of the
optimal assignments scales polynomially, sub-exponentially, or exponentially
depending on the value gap. We numerically reproduce a previously hypothesized
hardness plot associated with the UGC. Our results indicate that the UGC is
likely true, subject to our proposed conjectures that link dynamical systems
theory with computational complexity.
