Generic exponential mixing for volume-preserving partially hyperbolic systems

Consider volume-preserving partially hyperbolic systems, meaning systems preserving a volume and admitting a partially hyperbolic invariant splitting. Generic exponential mixing conjecture. Exponential mixing holds for an open and dense set of volume-preserving partially hyperbolic systems. Exponential mixing would provide strong statistical properties beyond ergodicity and the $K$-property; the source presents genericity of this behavior as an outstanding challenge, including in the partially hyperbolic setting.