The complexity hypothesis for typical three-dimensional billiards

Let $K_n$ be the upper bound on the number of smooth pieces into which the $n$th iterate of a smooth convex expanding submanifold is chopped by the billiard map. Complexity hypothesis. For typical three-dimensional billiards with smooth scatterers and finite horizon, there exists a number $\lambda>1$, strictly less than the smallest expansion rate on the unstable cones of the billiard, such that

$$K_n=o(\lambda^n).$$

Here typicality may be understood, for example, with respect to the $C^r$ topology on the scatterers for $r\geq 3$. The hypothesis is intended to control the effect of singularities relative to hyperbolicity and is presented as an open assumption needed for extending correlation-decay results to higher-dimensional billiards.