Higher order terms of Mather's β-function for symplectic and outer billiards

We compute explicitly the formal Taylor expansion of Mather's β-function up to seventh order terms for symplectic and outer billiards in a strictly-convex planar domain C. In particular, we specify the third terms of the asymptotic expansions of the distance (in the sense of the symmetric difference metric) between C and its best approximating inscribed or circumscribed polygons with at most n vertices. We use tools from affine differential geometry.