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We consider billiards with several possibly non-isometric and asymmetriccusps at flat points; the case of a single symmetric cusp was studiedpreviously in Zhang (2017) and Jung & Zhang (2018). In particular, we show thatproperly normalized Birkhoff sums of H\"older observables, with respect to thebilliard map, converge in Skorokhod's $M_1$-topology to an $\alpha$-stableL\'evy motion, where $\alpha$ depends on the `curvature' of the flattest pointsand the skewness parameter $\xi$ depends on the values of the observable atthose same points. Previously, Jung & Zhang (2018) proved convergence of theone-point marginals to totally skewed $\alpha$-stable distributions for asingle symmetric cusp. The limits we prove here are stronger, since they are inthe functional sense, but also allow for more varied behaviour due to thepresence of multiple cusps. In particular, the general limits we obtain allowfor any skewness parameter, as opposed to just the totally skewed cases. Wealso show that convergence in the stronger $J_1$-topology is not possible.

Convergence to $\alpha$-stable L\'evy motion for chaotic billiards with several cusps at flat points