Decay of correlations for billiards with flat points II: cusps effect

In this paper we continue to study billiards with flat points, by constructing a spec family of dispersing billiards with cusps. All boundaries of the table have positive curvature except that the curvature vanishes at the vertex of a cusp, i.e. there is a pair of boundaries intersection at the flat point tangentially. We study the mixing rates of this one-parameter family of billiards with parameter $β\in (2,\infty)$, and show that the correlation functions of the collision map decay polynomially with order $\cO(n^{-\frac{1}{β-1}})$ as $n\to\infty$. In particular, this solves an open question raised by Chernov and Markarian \cite{CM05}.

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