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arXiv · 1605.07714

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

Hong-Kun Zhang
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Abstract

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 $\beta\in (2,\infty)$, and show that the correlation functions of the collision map decay polynomially with order $\cO(n^{-\frac{1}{\beta-1}})$ as $n\to\infty$. In particular, this solves an open question raised by Chernov and Markarian \cite{CM05}.

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Hong-Kun Zhang. 2016-05-25. Decay of correlations for billiards with flat points II: cusps effect. https://arxiv.org/abs/1605.07714

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