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Dichotomy for the Hausdorff dimension of the set of nonergodic directions. (English) Zbl 1220.37039
In this very interesting paper the authors consider billiards in the table \(P_\lambda\) given by a \(1/2 \times 1\) rectangle with horizontal barrier of length \((1-\lambda)/2\) such that one end touches the midpoint of one of the vertical sides for \(0<\lambda <1\). The main result is that the Hausdorff dimension of the set of directions for which the resulting flow on \(P_\lambda\) is not ergodic is always equal to either \(0\) or \(1/2\). Which of these two possibilities occurs depends exclusively on the summability of the series \(\sum_{k=1}^{\infty} q_k^{-1} \log \log q_{k+1}\), where \((q)\) refers to the sequence of denominators of the approximants of \(\lambda\). More precisely, it is shown that the Hausdorff dimension is equal to \(1/2\) if the series converges, and it is equal to \(0\) otherwise. This rather beautiful result is a significant extension of earlier results of M. Boshernitzan and Y. Cheung [Ann. Math. (2) 158, No. 2, 661–678 (2003; Zbl 1037.37018)].

MSC:
37A25 Ergodicity, mixing, rates of mixing
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
28A78 Hausdorff and packing measures
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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