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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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##### References:
 [1] Cheung, Y.: Hausdorff dimension of the set of nonergodic directions. With an appendix by M. Boshernitzan. Ann. Math. (2) 158(2), 661–678 (2003) · Zbl 1037.37018 · doi:10.4007/annals.2003.158.661 [2] Cheung, Y.: Slowly divergent geodesics in moduli space. Conform. Geom. Dyn. 8, 167–189 (2004) · Zbl 1067.37036 · doi:10.1090/S1088-4173-04-00113-4 [3] Cheung, Y.: Hausdorff dimension of the set of singular pairs. Ann. Math. (to appear). arXiv:0709.4534 · Zbl 1241.11075 [4] Cheung, Y., Eskin, A.: Slow Divergence and Unique Ergodicity. Preprint. arXiv:0711.0240v1 · Zbl 1145.37003 [5] Cheung, Y., Masur, H.: A divergent Teichmuller geodesic with uniquely ergodic foliation. Isr. J. Math. 152, 1–15 (2006) · Zbl 1122.37027 · doi:10.1007/BF02771972 [6] Eskin, A., Masur, H.: Asymptotic formulas on flat surfaces. Ergod. Theory Dyn. Syst. 21, 443–478 (2001) · Zbl 1096.37501 · doi:10.1017/S0143385701001225 [7] Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications. Wiley, Chichester (1990) · Zbl 0689.28003 [8] Khintchin, A.Ya.: Continued Fractions. University of Chicago Press, Chicago (1964). English Translation. First Russian edition published 1935 [9] Keynes, H., Newton, D.: A minimal non uniquely ergodic interval exchange. Math. Z. 148, 101–106 (1976) · Zbl 0308.28014 · doi:10.1007/BF01214699 [10] Masur, H.: The growth rate of trajectories of a quadratic differential. Ergod. Theory Dyn. Syst. 10, 151–176 (1990) · Zbl 0706.30035 · doi:10.1017/S0143385700005459 [11] Masur, H.: Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66(3), 387–442 (1992) · Zbl 0780.30032 · doi:10.1215/S0012-7094-92-06613-0 [12] Milnor, J.W.: Dynamics in One Complex Variable. Vieweg, Wiesbaden (1999), (2000); Princeton U. Press (2006) · Zbl 0946.30013 [13] Masur, H., Smillie, J.: Hausdorff dimension of sets of nonergodic foliations. Ann. Math. 134, 455–543 (1991) · Zbl 0774.58024 · doi:10.2307/2944356 [14] Masur, H., Tabachnikov, S.: Rational Billiards and Flat Structures. Handbook of Dynamical Systems, vol. 1A, pp. 1015–1089. North-Holland, Amsterdam (2002) · Zbl 1057.37034 [15] Pérez Marco, R.: Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnol’d. (French. English summary) [Nonlinearizable holomorphic dynamics and a conjecture of V. I. Arnol’d]. Ann. Sci. Ecole Norm. Sup. (4) 26(5), 565–644 (1993) · Zbl 0812.58051 [16] Poincaré, H.: Sur un mode nouveau de représentation géométrique des formes quadratiques dénies et indénies (1880). In: Oeuvres complètes de Poincaré, Tome V, pp. 117–183 (1952) [17] Veech, W.: Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2. Trans. Am. Math. Soc. 140, 1–33 (1969) · Zbl 0201.05601 [18] Veech, W.: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3), 553–583 (1989) · Zbl 0676.32006 · doi:10.1007/BF01388890 [19] Vorobets, Y.: Periodic Geodesics on Translation Surfaces. Contemporary Math., vol. 385. Am. Math. Soc., Providence (2005) · Zbl 1130.37015
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