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Estimating the fractal dimension of sets determined by nonergodic parameters

This work was supported by NSF grant DGE-0841164
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  • Given fixed and irrational $0<α, θ<1$, consider the billiard table $B_{α}$ formed by a $\frac{1}{2}×1$ rectangle with a horizontal barrier of length $α$ emanating from the midpoint of a vertical side and a billiard flow with trajectory angle $θ$. In 1969, Veech introduced two subsets $K_{0}(θ)$ and $K_{1}(θ)$ of $\mathbb{R}/\mathbb{Z}$ that are defined in terms of the continued fraction representation of $θ∈\mathbb{R}/\mathbb{Z}$, and Veech showed that these sets have Hausdorff dimension $0$ when $θ$ is rational. Moreover, the set $K_{1}(θ)$ describes the set of all $α$ such that the billiard flow on $B_{α}$ in direction $θ$ is nonergodic. We show that the Hausdorff dimension of the sets $K_{0}(θ)$ and $K_{1}(θ)$ can attain any value in $[0, 1]$ by considering the continued fraction expansion of $θ$. This result resolves an analogue of work completed by Cheung, Hubert, and Pascal in which they consider, for fixed $α$, the set of $θ$ such that the flow on $B_{α}$ in direction $θ$ is nonergodic.

    Mathematics Subject Classification: 37C45, 37E99.


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  • Figure 1.  A billiard table $B_{\alpha}$ with a barrier of length $\alpha$.

    Figure 2.  Unfolding of billiard table $B_{\alpha}$.

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    [2] Y. Cheung, Hausdorff dimension of the set of singular pairs, Annals of Mathematics, 173 (2011), 127-167.  doi: 10.4007/annals.2011.173.1.4.
    [3] Y. Cheung and A. Eskin, Slow divergence and unique ergodicity, Fields Institute Communications, 51 (2007), 213-222. 
    [4] Y. CheungP. Hubert and H. Masur, Dichotomy for the Hausdorff dimension of the set of nonergodic directions, Inventiones, 183 (2001), 337-383.  doi: 10.1007/s00222-010-0279-2.
    [5] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, Chichester, 1990.
    [6] H. Masur and S. Tabachnikov, Rational billiards and flat surfaces, Handbook of Dynamical Systems, 1A (2002), 1015-1089.  doi: 10.1016/S1874-575X(02)80015-7.
    [7] L. Narins, Oral communication, 2013.
    [8] W. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem modulo 2, Trans. Amer. Math. Soc., 140 (1969), 1-33.  doi: 10.2307/1995120.
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