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Energy transfer model for the derivative nonlinear Schrödinger equations on the torus
Estimating the fractal dimension of sets determined by nonergodic parameters
University of California Irvine, Department of Mathematics, 440V Rowland Hall, Irvine, CA 92697-3875, USA |
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.
References:
[1] |
Y. Cheung,
Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space, Ergod. Th. Dynam. Sys., 27 (2007), 65-85.
doi: 10.1017/S0143385706000678. |
[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. Cheung, P. 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] | |
[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. |
show all references
References:
[1] |
Y. Cheung,
Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space, Ergod. Th. Dynam. Sys., 27 (2007), 65-85.
doi: 10.1017/S0143385706000678. |
[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. Cheung, P. 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] | |
[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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