American Institute of Mathematical Sciences

Advanced
November  2017, 37(11): 5843-5859. doi: 10.3934/dcds.2017254

Estimating the fractal dimension of sets determined by nonergodic parameters

Joseph Squillace

University of California Irvine, Department of Mathematics, 440V Rowland Hall, Irvine, CA 92697-3875, USA

Received  September 2016 Revised  June 2017 Published  July 2017

Fund Project: This work was supported by NSF grant DGE-0841164.

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.

Keywords: Continued fraction, Hausdorff dimension, nonergodic, self-similar covering.
Mathematics Subject Classification: 37C45, 37E99.
Citation: Joseph Squillace. Estimating the fractal dimension of sets determined by nonergodic parameters. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254
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.  Google Scholar

[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.  Google Scholar

[3]

Y. Cheung and A. Eskin, Slow divergence and unique ergodicity, Fields Institute Communications, 51 (2007), 213-222.   Google Scholar

[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.  Google Scholar

[5]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, Chichester, 1990.  Google Scholar

[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.  Google Scholar

[7]

L. Narins, Oral communication, 2013. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

Y. Cheung and A. Eskin, Slow divergence and unique ergodicity, Fields Institute Communications, 51 (2007), 213-222.   Google Scholar

[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.  Google Scholar

[5]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, Chichester, 1990.  Google Scholar

[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.  Google Scholar

[7]

L. Narins, Oral communication, 2013. Google Scholar

[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.  Google Scholar

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[2]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
[3]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
[4]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073
[5]

Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052
[6]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162
[7]

Vadim Azhmyakov, Juan P. Fernández-Gutiérrez, Erik I. Verriest, Stefan W. Pickl. A separation based optimization approach to Dynamic Maximal Covering Location Problems with switched structure. Journal of Industrial & Management Optimization, 2021, 17 (2) : 669-686. doi: 10.3934/jimo.2019128
[8]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[9]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[10]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001
[11]

Shin-Ichiro Ei, Masayasu Mimura, Tomoyuki Miyaji. Reflection of a self-propelling rigid disk from a boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229
[12]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[13]

Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054
[14]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[15]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460
[16]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283
[17]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[18]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018
[19]

Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127
[20]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (53)
  • HTML views (75)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences