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]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[2]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[3]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
[4]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[5]

Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323
[6]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[7]

Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401
[8]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[9]

D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685
[10]

G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131
[11]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[12]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[13]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[14]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[15]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020168
[16]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[17]

L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799
[18]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703
[19]

Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
[20]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (44)
  • HTML views (74)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences