Discrete and Continuous Dynamical Systems
2021, Volume 41, Issue 8: 3781-3796. Doi: 10.3934/dcds.2021016
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Chaotic Delone sets
Jesús A. Álvarez López1, ,
Ramón Barral Lijó2, , ,
John Hunton3, ,
Hiraku Nozawa4, and
John R. Parker5,
Departamento e Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain
Research Organization of Science and Technology, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan
Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK
Department of Mathematical Sciences, Colleges of Science and Engineering, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan
* Corresponding author: Ramón Barral Lijó (ramonbarrallijo@gmail.com)
Revised: October 31, 2020
Early access: January 2021
We present a definition of chaotic Delone set and establish the genericity of chaos in the space of $ (\epsilon,\delta) $-Delone sets for $ \epsilon\geq \delta $. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.
Delone set,
chaos,
tiling,
foliated space,
hyperbolic dynamical system,
geodesic flow.
Mathematics Subject Classification: 37D45, 52C23, 37B51.
Figure 1. Construction of $ S_{\ell} $ in $ \mathbb{H}^2 $. The black dots represent points in $ \Gamma x $, the blue area is $ E_{\ell} $, the red dots represent points in $ S_{\ell} $.
Figure 2. The disks represent the inverse image of $ \Delta $. The projection of $ k_{1} $ to $ \Sigma $ has one-sided tangency, while the projection of $ k_{2} $ to $ \Sigma $ does not.
Figure 3. A 12-gon P
Figure 4. A triangle T
Figure 5. The picture on the left represents $ T\subset \mathbb{T}^n $; the right one its lift to $ \mathbb{R}^n $ following a grid pattern
Figure 6. Approximation of $ S^{+}_{\ell} $ by $ S^{+}_{k} $: The vectors $ \nu_{+}(\ell) $ and $ \nu_{+}(k) $ represent the orientations of the normal bundles of $ \ell $ and $ k $, respectively. Two circles with dotted lines represent the boundary of the $ \rho $-neighbourhoods of $ I $ and $ J $, respectively. The dots represent points in $ \Gamma x $. The blue dots belong to both $ E^{+}_{\ell} $ and $ E^{+}_{k} $. But the black dots do not because they belong to the negative side of the boundary of $ E_{\ell} $ or $ E_{k} $, respectively
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Jesús A. Álvarez López
Ramón Barral Lijó
John Hunton
Hiraku Nozawa
John R. Parker
Construction of $ S_{\ell} $ in $ \mathbb{H}^2 $. The black dots represent points in $ \Gamma x $, the blue area is $ E_{\ell} $, the red dots represent points in $ S_{\ell} $.
The disks represent the inverse image of $ \Delta $. The projection of $ k_{1} $ to $ \Sigma $ has one-sided tangency, while the projection of $ k_{2} $ to $ \Sigma $ does not.
A 12-gon P
A triangle T
The picture on the left represents $ T\subset \mathbb{T}^n $; the right one its lift to $ \mathbb{R}^n $ following a grid pattern
Approximation of $ S^{+}_{\ell} $ by $ S^{+}_{k} $: The vectors $ \nu_{+}(\ell) $ and $ \nu_{+}(k) $ represent the orientations of the normal bundles of $ \ell $ and $ k $, respectively. Two circles with dotted lines represent the boundary of the $ \rho $-neighbourhoods of $ I $ and $ J $, respectively. The dots represent points in $ \Gamma x $. The blue dots belong to both $ E^{+}_{\ell} $ and $ E^{+}_{k} $. But the black dots do not because they belong to the negative side of the boundary of $ E_{\ell} $ or $ E_{k} $, respectively