American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021016

Chaotic Delone sets

 1 Departamento e Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain 2 Research Organization of Science and Technology, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan 3 Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK 4 Department of Mathematical Sciences, Colleges of Science and Engineering, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan 5 Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK

* Corresponding author: Ramón Barral Lijó (ramonbarrallijo@gmail.com)

Received  August 2020 Revised  November 2020 Published  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.

Citation: Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa, John R. Parker. Chaotic Delone sets. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2021016
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References:
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
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