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Chaotic Delone sets

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

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

    Mathematics Subject Classification: 37D45, 52C23, 37B51.

    Citation:

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  • 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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