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We define a partition $mathcal{P}_0$ and a $mathbb{Z}^2$-rotation ($mathbb{Z}^2$-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition $mathcal{P}_mathcal{U}$ and a $mathbb{Z}^2$-rotation on $mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that $mathcal{P}_mathcal{U}$ is a Markov partition for the $mathbb{Z}^2$-rotation on $mathbb{T}^2$. We prove in both cases that the toral $mathbb{Z}^2$-rotation is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is ${1,2,8}$. The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the toral $mathbb{Z}^2$-rotations. It provides a construction of these Wang shifts as model sets of 4-to-2 cut and project schemes. A do-it-yourself puzzle is available in the appendix to illustrate the results.
We propose a method for proving that a toral partition into polygons is a Markov partition for a given toral $mathbb{Z}^2$-rotation, i.e., $mathbb{Z}^2$-action defined by rotations on a torus. If $mathcal{X}_{mathcal{P},R}$ denotes the symbolic dynam
Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a
A group $G$ is said to be periodic if for any $gin G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $mu$ is finite. Moreover if the group
We provide a proof of the (well-known) result that the Poincare exponent of a non-elementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof only uses elementary hyperbolic and fractal geometry.
We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the