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Register a new userMarkov partitions for toral $mathbb{Z}^2$-rotations featuring Jeandel-Rao Wang shift and model sets
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Authors
Sebastien Labbe
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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.
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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 dynamical system corresponding to a partition $mathcal{P}$ and $mathbb{Z}^2$-action $R$ such that $R$ is Cartesian on a sub-domain $W$, we express the 2-dimensional configurations in $mathcal{X}_{mathcal{P},R}$ as the image under a $2$-dimensional morphism (up to a shift) of a configuration in $mathcal{X}_{widehat{mathcal{P}}|_W,widehat{R}|_W}$ where $widehat{mathcal{P}}|_W$ is the induced partition and $widehat{R}|_W$ is the induced $mathbb{Z}^2$-action on the sub-domain $W$. The induced $mathbb{Z}^2$-action extends the notion of Rauzy induction of IETs to the case of $mathbb{Z}^2$-actions where subactions are polytope exchange transformations. This allows to describe $mathcal{X}_{mathcal{P},R}$ by a $S$-adic sequence of 2-dimensional morphisms. We apply the method on one example and we obtain a sequence of 2-dimensional morphisms which is eventually periodic leading to a self-induced partition. We prove that its substitutive structure is the same as the substitutive structure of the minimal subshift $X_0$ of the Jeandel-Rao Wang shift computed in an earlier work by the author. As a consequence, we deduce the equality of the two subshifts and it implies that the partition is a Markov partition for the associated toral $mathbb{Z}^2$-rotation since $X_0$ is a shift of finite type. It also implies that $X_0$ is uniquely ergodic and is isomorphic to the toral $mathbb{Z}^2$-rotation $R_0$ which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and the code to reproduce the proofs are provided.
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 nontrivial translation. We study herein the Wang shift $Omega_0$ made of all valid tilings using the set $mathcal{T}_0$ of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift $X_0$ of $Omega_0$ such that every tiling in $X_0$ can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112 that are equivalent to a set of 19 self-similar and aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel-Rao tilings, as we believe that $Omega_0setminus X_0$ is a null set for any shift-invariant probability measure on $Omega_0$. The proof is based on 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles. The 2 other steps involve the addition of decorations to deal with fault lines and changing the base of the $mathbb{Z}^2$-action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions, and shear conjugacy from a set of Wang tiles.
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 consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.
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 argument fails only if the automorphisms have strong algebraic relations.
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