Corpus ID: 218571017

Rauzy induction of polygon partitions and toral $\mathbb{Z}^2$-rotations.

@article{Labbe2020RauzyIO,
  title={Rauzy induction of polygon partitions and toral \$\mathbb\{Z\}^2\$-rotations.},
  author={S'ebastien Labb'e},
  journal={arXiv: Dynamical Systems},
  year={2020}
}
We propose a method for proving that a toral partition into polygons is a Markov partition for a given toral $\mathbb{Z}^2$-rotation ($\mathbb{Z}^2$-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… Expand
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References

SHOWING 1-10 OF 69 REFERENCES
A self-similar aperiodic set of 19 Wang tiles
We define a Wang tile set $$\mathcal {U}$$U of cardinality 19 and show that the set $$\Omega _\mathcal {U}$$ΩU of all valid Wang tilings $$\mathbb {Z}^2\rightarrow \mathcal {U}$$Z2→U is self-similar,Expand
Renormalization of polygon exchange maps arising from corner percolation
We describe a family {Ψα,β} of polygon exchange transformations parameterized by points (α,β) in the square $[0, {\frac{1}{2}}]\times[0, {\frac{1}{2}}]$. Whenever α and β are irrational, Ψα,β hasExpand
Substitutive Structure of Jeandel–Rao Aperiodic Tilings
  • S. Labbé
  • Computer Science, Mathematics
  • Discret. Comput. Geom.
  • 2021
TLDR
It is shown that there exists a minimal subshift X 0 of X 0 such that every tiling in X 0 can be decomposed uniquely into 19 distinct patches of size ranging from 45 to 112 that are equivalent to a set of 19 self-similar aperiodic Wang tiles. Expand
Arithmetic construction of sofic partitions of hyperbolic toral automorphisms
For each irreducible hyperbolic automorphism $A$ of the $n$-torus we construct a sofic system $(\Sigma,\sigma)$ and a bounded-to-one continuous semiconjugacy from $(\Sigma,\sigma)$ to $({\BbbExpand
Geometry, dynamics, and arithmetic of $S$-adic shifts
This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iteratingExpand
Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions
Nous definissons des substitutions bi-dimensionnelles ; ces substitutions engendrent des suites doubles reliees a des approximations discretes de plans irrationnels. Elles sont obtenues au moyen deExpand
Continued Fraction Algorithms for Interval Exchange Maps: an Introduction ?
Rotations on the circle T = R/Z are the prototype of quasiperiodic dynamics. They also constitute the starting point in the study of smooth dynamics on the circle, as attested by the concept ofExpand
Reversing and extended symmetries of shift spaces
The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidityExpand
Sequences with minimal block growth II
  • E. Coven
  • Mathematics, Computer Science
  • Mathematical systems theory
  • 2005
TLDR
The class of sequences with minimal block growth is closed under flow isomorphism and contains all sequences x satisfying P(x, n) = n + 1 for all n _> 1. Expand
SYMBOLIC DYNAMICS AND MARKOV PARTITIONS
The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolicExpand
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