10 research outputs found
Rauzy induction of polygon partitions and toral -rotations
We extend the notion of Rauzy induction of interval exchange transformations
to the case of toral -rotation, i.e., -action
defined by rotations on a 2-torus. If denotes the
symbolic dynamical system corresponding to a partition and
-action such that is Cartesian on a sub-domain , we
express the 2-dimensional configurations in as
the image under a -dimensional morphism (up to a shift) of a configuration
in where
is the induced partition and is the
induced -action on .
We focus on one example for which we obtain
an eventually periodic sequence of 2-dimensional morphisms. We prove that it is
the same as the substitutive structure of the minimal subshift of the
Jeandel-Rao Wang shift computed in an earlier work by the author. As a
consequence, is a Markov partition for the associated toral
-rotation . It also implies that the subshift is
uniquely ergodic and is isomorphic to the toral -rotation
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 code to reproduce the proofs are
provided.Comment: v1:36 p, 11 fig; v2:40 p, 12 fig, rewritten before submission;
v3:after reviews; v4:typos and updated references; v5:typos and abstract; v6:
added a paragraph commenting that Algo 1 may not halt. Jupyter notebook
available at
https://nbviewer.jupyter.org/url/www.slabbe.org/Publications/arXiv_1906_01104.ipyn
Nonexpansive directions in the Jeandel-Rao Wang shift
We show that is the set of
slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao
Wang shift, where is the golden mean. This set is a
topological invariant allowing to distinguish the Jeandel-Rao Wang shift from
other subshifts. Moreover, we describe the combinatorial structure of the two
resolutions of the Conway worms along the nonexpansive directions in terms of
irrational rotations of the unit interval. The introduction finishes with
pictures of nonperiodic Wang tilings corresponding to what Conway called the
cartwheel tiling in the context of Penrose tilings. The article concludes with
open questions regarding the description of octopods and essential holes in the
Jeandel-Rao Wang shift.Comment: v1: 24 pages, 19 figures; v2: 30 pages, 23 figures, new section with
open questions on octopods and essential holes; v3: small fixe
A Numeration System for Fibonacci-like Wang Shifts
Motivated by the study of Fibonacci-like Wang shifts, we define a numeration
system for and based on the binary alphabet
. We introduce a set of 16 Wang tiles that admits a valid tiling of
the plane described by a deterministic finite automaton taking as input the
representation of a position and outputting a Wang tile.Comment: 17 pages, 5 figures, submitted to WORDS 202
A counterexample to the periodic tiling conjecture
The periodic tiling conjecture asserts that any finite subset of a lattice
which tiles that lattice by translations, in fact tiles
periodically. In this work we disprove this conjecture for sufficiently large
, which also implies a disproof of the corresponding conjecture for
Euclidean spaces . In fact, we also obtain a counterexample in a
group of the form for some finite abelian -group
. Our methods rely on encoding a "Sudoku puzzle" whose rows and other
non-horizontal lines are constrained to lie in a certain class of "-adically
structured functions", in terms of certain functional equations that can be
encoded in turn as a single tiling equation, and then demonstrating that
solutions to this Sudoku puzzle exist but are all non-periodic.Comment: 50 pages, 13 figures. Minor changes and additions of new reference
Around the Domino Problem – Combinatorial Structures and Algebraic Tools
Given a finite set of square tiles, the domino problem is the question of whether is it possible to tile the plane using these tiles. This problem is known to be undecidable in the planar case, and is strongly linked to the question of the periodicity of the tiling. In this thesis we look at this problem in two different ways: first, we look at the particular case of low complexity tilings and second we generalize it to more general structures than the plane, groups.
A tiling of the plane is said of low complexity if there are at most mn rectangles of size m × n appearing in it. Nivat conjectured in 1997 that any such tiling must be periodic, with the consequence that the domino problem would be decidable for low complexity tilings. Using algebraic tools introduced by Kari and Szabados, we prove a generalized version of Nivat’s conjecture for a particular class of tilings (a subclass of what is called of algebraic subshifts). We also manage to prove that Nivat’s conjecture holds for uniformly recurrent tilings, with the consequence that the domino problem is indeed decidable for low-complexity tilings.
The domino problem can be formulated in the more general context of Cayley graphs of groups. In this thesis, we develop new techniques allowing to relate the Cayley graph of some groups with graphs of substitutions on words. A first technique allows us to show that there exists both strongly periodic and weakly-but-not-strongly aperiodic tilings of the Baumslag-Solitar groups BS(1, n). A second technique is used to show that the domino problem is undecidable for surface groups. Which provides yet another class of groups verifying the conjecture saying that the domino problem of a group is decidable if and only if the group is virtually free
Subshifts with Simple Cellular Automata
A subshift is a set of infinite one- or two-way sequences over a fixed finite set, defined by a set of forbidden patterns. In this thesis, we study subshifts in the topological setting, where the natural morphisms between them are ones defined by a (spatially uniform) local rule. Endomorphisms of subshifts are called cellular automata, and we call the set of cellular automata on a subshift its endomorphism monoid. It is known that the set of all sequences (the full shift) allows cellular automata with complex dynamical and computational properties. We are interested in subshifts that do not support such cellular automata. In particular, we study countable subshifts, minimal subshifts and subshifts with additional universal algebraic structure that cellular automata need to respect, and investigate certain criteria of ‘simplicity’ of the endomorphism monoid, for each of them. In the case of countable subshifts, we concentrate on countable sofic shifts, that is, countable subshifts defined by a finite state automaton. We develop some general tools for studying cellular automata on such subshifts, and show that nilpotency and periodicity of cellular automata are decidable properties, and positive expansivity is impossible. Nevertheless, we also prove various undecidability results, by simulating counter machines with cellular automata. We prove that minimal subshifts generated by primitive Pisot substitutions only support virtually cyclic automorphism groups, and give an example of a Toeplitz subshift whose automorphism group is not finitely generated. In the algebraic setting, we study the centralizers of CA, and group and lattice homomorphic CA. In particular, we obtain results about centralizers of symbol permutations and bipermutive CA, and their connections with group structures.Siirretty Doriast