Alex Clark, Lorenzo Sadun
We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate.
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We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily finite. We also prove that a subshift over any finitely generated group that consists of finite orbits is finite, and related results for tilings of Euclidean space.
We present a technique to lift some tilings of the discrete hyperbolic plane -- tilings defined by a 1D substitution -- into a zero entropy subshift of finite type (SFT) on non-abelian amenable Baumslag-Solitar groups $BS(1,n)$ for $n\geq2$. For well chosen hyperbolic tilings, this SFT is also aperiodic and minimal. As an application we construct a strongly aperiodic SFT on $BS(1,n)$ with a hierarchical structure, which is an analogue of Robinson's construction on $\mathbb{Z}^2$ or Goodman-Strauss's on $\mathbb{H}_2$.
We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obta…
Domino tilings have been studied extensively for both their statistical properties and their dynamical properties. We construct a subshift of finite type using matching rules for several types of dominos. We combine the previous results about domino tilings to show that our subshift of finite type has a measure of maximal entropy with which the subshift has completely positive entropy but is not isomorphic to a Bernoulli shift.
We investigate substitution subshifts and tiling dynamical systems arising from the substitutions (1) θ: 0 \rightarrow 001,1 \rightarrow 11001 and (2) η: 0 \rightarrow 001,1 \rightarrow 11100. We show that the substitution subshifts arising from θand ηhave minimal self-joinings and are mildly mixing. We also give a criterion for 1-dimensional tiling systems arising from θor ηto have minimal self-joinings. We apply this to obtain examples of mildly mixing 1-dimensional tiling systems.
Local generation of tilings
本文提出两种局部生成定义，探讨其性质并开发判断子移位是否满足的方法，通过实例验证结果。
We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This description allows to compute its cohomology groups, and prove that it is a model set.
We give a new proof of the Cobham's first theorem using ideas from symbolic dynamics and of the Cobham-Semenov theorem (in the primitive case) using ideas from tiling dynamics.
We explicitly construct a strongly aperiodic subshift of finite type for the discrete Heisenberg group. Our example builds on the classical aperiodic tilings of the plane due to Raphael Robinson. Extending those tilings to the Heisenberg group by exploiting the group's structure and posing additional local rules to prune out remaining periodic behavior we maintain a rich projective subdynamics on $\mathbb Z^2$ cosets. In addition the obtained subshift factors onto a strongly aperiodic, minimal sofic shift via a map that is invertible on a dense set of configurations.
In this paper we use fixed point tilings to answer a question posed by Michael Hochman and show that every one-dimensional effectively closed subshift can be implemented by a local rule in two dimensions. The proof uses the fixed-point construction of an aperiodic tile set and its extensions.
The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this paper, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift. arXiv:1805.08829 proved that this problem is co-recursively enumerable ($Π_0^1$-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic ($Σ_1^1$-complete), in higher dimension: $d…
We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a Z^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that 1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), 2) only a finite number of tile types are allowed, and 3…
Finitely Dependent Processes on Subshifts
本文研究子移位上平稳有限依赖过程的存在性，通过正反两方面结果揭示强混合性质与上同调障碍的作用，并利用Conway-Lagarias-Thurston高度函数刻画$\mathbb Z^2$上盒子拼图空间中的有限依赖过程。
Substitution tilings with transcendental inflation factor
本文针对任意λ>2，在无限字母表上构造替换，生成具有超越膨胀因子的替换平铺，证明相关动力系统严格遍历，并提供显式构造。
An embedding theorem for subshifts over amenable groups with the comparison property
本文证明了具有比较性质的可均群上符号动力系统的嵌入定理，在拓扑熵与因子条件下将子移位嵌入有限型移位，部分推广了Krieger和Lightwood的经典结论。
Arithmetical Hierarchy of the Besicovitch-Stability of Noisy Tilings
该研究针对噪声有限型子移位的Besicovitch稳定性的算法复杂度展开，通过构造不稳定非周期平铺完成图灵机不可判定问题归约，证明其不可判定且算术分层下界为$Π_2$-难，同时给出$Π_4$上界。
A general framework for quasi-isometries in symbolic dynamics beyond groups
本文提出一种代数结构，用于通过状态、生成元和关系编码可数图集，用于研究蓝图上的符号动力学，并通过准等距转移不变量。
We study homeomorphisms of tiling spaces with finite local complexity (FLC), of which suspensions of $d$-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms, and show that all homeomorphisms are a combination of tiling deformations, translations, and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology, and show that all orbit equivalences are combinations of tiling deformations, transla…
We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map ${\mathcal I}$ from the parameter space of possible shapes of tiles to $H^1$ of a model tiling space, with values in ${\mathbb R}^d$. Two tiling spaces that have the same image under ${\mathcal I}$ are mutually locally derivable (MLD). When the difference of the images is `asymptotically negligible', then the tiling dynamics are topologically conjugate, but g…
In this paper we study the combinatorics of free Borel actions of the group $\mathbb Z^d$ on Polish spaces. Building upon recent work by Chandgotia and Meyerovitch, we introduce property $F$ on $\mathbb Z^d$-shift spaces $X$ under which there is an equivariant map from any free Borel action to the free part of $X$. Under further entropic assumptions, we prove that any subshift $Y$ (modulo the periodic points) can be Borel embedded into $X$. Several examples satisfy property $F$ including, but not limited to, the space of proper $3$-colourings, tilings by rectangles (under a natural arithmetic …
It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). This theorem can be reworded in terms of $d$-dimensional infinite words: if two $\mathbb{Z}^d$-subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent. An analogue theorem is proved for the repetitivity function, which …
From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $β$ and the greedy $β$-transformation. In this paper, we consider different transformations generating expansions in base $β$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $β$-expansions. Remarkably, the symmetric $β$-transformation does …
For each $Π^0_1$ $S\subseteq \mathbb{N}$, let the $S$-square shift be the two-dimensional subshift on the alphabet $\{0,1\}$ whose elements consist of squares of 1s of various sizes on a background of 0s, where the side length of each square is in $S$. Similarly, let the distinct-square shift consist of seas of squares such that no two finite squares have the same size. Extending the self-similar Turing machine tiling construction of Durand, Romashchenko and Shen, we show that if $X$ is an $S$-square shift or any effectively closed subshift of the distinct square shift, then $X$ is sofic.
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 $Ω_0$ 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 $Ω_0$ such that every tiling in $X_0$ can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112…
Consider a subshift over a finite alphabet, $X\subset Λ^{\mathbb Z}$ (or $X\subsetΛ^{\mathbb N_0}$). With each finite block $B\inΛ^k$ appearing in $X$ we associate the empirical measure ascribing to every block $C\inΛ^l$ the frequency of occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $μ$ on $X$, whose restriction to the space of measures is compatible with the weak-$\star$ topology. Next, in this combined metric space we fix an open set $\mathcal U$ containing all ergodic measures, and we…
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