Related papers: Fractal tiles induced by tent maps
In this paper, we study arithmetical and topological properties for a class of Rauzy fractals ${\mathcal R}_a$ given by the polynomial $x^3- ax^2+x-1$ where $a \geq 2$ is an integer. In particular, we prove the number of neighbors of…
Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural…
In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical…
In this paper, a class of fractals, called quadrilateral labyrinth fractals, are introduced and studied. They are a special kind of fractals on any quadrilateral on the plane. This type of fractal is motivated by labyrinth fractal on the…
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
We count tilings of a rectangle of integer sides m-1 and n-1 by a special set of tiles. The result is obtained fron the study of the kernel of the adjacency matrix of an n x n rectangular graph of Z x Z.
Given a metric measure space $X$, we consider a scale of function spaces $T^{p,q}_s(X)$, called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on…
The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example…
We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are…
We provide a proof of Pisot conjecture, a classification problem in Ergodic Theory on recurrent sequences generated by irreducible Pisot substitutions.
Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the…
Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of…
Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three…
Topological conjugateness of one dimensional unimodal dynamical systems, which are generated by interval [0, 1] into itself maps are studied. We study the smoothness and differentiability of the conjugacy of symmetrical and non-symmetrical…
Here I present several theorems about trapezoids tilings. The first one is related to trapezoids with rational base relation, the other ones are related to those with base relation from quadratic number field.
We introduce a new dynamical system: the wind-tree tiling billiards. This system studies trajectories of a ray in Euclidean space which has a negative refractive index when encountering rectangular obstacles located at lattice points. We…
In this article we show that there exist measurable sets W in the plane with finite measure that tile the plane in a measurable way under the action of a expansive matrix A, an affine Weyl group W, and a full rank lattice G. This note is…
We explore a deep connection between fracton order and product codes. In particular, we propose and analyze conditions on classical seed codes which lead to fracton order in the resulting quantum product codes. Depending on the properties…
Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some 'topological' features: they support fractional bulk excitations (dubbed fractons), and a ground state…
In a previous paper we considered a sequence of maps on a complete metric space $(X,d)$ and derived an extension of the Banach fixed point theorem. We showed that backward trajectories of maps $X\to X$ converge under mild conditions and…