Related papers: When Shape Matters: Deformations of Tiling Spaces
We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of $\mathbb{T}^2$ in the homotopy class of any linear expanding map with integer eigenvalues. These examples exhibit branching of centre curves along…
We relate a balancing property of letters for bi-infinite sequences to the invariance of the resulting 1-dimensional tiling dynamics under changes in the lengths of the tiles. If the language of the sequence space is finitely balanced, then…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
A particular, two-dimensional, tiling model, composed by the so called Wang tiles has been studied at finite temperature by Monte Carlo numerical simulations. In absence of any thermal bath the Wang tiles give the opportunity of building a…
This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works…
We study single-flip dynamics in sets of three-dimensional rhombus tilings with fixed polyhedral boundaries. This dynamics is likely to be slowed down by so-called ``cycles'': such structures arise when tilings are encoded via the…
The topological properties of a material depend on its symmetries, parameters, and spatial dimension. Changes in these properties due to parameter and symmetry variations can be understood by computing the corresponding topological…
Combinatorial mechanical metamaterials feature spatially textured soft modes that yield exotic and useful mechanical properties. While a single soft mode often can be rationally designed by following a set of tiling rules for the building…
Alignment interactions in active matter are typically modeled as relaxational dynamics toward local consensus. In unbounded systems, this makes alignment effectively decoupled from local density and therefore unable to sustain self-confined…
We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes,…
We study the topological structure and the topological dynamics of groups of homeomorphisms of scattered spaces. For a large class of them (including the homeomorphism group of any ordinal space or of any locally compact scattered space),…
I investigate a class of dynamical systems in which finite pieces of spacetime contain finite amounts of information. Most of the guiding principles for designing these systems are drawn from general relativity: the systems are…
We consider a net of *-algebras, locally around any point of observation, equipped with a natural partial order related to the isotony property. Assuming the underlying manifold of the net to be a differentiable, this net shall be…
In this paper we prove that if two self-similar tiling systems, with respective stretching factors $\lambda_1$ and $\lambda_2$, have a common factor which is a non periodic tiling system, then $\lambda_1$ and $\lambda_2$ are…
To preserve previously learned representations, continual learning systems must strike a balance between plasticity, the ability to acquire new knowledge, and stability. This stability-plasticity dilemma affects how representations can be…
The study of algebraic properties of groups of transformations of a manifold gives rise to an interplay between different areas of mathemathics such as topology, geometry, and dynamical systems. Especially, in this paper, we point out some…
We investigate topological mixing of compatible random substitutions. For primitive random substitutions on two letters whose second eigenvalue is greater than one in modulus, we identify a simple, computable criterion which is equivalent…
The discovery of topological matter has revolutionized the field of condensed matter physics giving rise to many interesting phenomena, and fostering the development of new quantum technologies. In this thesis we study the quantum dynamics…
We propose a formalism for tilings with infinite local complexity (ILC), and especially fusion tilings with ILC. We allow an infinite variety of tile types but require that the space of possible tile types be compact. Examples include…
A fundamental objective of materials modeling is identifying atomic structures that align with experimental observables. Conventional approaches for disordered materials involve sampling from thermodynamic ensembles and hoping for an…