Related papers: Substitution Rules for Higher-Dimensional Paperfol…

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove…

This paper is about the tiling dynamical systems approach to the study of aperiodic order. We compare and contrast four related types of systems: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional…

We present a new correspondence between a d-dimensional dynamical system and a whole family of (d+1)-dimensional systems. This new scale-holographic relation is built by the explicit introduction of a dimensionful constant which determines…

There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete…

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include…

Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…

The construction of supersymmetric invariant actions on a spacetime manifold with a boundary is carried out using the "ectoplasm" formalism for the construction of closed forms in superspace. Non-trivial actions are obtained from the…

Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished "root" nodes and with matrix edge weights, often taken from Bose-Mesner…

We consider the structure of Pisot substitution tiling spaces, in particular, the structure of those spaces for which the translation action does not have pure discrete spectrum. Such a space is always a measurable m-to-one cover of an…

This paper is a sharp and focussed exploration of the Fibonacci substitution and the mathematical entity it gives rise to, the Fibonacci word. Our investigations are both of an algebraic and a geometric nature. Indeed, it is the combination…

We present a general formalism which allows us to derive the evolution equations describing one-dimensional (1D) and isotropic 2D interfacelike systems, that is based on symmetries, conservation laws, multiple scale arguments, and exploits…

If two control systems on manifolds of the same dimension are dynamic equivalent, we prove that either they are static equivalent --i.e. equivalent via a classical diffeomorphism-- or they are both ruled; for systems of different…

It is well-known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as…

We consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space $\mathbb R^d$. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a…

In this work, we begin the study of a new class of dynamical systems determined by interval maps generated by the symbolic action of erasing substitution rules. We do this by discussing in some detail the geometric, analytical, dynamical…

In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how…

This paper offers a geometric framework for modeling port-Hamiltonian systems on discrete manifolds. The simplicial Dirac structure, capturing the topological laws of the system, is defined in terms of primal and dual cochains related by…

Topological techniques are powerful tools for characterizing the complexity of many dynamical systems, including the commonly studied area-preserving maps of the plane. However, the extension of many topological techniques to higher…

We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses…

A simple discontinuous map is proposed as a generic model for nonlinear dynamical systems. The orbit of the map admits exact solutions for wide regions in parameter space and the method employed (digit manipulation) allows the mathematical…