Related papers: A combinatorial approach to products of Pisot subs…
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…
This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete…
We provide a proof for one version of Pisot conjecture. We make use of the weak mixing property of the subshift of finite type derived from the prefix-suffix automaton to conclude that the substitution dynamical system has pure discrete…
Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the…
We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and…
It has been a long standing problem to find good symbolic codings for translations on the $d$-dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties.…
We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach…
This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval…
The matrix of a substitution is not sufficient to completely determine the dynamics associated, even in simplest cases since there are many words with the same abelianization. In this paper we study the common points of the canonical broken…
Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact sets, where each set is a…
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…
We prove the Pisot Conjecture for beta-substitutions: If beta is a Pisot number, the tiling dynamical system associated with the beta-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic…
We introduce a procedure for establishing pure discrete spectrum for substitution tiling systems of Pisot family type and illustrate with several examples.
We provide a survey of results from symbolic dynamics and algebraic topology relating to Grout, a new user-friendly program developed to calculate combinatorial properties and topological invariants of a large class of symbolic…
We develop the dichotomy spectrum for random dynamical system and demonstrate its use in the characterization of pitchfork bifurcations for random dynamical systems with additive noise. Crauel and Flandoli had shown earlier that adding…
We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars,…
Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emph{Rauzy fractals}, with $\sigma$. In our setting, these fractals are…
We consider a two-parameter family of random substitutions and show certain combinatorial and topological properties they satisfy. We establish that they admit recognisable words at every level. As a consequence, we get that the subshifts…
We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of…
We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the…