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A matrix formalism for conjugacies of higher-dimensional shifts of finite type

Michael Schraudner (2008)

Colloquium Mathematicae

We develop a natural matrix formalism for state splittings and amalgamations of higher-dimensional subshifts of finite type which extends the common notion of strong shift equivalence of ℤ⁺-matrices. Using the decomposition theorem every topological conjugacy between two d -shifts of finite type can thus be factorized into a finite chain of matrix transformations acting on the transition matrices of the two subshifts. Our results may be used algorithmically in computer explorations on topological...

Embedding tiling spaces in surfaces

Charles Holton, Brian F. Martensen (2008)

Fundamenta Mathematicae

We show that an aperiodic minimal tiling space with only finitely many asymptotic composants embeds in a surface if and only if it is the suspension of a symbolic interval exchange transformation (possibly with reversals). We give two necessary conditions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of substitutions on two symbols our classification is nearly complete. The results characterize the codimension one hyperbolic attractors of surface diffeomorphisms...

Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to β -shifts

Veronica Baker, Marcy Barge, Jaroslaw Kwapisz (2006)

Annales de l’institut Fourier

This article is devoted to the study of the translation flow on self-similar tilings associated with a substitution of Pisot type. We construct a geometric representation and give necessary and sufficient conditions for the flow to have pure discrete spectrum. As an application we demonstrate that, for certain beta-shifts, the natural extension is naturally isomorphic to a toral automorphism.

Maximal equicontinuous factors and cohomology for tiling spaces

Marcy Barge, Johannes Kellendonk, Scott Schmieding (2012)

Fundamenta Mathematicae

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that in degree one this map is injective and has torsion free cokernel. We show by example, however, that, in degree one, the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.

Mixing properties of nearly maximal entropy measures for d shifts of finite type

E. Robinson, Ayşe Şahin (2000)

Colloquium Mathematicae

We prove that for a certain class of d shifts of finite type with positive topological entropy there is always an invariant measure, with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.

On the helix equation

Mohamed Hmissi, Imene Ben Salah, Hajer Taouil (2012)

ESAIM: Proceedings

This paper is devoted to the helices processes, i.e. the solutions H : ℝ × Ω → ℝd, (t, ω) ↦ H(t, ω) of the helix equation H ( 0 ) = 0 ; H ( s + t,ω ) = H ( s, Φ ( t,ω ) ) + H ( t,ω ) where Φ : ℝ × Ω → Ω, (t, ω) ↦ Φ(t, ω) is a dynamical system on a measurable space (Ω, ℱ).More precisely, we investigate dominated solutions and non differentiable solutions of the helix equation. For the last case, the Wiener helix plays a fundamental role. Moreover, some relations with the cocycle equation defined...

Proximality in Pisot tiling spaces

Marcy Barge, Beverly Diamond (2007)

Fundamenta Mathematicae

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space φ has pure discrete spectrum, we describe the collection φ P of pairs of proximal tilings in φ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then φ and ψ are homeomorphic if and...

Shadowing and internal chain transitivity

Jonathan Meddaugh, Brian E. Raines (2013)

Fundamenta Mathematicae

The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with...

Shadowing in multi-dimensional shift spaces

Piotr Oprocha (2008)

Colloquium Mathematicae

We show that the class of expansive d actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction ν d . Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.

The branch locus for one-dimensional Pisot tiling spaces

Marcy Barge, Beverly Diamond, Richard Swanson (2009)

Fundamenta Mathematicae

If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space Φ factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Φ-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.

Tilings associated with non-Pisot matrices

Maki Furukado, Shunji Ito, E. Arthur Robinson (2006)

Annales de l’institut Fourier

Suppose A G l d ( ) has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called “good star”, and has A * 0 , where A * is an oriented compound of A . A morphism θ of the free group on { 1 , 2 , , d } is called a non-abelianization of A if it has structure matrix A . We show that there is a tiling substitution Θ whose “boundary substitution” θ = Θ is a non-abelianization of A . Such a tiling substitution Θ leads to a self-affine tiling of E u 2 with A u : = A | E u G L 2 ( ) as its expansion. In the last section we find conditions on A so...

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