arXiv subjects

Marcy Barge

Publications and source records attributed to Marcy Barge.

18 recordsLinked to original sources

arXiv2020

Complete regularity of Ellis semigroups of $\mathbb Z$-actions

It is shown that the Ellis semigroup of a $\mathbb Z$-action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$- or $\mathbb R$-action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

math.DS↗

arXiv2018

Cohomology of Substitution Tiling Spaces

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

math.DS↗

arXiv2018

Homological Pisot Substitutions and Exact Regularity

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.

math.DS↗

arXiv2018

Quotient Cohomology for Tiling Spaces

We define a relative version of tiling cohomology for the purpose of comparing the topology of tiling spaces when one is a factor of the other. We illustrate this with examples, and outline a method for computing the cohomology of tiling spaces of finite type.

math.DS↗

arXiv2015

Pure discrete spectrum for a class of one-dimensional substitution tiling systems

We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the R-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all beta-substitutions for beta a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from the Jacobi-Perron and Brun continued fraction expansions.

math.DS↗

arXiv2015

The Pisot Conjecture for $β$-substitutions

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 solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) every Pisot number is weakly finitary.

math.DS↗

arXiv2014

Equicontinuous factors, proximality and Ellis semigroup for Delone sets

We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. We consider in particular, the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.

math.DS↗

arXiv2013

Factors of Pisot tiling spaces and the coincidence rank conjecture

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 action by translation on a group called the maximal equicontinuous factor. The integer m is the coincidence rank of the substitution and equals one if and only if translation on the tiling space has pure discrete spectrum. By considering factors intermediate between a tiling space and its maximal equicontinuous factor, we establish a lower bound on the cohomology of a one-dimensional Pisot substitution tillng space with coincidence rank two and dilation of odd norm. The Coincidence Rank Conjecture, for coincidence rank two, is a corollary.

math.DS↗

arXiv2013

Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of $\nats$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone-Čech compactification $β\nats .$ This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone-Čech compactification of $\nats.$

math.CO↗

arXiv2012

Maximal equicontinuous factors and cohomology for tiling spaces

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

math.DS↗

arXiv2011

Geometric realization for substitution tilings

Given an n-dimensional substitution whose associated linear expansion is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the associated tiling space to construct a finite-to-one semi-conjugacy, called geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If the linear expansion satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of the linear expansion, the image of geometric realization is the entire torus and coincides with the map onto the maximal equicontinuous factor of the translation action on the tiling space. We are led to formulate a higher-dimensional generalization of the Pisot Substitution Conjecture: If the linear expansion satisfies the Pisot family condition and the rank of the one-dimensional cohomology of the tiling space equals the generalized degree of the linear expansion, then the translation action on the tiling space has pure discrete spectrum.

math.DS↗

arXiv2011

Proximality and pure point spectrum for tiling dynamical systems

We investigate the role of the proximality relation for tiling dynamical systems. Under two hypotheses, namely that the minimal rank is finite and the set of fiber distal points has full measure we show that the following conditions are equivalent: (i) proximality is topologically closed, (ii) the minimal rank is one, (iii) the continuous eigenfunctions of the translation action span the L^2-functions over the tiling space. We apply our findings to model sets and to Meyer substitution tilings. It turns out that the Meyer property is crucial for our analysis as it allows us to replace proximality by the a priori stronger notion of strong proximality.

math.DS↗

arXiv2011

Asymptotic structure in substitution tiling spaces

Every sufficiently regular space of tilings of $\R^d$ has at least one pair of distinct tilings that are asymptotic under translation in all the directions of some open $(d-1)$-dimensional hemisphere. If the tiling space comes from a substitution, there is a way of defining a location on such tilings at which asymptoticity `starts'. This leads to the definition of the {\em branch locus} of the tiling space: this is a subspace of the tiling space, of dimension at most $d-1$, that summarizes the `asymptotic in at least a half-space' behavior in the tiling space. We prove that if a $d$-dimensional self-similar substitution tiling space has a pair of distinct tilings that are asymptotic in a set of directions that contains a closed $(d-1)$-hemisphere in its interior, then the branch locus is a topological invariant of the tiling space. If the tiling space is a 2-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a 1-dimensional simplicial complex.

math.DS↗

arXiv2010

Classification of expansive attractors on surfaces

We prove the conjecture of F. Rodriguez Hertz and J. Rodriguez Hertz (2006) that every nontrivial transitive expansive attractor of a homeomorphism of a compact surface is a derived from pseudo-Anosov attractor.

math.DS↗

arXiv2008

The Branch Locus for One-Dimensional Pisot Tiling Spaces

If phi is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Phi on the tiling space T_Phi factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Phi-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.

math.DS↗

arXiv2007

Cohomology in one-dimensional substitution tiling spaces

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ``forces its border.'' One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. For one-dimensional substitution tiling spaces, we describe a modification of the Anderson-Putnam complex on collared tiles that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology.

math.DS↗

arXiv2005

Proximality in Pisot Tiling Spaces

A substitution $\vp$ is strong Pisot if its abelianization matrix is non-singular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot $\vp$ that satisfies a no cycle condition and for which the translation flow on the tiling space $\T_\vp$ has pure discrete spectrum, we describe the collection $\T^P_\vp$ of pairs of proximal tilings in $\T_\vp$ in a natural way as a substitution tiling space. We show that if $ψ$ is another such substitution, then $\T_\vp $ and $\T_ψ$ are homeomorphic if and only if $\T^P_\vp$ and $\T^P_ψ$ are homeomorphic. We make use of this invariant to distinguish tiling spaces for which other known invariants are ineffective. In addition, we show that for strong Pisot substitutions, pure discrete spectrum of the flow on the associated tiling space is equivalent to proximality being a closed relation on the tiling space.

math.DS↗

arXiv1999

Homeomorphisms of One-dimensional Inverse Limits with Applications to Substitution Tilings, Unstable Manifolds, and Tent Maps

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limit spaces are homeomorphic if and only if the substitutions associated with f and g are weakly equivalent. This, and related results, are applied to one-dimensional substitution tiling spaces, one-dimensional unstable manifolds of hyperbolic sets, and inverse limits of tent maps with periodic critical points.

math.GT↗