We give a short discussion about a weaker form of minimality (called quasi-minimality). We call a system quasi-minimal if all dense orbits form an open set. It is hard to find examples which are not already minimal. Since elliptic behaviour makes them minimal, these systems are regarded as parabolic systems. Indeed, we show that a quasi-minimal homeomorphism on a manifold is not expansive (hyperbolic).
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.
We prove some consistency results concerning the Moving Off Property for locally compact spaces and thus the question of whether their function spaces are Baire.
The results of Iv. Prodanov on abstract spectra and separative algebras were announced in the journal Trudy Mat. Inst. Steklova, 154, 1983, 200--208, but their proofs were never written by him in the form of a manuscript, preprint or paper. Since the untimely death of Ivan Prodanov withheld him from preparing the full version of this paper and since, in our opinion, it contains interesting and important results, we undertook the task of writing a full version of it and thus making the results from it known to the mathematical community. So, the aim of this paper is to supply with proofs the results of Ivan Prodanov announced in the cited above paper, but we added also a small amount of new results. The full responsibility for the correctness of the proofs of the assertions presented below in this work is taken by us; just for this reason our names appear as authors of the present paper.
We study injecti
We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.
We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field $mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present
Answering a question of P. Bankston, we show that the pseudoarc is a co-existentially closed continuum. We also show that $C(X)$, for $X$ a nondegenerate continuum, can never have quantifier elimination, answering a question of the the first and third named authors and Farah and Kirchberg.
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $Game (D_n({bfSigma}^0_2))$ for some natural number $ngeq 1$, where $Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^omega$ into the class ${bfDelta}^1_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${bfDelta}^1_2$.
Let $A$ be an abelian topological $G$-module. We give an interpretion for the second cohomology, $H^{2}(G,A)$, of $G$ with coefficients in $A$. As a result we show that if $P$ is a projective topological group, then $H^{2}(P,A)=0$ for every abelian topological $P$-module $A$.
