On properties of the set of invariant lines of a Brouwer homeomorphism

Abstract

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points.     

On fractional iterates of a Brouwer homeomorphism embeddable in a flow

We present a method for finding continuous (and consequently homeomorphic) orientation preserving iterative roots of a Brouwer homeomorphism which is embeddable in a flow. To obtain the roots we use a countable family of maximal parallelizable regions of the flow which is a cover of the plane. The maximal parallelizable regions are unions of equivalence classes of an appropriate equivalence relation. We show that if an equivalence class is invariant under the nth iterate of a Brouwer homeomorphism g, then it is invariant under g. We use this fact to prove that each maximal parallelizable region of the flow must be invariant under all homeomorphic orientation preserving iterative roots of the given Brouwer homeomorphism.     

An Equivalence Relation between Multiresolution Analysis ofL(R)

This paper is concerned with the development of an equivalence relation between two multiresolution analysis ofL²(R). The relation called unitary equivalence is created by the action of a unitary operator in such a way that the multiresolution structure and the decomposition and reconstruction algorithms remain invariant. A characterization in terms of the scaling functions of the multiresolution analysis is given. Distinct equivalence classes of multiresolution analysis are derived. Finally, we prove that B-splines give rise to nonequivalent examples.     

The full power of the half-power

We use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully unveiled.     

On the topological equivalence of flows of Brouwer homeomorphisms

We prove that the set of all regular points of a flow of Brouwer homeomorphisms is invariant under topological equivalence of flows. We also show that a similar result holds for the first prolongational limit set.     

Automorphic equivalence in the varieties of representations of Lie algebras

Abstract

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.     

Topological dynamics and the complexity of strong types

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g., we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup.We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over Ø is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let \(\mathfrak{C}\) be a monster model of a countable theory, p ∈ S(Ø), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p(\(\mathfrak{C}\)). Then, exactly one of the following holds: (1) E is relatively definable (on p(\(\mathfrak{C}\))), smooth, and has finitely many classes, (2) E is not relatively definable, but it is type-definable, smooth, and has 2^?0 classes, (3) E is not type definable and not smooth, and has 2^?0 classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.     

Applications of periodic unfolding on manifolds

Abstract

We show how the newly developed method of periodic unfolding on Riemannian manifolds can be applied to PDE problems: we consider the homogenization of an elliptic model problem. In the limit, we obtain a generalization of the well-known limit- and cell-problem. By constructing an equivalence relation of atlases, one can show the invariance of the limit problem with respect to this equivalence relation. This implies e.g. that the homogenization limit is independent of change of coordinates or scalings of the reference cell. These type of problems emerge for example when modeling surface diffusion and reactions in heterogeneous catalysts, or in processes involved in crystal formation.     

Real numbers and projective spaces: Intuitionistic reasoning with undecidable basic relations

Brouwer introduced in 1924 the notion of an apartness relation for real numbers, with the idea that whenever it holds, a finite computation verifies it in contrast to equality. The idea was followed in Heyting’s axiomatization of intuitionistic projective geometry. Brouwer in turn worked out an intuitionistic theory of “virtual order.” It is shown that Brouwer’s proof of the equivalence of virtual and maximal order goes only in one direction, and that Heyting’s axiomatization needs to be made a bit stronger.     

Structural matrix algebras related to finite distributive lattices

Abstract

In this paper, we investigate the relation between a structural matrix algebra and the lattice properties of its lattice of invariant subspaces, and reprove known results in a fresh and an explanatory way. Moreover, we also prove the theorem which is partially converse of Proposition 2.6 of [15].     

Foliations of polynomial growth are hyperfinite

We define a class of equivalence relations with polynomial growth and show that such relations always support finite invariant measures and are hyperfinite. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. We also extend a result of Dye for countable groups to show that if a locally compact second countable groupG acts freely on a Lebesgue spaceX with finite invariant measure, so that the orbit relation onX is hyperfinite, thenG is amenable.     

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Summary For Gibbsian systems of particles inR ^d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.     

Une version feuilletée du théorème de translation de Brouwer

The Brouwers plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a proper topological imbedding C of R, disjoint from its image and separating $f(C)$ and $f^{-1}(C)$. Such a curve is called a Brouwer line. We prove that we can construct a foliation of the plane by Brouwer lines.      

Intuitionistic sequential compactness?

Perhaps because the classical notion of sequential compactness fails to apply constructively even to $\{0,1\}$, Brouwer and his successors have paid little attention to the possibility of a constructive counterpart that is classically equivalent to sequential compactness and has serious potential for applications in analysis. We discuss such a notion – the anti-Specker property – and its equivalence, over Bishop-style constructive mathematics, to Brouwer’s fan theorem for $c$-bars.     

Coût des relations d’équivalence et des groupes

We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups. Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions over amenable groups and for direct product situations. For instance, the cost of the free group on n generators equals n. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of the paper, we give a mercuriale, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable, but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra. Oblatum 27-I-1999 & 4-IV-1999 / Published online: 22 September 1999     

Ordered K-groups associated to substitutional dynamics

The Matsumoto K₀-group is an interesting invariant of flow equivalence for symbolic dynamical systems. Because of its origin as the K-theory of a certain C^∗-algebra, which is also a flow invariant, this group comes equipped with a flow invariant order structure. We emphasize this order structure and demonstrate how methods from operator algebra and symbolic dynamics combine to allow a computation of it in certain cases, including Sturmian and primitive substitutional shifts. In the latter case we show by example that the ordered group is a strictly finer invariant than the group itself.     

Martin’s conjecture and strong ergodicity

In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set.     

Shift equivalence and the Conley index

In this paper we introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

     

Approximation of analytic by Borel sets and definable countable chain conditions

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ ₁ ¹ countable chain condition if there is aΣ ₁ ¹ equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ ₁ ¹ set is equal to a Borel set modulo a set fromI iffI fulfils theΣ ₁ ¹ countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ ₁ ¹ sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition. Research partially supported by NSF grant DMS-9317509.     

Homotopy invariants of Gauss words

By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopic to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.