On families of invariant lines of a Brouwer homeomorphism

ABSTRACT

We present properties of equivalence classes of the codivergency relation defined for a Brouwer homeomorphism for which there exists a family of invariant pairwise disjoint lines covering the plane. In particular, using the codivergency relation we describe the sets of regular and irregular points for such Brouwer homeomorphisms. Moreover, we show that under this assumption the interior of each equivalence class of this relation is invariant and simply connected.     

An orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line

Though fixed point free homeomorphisms of the plane would appear to exhibit the simplest dynamical behavior, we show that the minimal sets can be quite complex. Every homeomorphism which is conjugate to a translation must have a closed invariant line. However we construct an orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line. We verify that no such line exists by considering the ‘fundamental regions” of our example. Fundamental regions, studied first by Stephen Andrea, are equivalence classes of points in the plane associated with a given homeomorphism. Two points are said to be in the same equivalence class if they can be connected by an arc which diverges to infinity under both the forward and backward iterates of the homeomorphism. Our example contains no invariant fundamental regions.     

Une version feuilletée équivariante du théorème de translation de Brouwer

The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a Brouwer line: a proper topological embedding C of R, disjoint from its image and separating f(C) and f^–1(C). Suppose that f commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines. We apply this result to give simple proofs of previous results about area-preserving homeomorphisms of surfaces and to prove the following theorem: any Hamiltonian homeomorphism of a closed surface of genus g ≥ 1 has infinitely many contractible periodic points.      

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

O-minimal Hauptvermutung for polyhedra I

Milnor discovered two compact polyhedra which are homeomorphic but not PL homeomorphic (a counterexample to the Hauptvermutung). He constructed the homeomorphism by a finite procedure repeated infinitely often. Informally, we call a procedure constructive if it consists of an explicit procedure that is repeated only finitely many times. In this sense, Milnor did not give a constructive procedure to define the homeomorphism between the two polyhedra. In the case where the homeomorphism is semialgebraic, the author and Yokoi proved that the polyhedra in R ⁿ are PL homeomorphic. In that article, the required PL homeomorphism was not constructively defined from the given homeomorphism. In the present paper we obtain the PL homeomorphism by a constructive procedure starting from the homeomorphism. We prove in fact that for any ordered field R equipped with any o-minimal structure, two definably homeomorphic compact polyhedra in R ⁿ are PL homeomorphic (the o-minimal Hauptvermutung theorem 1.1). Together with the fact that any compact definable set is definably homeomorphic to a compact polyhedron we can say that o-minimal topology is “tame”.     

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.     

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.      

Homeomorphic measures on stationary Bratteli diagrams

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation R. Equivalently, the set S is formed by ergodic probability measures invariant with respect to aperiodic substitution dynamical systems. The paper is devoted to the classification of measures μ from S with respect to a homeomorphism. The properties of the clopen values set S(μ) are studied. It is shown that for every measure μ∈S there exists a subgroup G⊂R such that S(μ)=G∩[0,1]. A criterion of goodness is proved for such measures. Based on this result, the measures from S are classified up to a homeomorphism. We prove that for every good measure μ∈S there exist countably many measures {μi}_i∈N⊂S such that the measures μ and μi are homeomorphic but the tail equivalence relations on the corresponding Bratteli diagrams are not orbit equivalent.     

Meromorphic iterative roots of linear fractional functions

Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are many results on iterative roots of monotone functions. However, this problem gets more diffcult in non-monotone cases. Therefore, it is interesting to find iterative roots of linear fractional functions (abbreviated as LFFs), a class of non-monotone functions on ℝ. In this paper, iterative roots of LFFs are studied on ℂ. An equivalence between the iterative functional equation for non-constant LFFs and the matrix equation is given. By means of a method of finding matrix roots, general formulae of all meromorphic iterative roots of LFFs are obtained and the precise number of roots is also determined in various cases. As applications, we present all meromorphic iterative roots for functions z and 1/z. This work was supported by the Youth Fund of Sichuan Provincial Education Department of China (Grant No. 07ZB042)     

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.     

A certain synchronizing property of subshifts and flow equivalence

We will study a certain synchronizing property of subshifts called λ-synchronization. The λ-synchronizing subshifts form a large class of irreducible subshifts containing irreducible sofic shifts. We prove that the λ-synchronization is invariant under flow equivalence of subshifts. The λ-synchronizing K-groups and the λ-synchronizing Bowen-Franks groups are studied and proved to be invariant under flow equivalence of λ-synchronizing subshifts. They are new flow equivalence invariants for λ-synchronizing subshifts.     

Nielsen classes of orientation reversing homeomorphisms and complementary domains of plane separating invariant continua

Let h be an orientation reversing planar homeomorphism and X be an invariant plane separating continuum. We show that there is a natural linear order on the invariant components of R²?X that resemble the one found in connected unions of circles invariant under the reflection r(x,y)=(−x,y). The main result relates to the Nielsen fixed point theory and work of Krystyna Kuperberg on fixed points of planar homeomorphisms in invariant continua.     

Maximal pseudocompact spaces and the Preiss-Simon property

We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.     

On the knot complement problem for non-hyperbolic knots

This paper explicitly provides two exhaustive and infinite families of pairs (M,k), where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.     

Smooth invariants of focus-focus singularities

We consider focus-focus singularities of integrable Hamiltonian systems with two degrees of freedom. It is known that if two singularities of this kind have the same number of singular points, then they are fiber-wise homeomorphic. However, it may happen that this homeomorphism cannot be made smooth. An invariant is constructed, which makes it possible to classify focus-focus singularities up to a C¹-diffeomorphism.     

Monotonicity in the damped pendulum type equations

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincaré map P^T is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincaré map has in the cylindrical phase space an invariant circle, on which P^T is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of P^T on this curve.     

Monotonicity in the damped pendulum type equations

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincaré map P^T is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincaré map has in the cylindrical phase space an invariant circle, on which P^T is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of P^T on this curve. Supported by National Natural Science Foundation of China (10771155, 10571131) and Natural Science Foundation of Jiangsu Province (BK 2006046).     

Free actions on handlebodies

The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g?1 can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g−1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+?(G)|G|, where ?(G) is the maximal length of a chain of subgroups of G. There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained.     

The evolution of the concept of homeomorphism

Topology, or analysis situs, has often been regarded as the study of those properties of point sets (in Euclidean space or in abstract spaces) that are invariant under “homeomorphisms.” Besides the modern concept of homeomorphism, at least three other concepts were used in this context during the late 19th and early 20th centuries, and regarded (by various mathematicians) as characterizing topology: deformations, diffeomorphisms, and continuous bijections. Poincaré, in particular, characterized analysis situs in terms of deformations in 1892 but in terms of diffeomorphisms in 1895. Eventually Kuratowski showed in 1921 that in the plane there can be a continuous bijection of P onto Q, and of Q onto P, without P and Q being homeomorphic.