Approximation of planar BV homeomorphisms by diffeomorphisms

We show that a planar BV homeomorphism can be approximated in the area strict sense, together with its inverse, with smooth or piecewise affine homeomorphisms.     

On fixed points of Knaster continua

Aarts and Fokkink [Proc. Amer. Math. Soc. 126 (1998) 881] have shown that any homeomorphism of the bucket handle has at least two fixed points. Using their methods, we determine the minimum number of fixed points homeomorphisms on generalized one-dimensional Knaster continua can have. We show that there is a class of these continua that admit homeomorphisms with a single fixed point. Among the examples is one that shows that Theorem 15 in [Proc. Amer. Math. Soc. 126 (1998) 881] is incorrect. We also show that there are generalized Knaster continua on which every homeomorphism has either uncountably many fixed points or uncountably many points of period two.     

On the Equivalence of Two Approaches to Problems of Quasiconformal Analysis

Siberian Mathematical Journal - We prove that the two approaches to describing homeomorphisms in modern quasiconformal analysis are equivalent: A homeomorphism changes under control the...     

Conformal barycenters and the Douady-Earle extension—a discrete dynamical approach

The Douady-Earle extension produces a homeomorphism of a disk from a homeomorphism of its bounding circle. It is based on a center of mass computation at the center and is extended to the disk by naturality. Quasiconformal homeomorphisms are the extensions of quasisymmetric ones. There are several approaches to the numerical computation of the extension defined by a redistribution of mass. The algorithms of Abikoff-Ye and Milnor turn out to be the same —even in the more general situation of nontrivial probability measures on the unit circle. The numerical computation uses measures with finite support; in that case, the iterator has rational square. We obtain an easy approach to the proof of the validity of the algorithm and to its calculation. New proofs and additional properties of the computation of the barycenter and the extension are also presented. Much of this work was done while the author was a Lady Davis Visiting Professor at the Technion— Israel Institute of Technology. I gratefully acknowledge the hospitality and support of the Faculty of Mathematics there.     

Metrics and entropy for non-compact spaces

We investigate Bowen’s metric definition of topological entropy for homeomorphisms of non-compact spaces. Different equivalent metrics may assign to the homeomorphism different entropies. We show that the infimum of the metric entropies is greater than or equal to the supremum of the measure theoretic entropies. An example shows that it may be strictly greater. If the entropy of the homeomorphism can vary as the metrics vary we see that the supremum is infinity.     

The topological entropy on a space of homeomorphisms does not belong to the first Baire class

A parametric family of Lipschitz homeomorphisms of a compact metric space continuously dependent on some parameter is studied. A family such that the topological entropy of homeomorphism considered as a function of the parameter does not belong to the first Baire class is constructed.     

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.     

A classification of minimal sets of torus homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Moreover, in case (1) bounded disks are non-periodic and in case (2) all disks are non-periodic. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.     

Homeomorphisms between limbs of the Mandelbrot set

Using a family of higher degree polynomials as a bridge, together with complex surgery techniques, we construct a homeomorphism between any two limbs of the Mandelbrot set of equal denominator. Induced by these homeomorphisms and complex conjugation, we obtain an involution between each limb and itself, whose fixed points form a topological arc. All these maps have counterparts at the combinatorial level relating corresponding external arguments. Assuming local connectivity of the Mandelbrot set we may conclude that the constructed homeomorphisms between limbs are compatible with the embeddings of the limbs in the plane. As usual we plough in the dynamical planes and harvest in the parameter space.     

直线同胚的嵌入流及其应用

设M是一个C~r(r≥0)流形,f:M→M为给定的同胚,f是否可以嵌入一个M上的C~r流?这是一个很有意义也很困难的问题。到目前为止已有不少人做了不少工作,但只对一维的情形才有较完整的结果。Sterberg较早地通过研究直线的局部同胚的共轭类,证明了直线的保向同胚在其双曲不动点附近可局部地嵌入一连续流。P.F.Lam对无不动点的可     

tvs锥度量空间上同胚映射的可扩性

设$f$是紧tvs锥度量空间上同胚映射. 本文证明了$f$是tvs锥可扩的当且仅当$f$有生成元. 进一步, 如果$f$是tvs锥可扩的,则具有收敛半轨的点集是可数集. 本文的这些结果改进了拓扑动力系统的一些可扩同胚定理, 将有助于研究tvs锥度量空间上同胚映射的动力性质.     

A dynamical property for planar homeomorphisms and an application to the problem of canonical position around an isolated fixed point

The first theorem in this paper gives a criterion for detecting a fixed point of a planar homeomorphism into a disc. We then use this result to re-prove the path-connectedness of the space of all planar homeomorphisms fixing only one point, with a given Lefschetz index.     

Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique

In [Rees, M., A minimal positive entropy homeomorphism of the 2-torus, J. London Math. Soc. 23 (1981) 537-550], Mary Rees has constructed a minimal homeomorphism of the n-torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotation and to control precisely the measurable dynamics of f. This yields in particular the following result: Any compact manifold of dimensiond?2which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy.More generally, given some homeomorphism R of a compact manifold and some homeomorphism hC of a Cantor set, we construct a homeomorphism f which “looks like” R from the topological viewpoint and “looks like” R×hC from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds?     

Locally continuously perfect groups of homeomorphisms

The notion of a locally continuously perfect group is introduced and studied. This notion generalizes locally smoothly perfect groups introduced by Haller and Teichmann. Next, we prove that the path connected identity component of the group of all homeomorphisms of a manifold is locally continuously perfect. The case of equivariant homeomorphism group and other examples are also considered.     

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.      

On Homeomorphisms of Effective Topological Spaces

We study effective presentations and homeomorphisms of effective topological spaces. By constructing a functor from the category of computable models into the category of effective topological spaces, we show in particular that there exist homeomorphic effective topological spaces admitting no hyperarithmetical homeomorphism between them and there exist effective topological spaces whose autohomeomorphism group has the cardinality of the continuum but whose only hyperarithmetical autohomeomorphism is trivial. It is also shown that if the group of autohomeomorphisms of a hyperarithmetical topological space has cardinality less than 2 then this group is hyperarithmetical. We introduce the notion of strong computable homeomorphism and solve the problem of the number of effective presentations of T ₀-spaces with effective bases of clopen sets with respect to strong homeomorphisms.     

Compactness theory and mappings with finite length distortion

The present paper is devoted to the study of mappings with finite length distortion introduced in 2004 by O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov. It is proved that the locally uniform limit of homeomorphisms with finite length distortion is a homeomorphism or a constant provided that the so-called inner dilatations of the sequence of homeomorphisms are almost everywhere (a.e.) majorized by a locally integrable function. In particular, it is studied the pointwise behavior of the so-called outer dilatations. For these dilatations, the pointwise semicontinuity and semicontinuty in the mean are proved. It is also proved some theorems on the convergence of matrix dilatations.     

Typical Recurrence for Lifts of mean Rotation Zero Annulus Homeomorphisms

This paper is about typical (uniform topology dense G) propertiesof homeomorphisms of the torus or annulus which preserve a fixedmeasure and have mean rotation zero. We first show that ergodicityis typical (Theorem 1). We then show that the lift (to the universalcovering space) of such a homeomorphism of the annulus is theskew product of the annulus homeomorphism with respect to askewing function of mean zero. Hence Atkinson's Theorem on skewproducts, together with Theorem 1, implies that it is typicalfor an annulus homeomorphism of mean rotation zero to have arecurrent lift (Theorem 3). Standard arguments then give thePoincaré-Birkhoff Fixed Point Theorem as a corollary.     

Dynamique des homéomorphismes du plan au voisinage d'un point fixe

We study the dynamics of a homeomorphism of a surface near a fixed point. We compute the sequence of the Lefschetz indices of the iterates of the map. We deduce the existence of an infinite number of periodic orbits for some conservative homeomorphisms of surfaces.     

Inverse limit spaces arising from problems in economics

In this paper we use tools from topology and dynamical systems to analyze the structure of solutions to implicitly defined equations that arise in economic theory, specifically in the study of so-called “backward dynamics”. For this purpose we use inverse limit spaces and shift homeomorphisms to describe solutions which are typical in that they are likely to be observed in future time. These predicted solutions corresponds to attractors in an inverse limit space under the shift homeomorphism(s).