Jan M. Aarts

The geometry of Julia sets

The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on intervals. Julia sets play a critical role in the understanding of the dynamics of families of mappings. In this paper we introduce another class of objects (called hairy objects) which share many properties with the Cantor set and the interval: they are topologically unique and admit only one embedding in the plane. These uniqueness properties explain the regular occurrence of hairy objects in pictures of Julia sets-hairy objects are ubiquitous. Hairy arcs will be used to give a complete topological description of the Julia sets of many members of the exponential family

Variations on a theorem of lusternik and schnirelmann

Abstract A fixed-point free map f: X → X is said to be colorable with k colors if there exists a closed cover β of X consisting of k elements such that C∩f(C) = φ for every C in β. It is shown that each fixed-point free involution of a paracompact Hausdorff space X with dim X ≤ n can be colored with n + 2 colors. Each fixed-point free homeomorphism of a metrizable space X with dim X ≤ n is colorable with n + 3 colors. Every fixed-point free continuous selfmap of a compact metrizable space X with dim X ≤ n can be colored with n + 3 colors

Chaotic homeomorphisms on manifolds

Abstract For n≥2 , every n -dimensional compact manifold X admits a chaotic homeomorphism. The set of all chaotic measure-preserving homeomorphisms on X is dense in the space of all measure-preserving homeomorphisms.

The dynamics of the Sierpiński curve

The Sierpinski curve X admits a homeomorphism with a dense orbit. However, X is not minimal and does not admit an expansive homeo- morphism. 1. Statement of the theorem

An addition theorem for the color number

There is a close relation between the color number of a continuous map f: X -* X without fixed points and the topological dimension. If f is an involution, the color number is also related to the co-index. An addition theorem for the color number is established thus underscoring the interrelations between color number, dimension and co-index.

Intersection properties for coverings of G-spaces

Abstract The Ljusternik–Schnirelmann–Borsuk theorem for antipodal maps on the sphere can be stated as an intersection property of coverings of the sphere. We generalized this theorem to free finite-group actions on paracompact Hausdorff spaces and discuss how this result might be improved. We illustrate our results by a free action of the Klein four-group on the torus.

Fixed points of the bucket handle

If a homeomorphism on the bucket handle has an invariant composant, it has a fixed point in that composant. It follows that a homeomorphism on the bucket handle has at least two fixed points. Our methods apply to general Knaster continua.

Coincidence theorems for involutions

Abstract Scepin (1974) and Izydorek and Jaworowski (1995, 1996) showed that for each k and n such that 2 k > n there exists a contractible k -dimensional simplicial complex Y and a continuous map ϑ :S n → Y without the antipodal coincidence property, i.e., ϑ ( x ) / ne ϑ (− x ) for all x ϵ S n . On the other hand, if 2 k ⩽ n then every map ϑ :S n → Y to a k -dimensional simplicial complex has an antipodal coincidence point. In this paper it is shown that, with some minor modifications, these results remain valid when S n and the antipodal map are replaced by any normal space and an involution with color number n + 2.

A compactification problem of J. De Groot

Abstract Recently, De Groots conjecture that cmp X = def X holds for every separable and metrizable space X has been negatively resolved by Pol. In previous efforts to resolve De Groots conjecture various functions like cmp have been introduced. A new inequality between two of these functions is established. Many examples which have been constructed so far in relation with the conjecture are obtained by attaching a locally compact space to a compact space. An upper bound for the compactness deficiency def of the resulting space is given.