We generalize the model theory of small profinite structures developed by Newelski to the case of compact metric spaces considered together with compact groups of homeomorphisms.

In this paper, we give an overview of some topics involving behavior of homeomorphisms and ways in which real analysis can arise in geometric settings.

The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. Minimal are constructed compact connected metric with any prescribed finitely generated $K$-theory or cohomology. In particular, although non-zero Euler characteristic...

Let Homeo(Ω) be the group of all homeomorphisms of a Cantor set Ω. We study topological properties of Homeo(Ω) and its subsets with respect to the uniform (τ) and weak (τ w) topologies. The classes of odometers and periodic, aperiodic, minimal, rank 1 homeomorphisms are considered and the closures of those classes in τ and τ w are found. 0 Introduction The present paper is a continuation of our...

Let F = η be arbitrary. In [5], the main result was the derivation of homeomorphisms. We show that

Let Homeo(Ω) be the group of all homeomorphisms of a Cantor set Ω. We study topological properties of Homeo(Ω) and its subsets with respect to the uniform (τ) and weak (τ w) topologies. The classes of odometers and periodic, aperiodic, minimal, rank 1 homeomorphisms are considered and the closures of those classes in τ and τ w are found. 0 Introduction The present paper is a continuation of our...

Let K be a closed subset of a smooth manifold M, and let f : K ! K be a continuous self-map of K. We say that f is smoothable if it is conjugate to the restriction of a smooth map by a homeomorphism of the ambient space M. We give a necessary condition for the smoothability of the faithfully innnitely interval-renormalizable homeomorphisms of Cantor sets in the unit interval. This class contain...

We initiate a study of homeomorphisms f with constant principal strains (cps) between smoothly bounded planar domains D, D′. An initial result shows that in order for there to be such a mapping of a given Jordan domainD ontoD′, a certain condition of an isoperimetric nature must be satisfied by the latter. Thereafter, we establish the fundamental fact that principal strain lines (characteristic...

The notation and terminology used in this paper are introduced in the following articles: [20], [21], [1], [3], [22], [4], [5], [19], [10], [18], [7], [17], [11], [2], [8], [9], [16], [13], [14], [15], [6], [23], and [12]. In this paper p1, p2 are points of E 2 T, C is a simple closed curve, and P is a subset of E T. Let n be a natural number, let A be a subset of En T, and let a, b be points o...