An example on movable approximations of an invariant set under a continuous flow in dimension 3
by
Petra Sindelarova
Auburn University

Our study of flows on n-manifolds in particular in dimension 3, e.g., R³, is motivated by the following question. Let A be a compact invariant set of a flow on X. Does every neigborhood of A contain a movable invariant set C containing A? The topological notion of movability (also called the UV-property) is in the sense of K. Borsuk and is closely related to the notion of stability in dynamics. A continuum C is said to be movable if for every neighborhood U of C there exists a neighborhood U₀ ⊂ U of C such that for every neighborhood W of C there is a map j:U₀×I→ U satisfying the condition j(x, 0)=x and j(x, 1) ∈ W for every point x ∈ U₀. It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighborhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the thesis consists in constructing an example in R³ which use Denjoy-like invariant approximating sets instead of periodic orbits. This gives a partial answer to the above question. The construction involves both, the adding machines and Denjoy maps, and the suspension of specially defined Cantor set homeomorphisms.

Date received: February 14, 2006