An example of movable approximations of a minimal set in a continuous flow
Type of DegreeDissertation
Mathematics and Statistics
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In the present dissertation the study of flows on n-manifolds in particular in dimension three, e.g., R^3, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? Here, a dynamical system (a flow) is the pair (X,p), where X, in general, is a manifold, p is a continuous map from the product of X and the real line R to X, such that p(x,0)=x and p(p(x,t_1),t_2)=p(x,t_1+t_2), for each x in X and each t_1, t_2 in R. A nonempty set A in X is invariant if p(A,t)=A for each t in R. A compact invariant set A in X is stable if for every neighbourhood U of A there exists a neighbourhood V of A with V a subset of U, such that p(V,t) is a subset of U for all t=0. 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 M in X is said to be movable if for every neighbourhood U of M there exists a neighbourhood U_0 of M contained in U such that for every neighbourhood W of M there is a continuous map f from the product of U_0 with the unit interval I to U satisfying the conditions f(x,0)=x and f(x,1) in W, for every point x in U_0. It is known that a stable solenoid (an intersection of a nested sequence of solid tori positioned one inside another in some regular way) in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. 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 dissertation consists of constructing an example in R^3 which uses 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.