An example on movable approximations of a minimal set in a continuous flow |
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Authors: | Petra Šindelá?ová |
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Institution: | Department of Mathematics, Auburn University, Parker Hall, Auburn, AL 36849, USA |
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Abstract: | In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R3, 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? It is known that a stable solenoid 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 paper consists of constructing an example of a set in R3 that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question. |
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Keywords: | primary 54H20 37B05 secondary 37B10 37B25 37E35 28A78 54C55 |
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