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Sections in Topological Dynamics
by
Krzysztof Ciesielski
University of West Virginia
For a topological dynamical system (i.e. a triplet (X, R, \pi) on a metric space X) we define a section S through any nonstationary point x as a set S containing x such that for some \lambda > 0 the set U = \pi((-\lambda, \lambda), S) is a neighbourhood of x and for every y in U there is a unique z in S and a unique t in (-\lambda, \lambda) with \pi(t, z) = y. According to the Whitney-Bebutov Theorem there exists a section through any nonstationary point x. Parallelizability of the system is fulfilled. This gives a very good tool for solving many problems as it is possible to describe very precisely the behaviour of a system in a neighbourhood of any nonstationary point (by a stationary point we mean a point x with \pi(t, x) = x for all t).
For a semidynamical system (i.e. a system where R is replaced by R+) the situation is much more complicated, as we do not have negative unicity guaranteed and a trajectory can join another one; also, it may happen that it is impossible to prolongate trajectories in negative direction far enough. Thus in general situation it is impossible to present a neighbourhood of a point as a union of "parallel" segments of trajectories. However, a natural questions arises: is it possible to find a neighbourhood of a given point which can be presented as the union of segments of trajectories going with the same "time length" in the same direction? One of the main problems is stating a definition of section in a semidynamical system.
The definition presented here allows to describe a neighbourhood of a nonstationary point x as a "box" in which the segments of trajectories go from one side of the box to the opposite side by the time interval 2\lambda. Moreover, any another trajectory does not join the trajectories in this "box", i.e. if any point y belongs to this box then the whole segment of any trajectory through y with a required time length is contained in the box. The existence of such a section is proved. This shows an opportunity of local presentation of semidynamical systems in a parallelizable way.
We denote: F(t, y) = { z: \pi(t, z) = y }, F(\Delta, A) = \cup { F(t, z): t in \Delta, z in A }.
Definition. Assume that a semidynamical system (X, R+, \pi) on a metric space X and a nonstationary point x in X are given. A closed set S containing x is called a section (a \lambda-section) through x if there is a closed set L such that:
In the case of dynamical systems the presented definition is equivalent to the classical one.
Then F([0, 2\lambda], L) is a parallelizable "box" in which all the segments of the trajectories go from one side to the opposite one in the time interval 2\lambda and begin their movement in this box at the base F(2\lambda, L) of the box. In some systems on not locally compact spaces it may happen that some trajectories "begin" inside the box and their time intervals are shorter.
We have
Theorem. For every nonstationary point x there exists a section containing x. For a system without start points on locally compact space there exists a compact section containing x.
Date received: June 24, 1996
Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-13.