Recent Advances in Topological Dynamics Proceedings of the Conferenc
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318 Recent Advances in Topological Dynamics Proceedings of the Conference on Topological Dynamics, held at Yale University, June 19-23, 1972, in honor of Professor Gustav Arnold Hedlund on the occasion of his retirement.
Edited by Anatole Beck University of Wisconsin, Madison, WI/USA
Springer-Verlag Berlin · Heidel berg · New York 19 73
AMS Subject Classifications (1970): 28A65, 34C35, 47A35, 54H20
ISBN 0-387-06187-8 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-06187-8 Springer-Verlag New York · Heidelberg · Berlin. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under§ 54 of the German Copyright Law where copies are made for other than private use. a fee is J> 0 .
Liapunov families can be paraphrased as
be a net in
X; we write
eventually in every F E 'f. If u 7 xn M if and only if V(xn) 0,
xn
7 +
M if {xn}
is
is an 7 -Liapunov family, then V E 1r.
for all
Using Liapunov families, the classical stability theorem can be generalized. Theorem.
Let
M be a closed invariant subset of the second countable
locally compact metric space hood filter of
M.
Then
7-Liapunov family for
X
M is
and let ~
be a sufficient neighbor-
~
stable if and only if there is an
M.
Our second topic is the study of dynamical systems which have no generalized recurrent points. order
a ,
if
x E A (x). a
points, and call
~ =U~
a a
A point We write
x EX ~
a
is said to be recurrent for the set of such
the generalized recurrent set.
the Poisson stable and non-wandering points.
~
includes
A compact dynamical
system always includes recurrent points (in fact, almost periodic points) so a dynamical system for which non-compact phenomenon.
~
is empty is an essentially
We call such systems gradient dynamical
systems; they are characterized by the existence of a continuous real valued function t
> 0,
[2].
f
such that
f(xt) < f(x),
for all
x
X
and
10
Now, if that
A (x)
¢
a
gradient.
=
J\(X)
for all
¢•
a
for all
and all
S x
then it follows easily and the flow is clearly
x,
In this case, the dynamical system is known to be parallel-
izable -there is a closed set to
x EX,
S
in
X
such that
X
is homeomorphic
and the flow corresponds to translation in the second
~.
coordinate, [1].
The general case (gradient, but not parallelizable)
is characterized by the existence of orbits which have non-empty prolongational limit sets.
These orbits are separatrices, as defined
X,
by Markus [?]-regarded as elements of the orbit space the points where Let above. Xr
=
f
X
they are
fails to be Hausdorff, or limits of such points.
be a function decreasing along every orbit, as described
We may suppose the range of 1
f- (r)
and
Zr
=
[xJxt E Xr'
f
is
J
Then i f
(-1,1).
for some
Z
t] (r E J),
r
is an
invariant set, and the flow
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