CHAPTER 1
BASIC NOTATION AN D BACKGROUND DEFINITIONS
A. Basic Notation. The symbols N, Z, R, R
+
, and R~ will respectively
denote the natural numbers (0 included), the integers, the reals, the non-negative
reals, and the non-positive reals. For a, b G R, ]a,6[ and [a, b] respectively denote
the open and closed intervals from a to 6; similarly, ]a,b] and [a,b[ denote half-
open intervals open respectively on the left and right. Throughout, M denotes a
second countable Cr manifold without boundary (0 r oo) of dimension m.
The second countability assumption ensures that M is a Euclidean neighborhood
retract (hereinafter abbreviated to ENR) and that M is metrizable. Further, M is
assumed oriented over the PID R.
A Ck Bow (0 k r) in M is a Ck map on an open 0 in M x R into M,
generally denoted by (u,£) H- U t, with the following properties: (1) (3 D M x {0}
and u 0 = u for each u M, (2) (u ti) t^ u (ti + £2) whenever (u,£i),
(u £1,^2)^ and (u, £1 -f £2) S (3) for each u G M, the trajectory of u, i.e., the
map t i- u t defined for all t satisfying (u,t) G 0, has closed, connected graph
in the topology inherited from R x M. It is immediate from the openness of 0
and properties (3) and (1) that the domain of each trajectory is an open interval
containing 0. Note that any locally Lipschitz vectorfield o n a C
1
manifold generates
a flow in the manifold assuming integral curves of the vectorfield are taken with
maximally extended domain. Also, any continuous Hamiltonian vectorfield on a
two-dimensional symplectic manifold generates a continuous flow in M.
Assume given a flow in M. The orbit through u G M is the image of the trajec-
tory of u and is complete if the trajectory has domain R; else it is non-complete.
The positive (resp. negative) semi-orbit of u is the image of the trajectory restricted
to the non-negative (resp. non-positive) reals in its domain and is complete if the
restricted trajectory has domain R + (resp. R~); else it is non-complete. Because
each trajectory has an open interval as domain, no non-complete semi-orbit can
lie in a compact set as a simple consequence of property (3) of a flow. A sub-
set of M is called invariant (resp. positively invariant, resp. negatively invariant)
if, and only if, it is the union, the empty union not excluded, of complete orbits
(resp. complete positive semi-orbits, resp. complete negative semi-orbits). Thus,
the union of invariant sets is invariant; hence, each N C M contains a unique max-
imal invariant set relative to the partial order induced by inclusion on the invariant
sets contained in N. It is a simple consequence of property (3) of a flow that the
closure of a relatively compact, positively (resp. negatively) invariant set (relative
compactness is not needed if (3 M x R) is again positively (resp. negatively)
invariant. It follows that the maximal invariant set in each compact N C M is
itself compact. Call a compact i V c M a n isolating neighborhood relative to the
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