Discrete Dynamical Systems
2.1. Limit sets.
Suppose X is a complete metric space and T: X —• X is a continuous map-
ping. For any x E X, the positive orbit 7+ (x) through a: is defined as 7+ (x) =
A negative orbit through x is a sequence {xj,j = 0, —1, —2,... } such
that xo = x, Txj-i = x^ for all j . A complete orbit through x is a sequence
{zj,j = 0, ±1 , ± 2 , . . . } such that xo = x and
= xy for all y.
Since the range of T need not be all of X, to say that there is a negative or
complete orbit through x may impose restrictions on x. In spite of this fact, it
will be clear from later discussions that some of the most interesting orbits are
complete orbits. Also, since T need not be one-to-one, it is not necessary for a
negative orbit through x to be unique if one exists. Let the negative orbit 7 " (x)
through x be defined to be the union of all negative orbits through x. Then
7~(z) = | J H(n,x),
H(n, x) = {y G X: there is a negative orbit {x-j,j = 0,1,2,... }
through x with x_
= y}
The complete orbit 7(x) through x is 7(X) = 7~(x) U 7 + (x). For any subset
BCX, let
+ ( 5 ) = U.
(*), 7"(5 ) =
I U B T ( * ) ,
7(5) = U e B ^ W be
respectively the positive orbit, negative orbit, complete orbit through B if the
latter exist.
For any set B C X define the uj-limit set OJ{B) of B and the a-limit set a(B)
0/ B a s
w{B) = f ) CI ( J
n0 kn
a(B)= f]Cl\jH(k,B).
n0 kn
For a point x, the w-limit set UJ(X) can be characterized by saying that y G u(x)
if and only if there is a sequence of integers rij —• 00 such that
—• y as
y —• 00. For a set B, y £ w(B) if and only if there is a sequence xy G B
and a sequence of integers rij * 00 such that
2/ as j —• 00. Similar
characterizations hold for a(x) and a(B).
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