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preimages may lead to basins with complex structures, such as multiply connected
or disconnected sets, sometimes formed by infinitely many disconnected portions
(see Mira et al. (1994), Mira and Rauzy (1995), Mira et al. (1996), Chap. 5 and
Abraham et al. (1997), Chap. 5). In the context of noninvertible maps it is useful to
define the
immediate basin
B
0
.A/, of an attracting set A, as the largest connected
component of the basin that contains A. Then the total basin can be expressed as
[
T
n
.
B
B
.A/
D
0
.A//
n
D
0
where T
n
.x/ represents the set of all the rank-n preimages of x,inotherwords
the set of points which are mapped into x after n iterations of the map T .The
backward iteration of a noninvertible map
repeatedly unfolds
the phase space, and
this implies that the basins may be disconnected, that is they are formed by several
disjoint portions. Also in this case, we first illustrate this property by using a one-
dimensional map based on an evolutionary game proposed in Bischi et al. (2003b).
In Fig. C.9a, b the graph of a Z
1
Z
3
Z
1
noninvertible map is shown, where
Z
3
is the portion of the co-domain bounded by the relative minimum value c
min
and the relative maximum value c
max
. In the situation shown in Fig. C.9a we have
three attractors: the fixed point
z
, with
.
z
/
D
.
1
;q
/, the attractor A around
B
x
, with basin
.A/
D
.q
;r
/ bounded by two unstable fixed points, and
C1
(attracting positively diverging trajectories) with basin
B
.
C1
/
D
.r
;
C1
/.In
this case all the basins are immediate basins, each being given by an open interval.
In the situation shown in Figure C.9(a), both basin boundaries q
B
and r
are in Z
1
,
H
−2
q
−2
q
−2
q
−1
B
(∞)
r
*
r
*
H
-
1
B
(
x
*
)
q
*
Z
1
Z
1
−1
x
*
c
max
x
*
c
max
Z
3
Z
3
H
0
c
min
q
*
q
*
c
min
B
(
z
*
)
Z
1
Z
1
z
*
z
*
(a)
(b)
Fig. C.9
The global bifurcation of a one-dimensional noninvertible Z
1
Z
3
Z
1
,map.(
a
)The
attractors of the map are
z
, x
and C1, and their basins are .1;q
/, .q
;r
/ and .r
; 1/
respectively. Note that c
min
is above q
.(
b
) After a parametric change c
min
moves below q
and
a global bifurcation has occurred. Now the basin of
z
includes the (countably infinite number
of) disconnected portions, H
1
;H
2
etc. on .x
;r
/. These are the preimages of the portion
.c
min
;q
/
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