
Theorem
6.7.3
Assume
that
f
is
a
chaotic
c
3
-mapping
from
a
nontrivial
interval
X
into
itself
satisfying
the
following
conditions:
(i)
f
has
a
nonpositive
Schwarzian
derivative,
i.e.,
(d'f(x)jdx')/(df(x)jdx)
-
(3/2)[(d'f(x)jdx')/(df(x)jdx)]'
S
0
for
all
x e X
with
df(x)jdx
+
o:
(ii)
The
set
of
points,
whose
orbits
do
not
converge
to
an
absorbing
boundary
point(s)
of
X
for
f,
is
a
nonempty
compact
set:
(iii)
The
orbit
of
each
critical
point
for
f
converges
to
an
asymptotically
stable
periodic
orbit
of
f
or
to
an
absorbing
boundary
point(s)
of
X
for
f:
(iv)
The
fixed
points
of
f'
are
isolated.
Then
we
have:
(a)
The
set
of
points
whose
orbits
do
not
converge
to
an
asymptotically
stable
periodic
orbit
of
f
or
to
an
absorbing
boundary
point(s)
of
X
for
f,
has
Lebesgue
measure
zero:
(b)
There
exists
a
positive
integer
p
such
that
almost
every
point
in
X
is
asymptotically
periodic
with
period
p,
provided
that
f(X)
is
bounded.
In
the
following,
we
will
give
some
simple
examples:
(A)
X=
[-1,1],
f:
X~
X
is
defined
by
f(x)
=
3.701x
3
-
2.701x.
It
can
be
shown
that
f
has
two
asymptotically
stable
periodic
orbits
with
period
three.
Since
f
has
a
negative
Schwarzian
derivative
and
dfjdx(x=1)
=
dfjdx(x=-1)
>
1,
by
Theorem
6.7.3
we
have
that
almost
every
point
in
X
is
asymptotically
periodic
with
period
three.
(B)
Let
f
be
a
chaotic
map
of
c
3
from
a
compact
interval
[a,b)
into
itself
with
the
following
properties
[Collet
and
Eckmann
1980):
(i)
f
has
one
critical
point
c
which
is
nondegenerate,
f
is
strictly
increasing
on
[a,c)
and
strictly
decreasing
on
[c,b]:
(ii)
f
has
negative
Schwarzian
derivative:
(iii)
the
orbit
of
c
converges
to
an
asymptotically
stable
periodic
orbit
of
f
with
smallest
period
p,
for
some
positive
integer
p.
But
since
the
existence
of
an
asymptotically
stable
fixed
point
in
[a,f'
(c))
has
not
been
excluded,
thus
the
following
cases
can
occur:
(1)
f
has
an
asymptotically
stable
fixed
point
in
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