Ergod.
Th. & Dynam. Sys. (1983), 3, 13-46
Printed
in
Great Britain
Positive Liapunov exponents and
absolute continuity for maps
of
the
interval
P.
COLLET ANDJ.-P. ECKMANN
Centre
de
Physique Theorique, Ecole Poly technique,
91128
Palaiseau, France;
Departement de Physique
Theorique,
Universite
de Geneve, 1211 Geneve
4,
Switzerland
(Received 16
September
1981 and
revised
22
September
1982)
Abstract.
We give a
sufficient condition for a unimodal map of the interval to have an
invariant measure absolutely continuous with respect to the Lebesgue
measure.
Apart
from some weak regularity assumptions,
the
condition requires positivity
of the
forward and backward Liapunov exponent
of
the critical point.
1.
Introduction
and
statement
of
results
Continuous maps
of
an interval to itself can be viewed as dynamical systems, whose
time evolution
is
given
by
iterating
a
given map. Despite their innocent looking
simplicity, iterated maps
can
serve
as
important models
for
testing general ideas
about dynamical systems.
One such circle
of
ideas concerns the existence
of
invariant measures which
are
absolutely continuous (with respect
to
Lebesgue measure).
If, in
addition,
the
measure
is
ergodic, then erratic behaviour can
be
expected
for
many orbits.
One
would like to argue that if a system has positive characteristic (Liapunov) exponents,
then
it
behaves erratically. One
is
still
far
from
a
complete understanding
of
these
matters, because
of
the presence
of
stable directions, see e.g. [9]. The analogue of
this question
for
maps
of
the interval is easier
to
handle because, when there
is an
unstable direction, then there
is no
space
for a
stable direction.
The
Liapunov
exponent
is
clearly positive
if the map is
everywhere expanding,
and
this
is the
easiest case
in
which existence
of an
absolutely continuous invariant measure
can
be shown [6], [12].
If the
map
of the
interval
has a
critical point,
it is of
course
not uniformly expanding.
It may
nevertheless possess
an
absolutely continuous
invariant measure [11].
It
was then discovered that this result can
be
generalized
to maps with the property that the critical points have orbits which eventually land
on unstable fixed points [8], [1].
One can
further generalize this
to
maps whose
critical points have orbits staying away from
the
critical points [7], [10].
In the
present paper,
we
give different conditions
for the
existence
of an
absolutely
continuous invariant measure.
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14
P.
Collet
and J.-P. Eckmann
For simplicity, we shall consider maps with
a
single critical point,
xo-
Apart from
technical conditions
to be
given below, we require that the map
/
satisfies
(Cl) liminf-log
n-.oo
n
j-
(f
n
)(f(x
0
))
ax
(C2)
liminf-
inf log
n
"
where
f" =f°
•
•
•
°f
(n times) and
/ "(x
0
)
denotes the set
of n'th
pre-images
of
x
0
-
These conditions
can be
fulfilled even
if the
orbit
of x
0
does
not
stay away
from
xo,
and
they
are
thus weaker than those mentioned above.
One can
show
that they
are met for a
large
set of
maps among
the
one-parameter family
8->f
s
given
by
.-8~(x
2
/8)
if|jc|<5,
for 0
<
8 <
\.
We
have studied these maps
in
[2],
and a
slight extension
of
that
work shows that there is a set
of
positive Lebesgue measure in 8 such that/
s
satisfies
all conditions
to be
enumerated below,
and
hence
f
s
has
an
absolutely continuous
invariant measure.
In
addition, this
set
of 8
has
a
Lebesgue point (i.e. full relative
measure)
at 5 =
0. Similar results
for
one-parameter families
of
maps have been
obtained earlier
in
[5].
If
we
consider
our
conditions
in the
general framework
of
dynamical systems
then
(Cl)
corresponds
to
requiring that
the
Liapunov exponent
is
positive, while
condition (C2) says that
the
inverse
of /
(which only exists
as a
set
function
in
our
case)
is
contracting.
It
is tempting
to
conjecture that (Cl) might imply (C2), maybe
with some additional convexity condition,
but our
insight into this question
is
incomplete. Note also that,
if
the orbit
of
the critical point stays
at a
distance from
the critical point, then Misiurewicz' conditions
are
stronger than
(Cl) and
(C2),
see the Appendix.
We now state our hypotheses
in
detail, followed by the statement
of
the theorem
and some remarks.
Hypotheses
(HI)
/
is defined onfl =
[/(I),
1]
and
takes values
in fi. It is
strictly increasing
on
[/(I),
0] and strictly decreasing
on
[0,1].
The function
/
is
of
class
"if
1
. In
addition,
(H2)
The
function
/' is
Lipschitz continuous,
and
|/T
J
is
convex
on
[/(I),
0]
and
on [0, 1].
(H3)
limsup|/'(x)/;t|<oo,
inf \f'(x)/x\>0.
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Positive
Liapunov
exponents
15
(H4) There is a d
> 0
and a d
> 0
such that
(1)
> Ci exp (n0) for all n > 0.
(2) If f
m
(z)
= 0 for some m >
0,
then
>Ciexp(m0).
Our main result is the following:
THEOREM. // / satisfies (H1)-(H4) then f has an invariant measure which is
absolutely continuous
with
respect to Lebesgue
measure.
The conditions (H1)-(H4) can be formulated more concisely under the assumption
that / is of class
<
#
3
. Namely, we can replace (H2), (H3) by the more readable
(H2') Sf(x)s
0
for x e fi, where
s/=/'"//'-§(/7/')
2
is the Schwarzian derivative.
(H3') /"(0) <
0
and /'(/(I)) * 0 and /'(I) * 0.
Although the Schwarzian derivative is not defined in the general case, the results
of the corresponding theory do apply, see e.g. [3]. In particular, (H1)-(H3) and
(H4.1) imply that the map / has no stable periodic orbit.
We next outline the main steps of the proof of the main theorem. Define the
operator
5E
by
where f'\E)
= {x
e Cl,f(x)eE}.
The density h of the invariant measure, if it exists, satisfies the equation
We shall consider the sequence of functions
h
n
=£6
n
\, «= 0,1,2
It is easy to see that
J My)
</y
=
W(l).
(1)
We shall show:
THEOREM 1.1. Define h
n
{y)
=
{£" \){y) if
y
eft, h
n
(y)
=
0 i/ygfl. Then, for all n,
one
has
J[ iMy)
-
My + e
)|
dy < exp
(-|log
eh (2)
for all e
> 0,
e
sufficiently
small.
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16 P.
Collet
and
J.-P.
Eckmann
The conclusions of theorem 1.1 together with (1) are the main input to Kolmogorov's
compactness criterion (see e.g. [13]). It then follows from
[4]
(Mazur's theorem) that
1 "
lim - £ hi
=
h
exists,
is in L\, is not zero and satisfies
5£h =
h.
Thus the main theorem follows
from theorem 1.1.
To discuss our method of
proof,
we define
Df
=£(/")•
The difficulty in analysing h
n
(y)
- h
n
(y +
e) comes from those regions in which
l/\Df"\
varies quickly. These regions are located near the pre-images of 0, where
\/\Df"\
is infinite. In other words, since
1
(X)\
h
n
varies rapidly around the points of the forward orbit of the critical point, i.e. of
zero.
This orbit is allowed to come close to the critical point, (it may be dense),
and we have to subdivide carefully the space into pieces where h
n
(y)
-
h
n
(y +
e) is
regular, and their complement where \h
n
will be small. To be more precise, fix
e >
0
sufficiently small and define the following e dependent quantities.
Definitions. The following symbols have fixed meaning throughout the paper:
U= sup
\f'{t)\.
lelf(l),
1]
L
2
will be a constant which is fixed in (18) on p.
25.
The constant T is defined by
6
We shall assume for simplicity that r<\. (This can always be achieved by making
6 smaller.) Our main definitions are now as follows:
, (l-3r/2)|log
e
|+L
2
n
e
=lOOO|loge|/0;
0 if m<n'
e
,
e
1+T
if n'
e
<m<n
e
, (4)
exp (-md/20) if n
e
<
m.
Sometimes we shall use conditions of the form / = 1,
2,...,
n
E
.
Then we tacitly
assume n
c
is rounded to that integer which allows for a larger set of /; similarly
for
n'
e
.
We next define m,
e-regular
points.
Let / be a maximal connected component
of r
m
([-e, el). The set / is called m,
e-regular
if
(these conditions imply that f
m
\J is strictly monotone);
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Positive
Liapunov
exponents
17
(2) let{z}
=
/"
m
({O})n/. Then,
if
m>n'.,
for;
=0,1,...,
m
-n'
e
.
Every point
in
a
m, e-regular set
is
called
an
m, e-regular point. We define
h'Z(y)=
I i* ,, (5)
x
6
r
m
(y)
\Df (x)\
x
is
m,e-regular
Our first main estimate is:
THEOREM
1.2.
For
all
sufficiently
small p
>
0,
all small e
>
0, and all n
>
0,
f /C
6
(y)rfy<4(p
T/4
+p/a
0
),
(6)
J
|y|=p
for
some universal
constant ao
>
0.
We shall then show that there
are so
many m, e-regular points that
h
n
can be
bounded
in
terms
of
A™f. This will lead
to:
THEOREM
1.3.
For
all
sufficiently
small e
>
0,
f h
n
(y)dy^e
T/40
.
(7)
This shows that
h
n
is
relatively well-behaved near
y =
0.
To
analyse
the
global
situation, we need other subsets
of
O.
We define new cut-off functions which are similar
to
the cr
m>E
. Namely,
let
1 ifm<0,
Pm,e
—
exp(-|loge|
3
) if0<m<n",
exp(-m<?/20)
if n"<m,
where
n"
=
7|log
e|
5
. Note that p
is
decreasing
in
m and increasing as
e|0.
Definition. We define,
for
given
e >
0,
£
£
m
,,={f€n||f|<
w
i,
e
};
(8)
F
e
m
.
y
={ten\\l-t\<
yPm
J;
(9)
G
E
m
,
y
={teil\\f(l)-t\^yp
m
J.
(10)
These
are
small sets around
the
critical point
of / and
near
the
endpoints
of
il =
[/(I),
1]. We define
\E
as the characteristic function
of
the set
E,
and set
n*
E
(x)=
f[
(l-Ar
E
'_
Pl
wF^
pl
^_
pl
)(/
p
U)),
(11)
p=0
and,
for
some e-independent constant yo>0,
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