A classical ergodic property for IFS: A simple
proof
B. Forte1;2 , F. Mendivil2
1 Facolta di Scienze MM. FF.
e NN. a Ca Vignal
Universita Degli Studi di Verona
Strada Le Grazie
37134 Verona, Italy
e-mail:
[email protected]
2 Department of Applied
Mathematics
Faculty of Mathematics
University of Waterloo
Waterloo, Ontario, Canada N2L 3G1
e-mail:
[email protected]
(February 12, 2000)
Abstract
Let wi ; pi be a contractive IFS with probabilities. We provide a
simple proof that
and for all x the
P for almost every address sequence
w
limit
lim
1 (x) exists and is equal to
n 1=n in f wn wn 1
R
X f (z ) d(z ) where is the invariant measure of the IFS. This is the
so called \ergodic property" for the IFS and was proved by Elton in [3].
However, the uniqueness of the invariant measure was not previously exploited. This provides considerable simplication to the proof.
f
g
Let X be a compact metric space and fwi gLi=1 a collection
P of L contraction
maps on X . Let fpi g be a collection of L probabilities (i.e. i pi = 1).
In [4] Hutchinson proves that there exists a unique measure invariant under
the Markov operator M dened by
X
M ( )(B ) = pi (wi 1 (B ))
i
where is a probability measure on X and B is a Borel subset of X . In fact
M n ( ) ! for any probability measure since the wi 's are contractive (see
[4, 1]).
The operator U dened as
X
U (f )(x) = pi f (wi (x))
i
1
(where x is a point in X and f is a continuous function on X ) is the adjoint to
M and will play an important role in what follows.
Let
Y
= f1; 2; : : :; Lg
N
be the code space (see [4, 1]) with P the product measure induced by the measure
p(fig) = pi on each factor.
We will need the projections n : ! n dened by n () = (n ; n 1 ; : : : ; 1 ).
We will denote n () by n .
For n 2 n we denote by p the product p p 1 p1 . Furthermore,
we denote by w the composition
n
n
n
n
w w
n
n
1
w 1 :
The following theorem was proved by Elton in [3]. We provide a simplied
proof of this result.
Theorem 1 For any continuous function f on X and any x 2 X we have
lim 1=n
n!1
X
in
f w w
i
i
1 w1 (x) =
Z
X
f (z ) d(z )
for P almost all address sequences 2 .
Proof: Let f a continuous function on X and x be a xed element of X .
We wish to show that
X
1=n f (w (x))
i
in
converges. Let -lim be a Banach Limit on l1 (IN ) (see [2], p. 82 for a nice
discussion of Banach Limits). Recall that a Banach Limit is a \generalized
limit" in the sense that it is a bounded linear functional on l1 which does not
depend on the rst terms of the sequence in l1 . We will show that any two
Banach limits will give the same value for P almost every 2 so that the
limit exists almost everywhere.
P
Now f 7! -limn1=n in f (w (x)) is a bounded linear functional on
C (X ). Thus, by the Riesz Representation Theorem it corresponds to a measure
on X . Since if f = 1, we get the limit equals to 1, we know that this measure
is a probability measure.
We show that = for P almost all . This will show that for almost all
the limit exists and is what we wish it to be.
Let S : ! denote the shift map on . Since each wi is contractive and
X is compact, we know that
i
1
=n
X
in
f (w (x)) 1=n
i
2
X
in
) f wS() (x
i
!0
as n ! 1. Thus,
X
X
-lim(1=n) f (w (x)) = -lim(1=n) f wS() (x) :
i
in
i
in
Since the shift map on is ergodic, we know that -limn1=n
is constant for P almost all .
To show that = it suces to show that
Z
X
f (z ) d (z ) =
Z
P
in f (wi (x))
Uf (z ) d (z )
X
which is the same as showing that
X
XX
pj f (wj w (x))
-limn1=n f (w (x)) = -limn1=n
i
in
Computing we get
XX
-limn1=n
in j
i
in j
pj f (wj w (x)) =
i
=
Z
2
X
j
-limn1=n
pj
Z
2
XX
in j
-limn1=n
pj f (wj w (x)) dP ()
i
X
in
f (wj w (x)) dP ()
i
Doing the change of variable (1 ; 2 ; : : :) ! (j; 1 ; 2 ; : : :) we get dP !
dP=pj so this integral becomes
X
j
pj =pj
Z
1 =j
-limn1=n
X
in
f (w +1 (x)) dP () =
i
Z
2
-limn1=n
= -limn 1=n
X
in
X
in
f (w (x)) dP ()
i
f (w (x))
i
for P almost all .
Therefore for P almost all we know that is invariant under M so = .
However, since was arbitrary this shows that for P almost all lim
1=n
n
for all f and all x 2 X .
X
in
f (w (x)) =
i
Acknowledgments
Z
X
f (z ) d(z )
The authors are grateful to E.R. Vrscay and C. Sempi for their comments and
suggestions. The authors would also like to thank the referees for their suggestions, which did much to improve the paper.
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References
[1] Barnsley, Michael, Fractals Everywhere, Academic Press, New York, 1988.
[2] Conway, John, A Course in Functional Analysis, Graduate Texts in
Mathematics; 96, Springer Verlag, New York, 1990.
[3] Elton, John, An Ergodic Theorem for Iterated Maps, Journal of Ergodic
Theory and Dynamical Systems 7 (1987), 481-488.
[4] Hutchinson, J. E., Fractals and Self-similarity, Indiana Univ. Math. J. 30
(1981), 713-747.
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