Dedicated to IOAN CUCULESCU on the occasion of his retirement
and to MARIUS IOSIFESCU on the occasion of his 70th birthday
EQUICONTINUITY AND EXISTENCE
OF ATTRACTIVE PROBABILITY MEASURES
FOR SOME ITERATED FUNCTION SYSTEMS
RADU ZAHAROPOL
Our goal in this paper is to offer two criteria for the existence of attractive probabilities for iterated function systems (i.f.s.) with constant probabilities. The first
criterion is related to Theorem 1 of Barnsley and Elton [3], and is obtained by
weakening one of the average contractivity conditions that emerged in Barnsley
and Elton (op. cit.) (see also Barnsley, Demko, Elton, and Geronimo [2]. The
second criterion is stated in terms of a notion of weak hyperbolicity of i.f.s. which
was introduced by Edalat [5] for i.f.s. defined on compact spaces, and by Lasota
and Myjak [11] for i.f.s. defined on locally compact spaces; the criterion strengthens and extends Corollary 3.3 of Edalat (op. cit.), and complements Theorem
5.1 of Lasota and Myjak (op. cit.). Both criteria are proved using results of our
monograph [29].
AMS 2000 Subject Classification: Primary 47A35; Secondary 26A18, 28A80,
37A30, 47B38, 60J05.
Key words: attractive probability measure, average contractivity condition, invariant probability measure, iterated function system with constant
probabilities, Markov-Feller operator, unique ergodicity, weak hyperbolicity.
1. INTRODUCTION
Our goal in this paper is to obtain two criteria for the existence of attractive probabilities for iterated function systems (i.f.s.) with constant probabilities. These i.f.s. are particular cases of random systems with complete
connections; the theory of random systems with complete connections can be
thought of as having been started in the pioneering paper of Onicescu and
Mihoc [19], and has been developed extensively (see, for example, the monographs by Iosifescu and Grigorescu [7], Iosifescu and Theodorescu [8], Norman
[18], and the more recent paper by Stenflo [23]). Results involving attractive
probabilities of i.f.s. with probabilities are used in the study of fractals (since
in many cases the supports of attractive probabilities are fractals), in image
REV. ROUMAINE MATH. PURES APPL., 52 (2007), 2, 259–286
260
Radu Zaharopol
2
processing (in creating computer pictures), in the theory of learning, etc. (see,
for example, Barnsley [1], Barnsley and Elton [3], and Barnsley, Demko, Elton, and Geronimo [2]). The number of works that deal with topics involving
attractive probabilities of i.f.s. with probabilities is rather large (see, for example, the survey paper of Stenflo [24], and the volume in which [24] appeared).
Our approach in this paper is closely related to the works by Barnsley, Demko,
Elton, and Geronimo [2], Barnsley and Elton [3], Edalat [5], Lasota and Myjak
[10], [11], [12], [13], [14], Lasota and Yorke [15], and our papers [25], [26], [27],
and [28].
Let (X, d) be a locally compact separable complete metric space.
Let k ∈ N, let wi : X → X, i = 1, 2, . . . , k, be k continuous functions, and
k
pi = 1.
let pi , i = 1, 2, . . . , k, be k strictly positive real numbers such that
i=1
Set w = (w1 , w2 , . . . , wk ) and p = (p1 , p2 , . . . , pk ). The ordered pair (w, p) is
called an i.f.s. with constant probabilities. The pi are called the probabilities
of (w, p). We say that the pi are constant probabilities because one may
consider a more general i.f.s. with probabilities where the probabilities are
place-dependent. That is, the pi are continuous real-valued functions on X,
pi : X → R, i = 1, 2, . . . , k, such that pi (x) ≥ 0 for every x ∈ X and i =
k
pi (x) = 1 for every x ∈ X. In this paper we deal with i.f.s.
1, 2, . . . , k, and
i=1
with constant probabilities, and only occasionally we will relate to results that
involve i.f.s. with place-dependent probabilities.
Let (w, p) be an i.f.s. with constant probabilities defined on (X, d),
w = (w1 , w2 , . . . , wk ), p = (p1 , p2 , . . . , pk ), and let M(X) be the Banach
lattice of all real-valued signed Borel measures on X (the norm on M(X) is
the usual one, namely, the total variation norm). Then we can associate with
k
pi µ(wi−1 (A))
(w, p) an operator T : M(X) → M(X) defined by T µ(A) =
i=1
for every µ ∈ M(X) and every A ∈ B(X), where B(X) stands for the σalgebra of all Borel measurable subsets of X. It is known (and not difficult
to see) that T is well-defined (that is, T µ is an element of M(X) for every
µ ∈ M(X)), and that T is a linear operator. Also known (and easy to see)
is that T µ ≥ 0 and T µ = µ whenever µ ∈ M(X), µ ≥ 0. Thus, T is a
Markov operator. (For the definition of a Markov operator, and for details on
these operators, see the next section.)
A probability measure µ ∈ M(X) (that is, an element µ of M(X) such
that µ ≥ 0 and µ = 1) is called an invariant probability of T (or a T invariant probability, or an invariant probability of (w, p)) if T µ = µ. If T
has only one invariant probability, then T (or (w, p)) is said to be uniquely
ergodic.
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Equicontinuity and existence of attractive probability measures
261
Let Bb (X) be the Banach lattice of all real-valued Borel measurable
bounded functions on X, where the norm on Bb (X) is the usual sup (uniform)
norm: f = sup |f (t)| for every f ∈ Bb (X). Let also Cb (X) be the Banach
t∈X
sublattice of Bb (X) that consists of all real-valued bounded continuous functions on X, and C0 (X) be the Banach sublattice of Cb (X) (and of Bb (X),
of course) that consists of all real-valued continuous
functions that vanish at
infinity. We will use the notation f, µ = f (x) dµ(x), where f ∈ Bb (X) and
µ ∈ M(X).
A probability measure µ∗ ∈ M(X) is called an attractive probability of
T (or of (w, p)) if the sequence (f, T n µ)n∈N∪{0} converges to f, µ∗ whenever f ∈ C0 (X) and µ is a probability measure, µ ∈ M(X). Since a probability measure µ∗ is an attractive probability of T if and only if the sequence
(f, T n µ)n∈N∪{0} converges to f, µ∗ for every f ∈ Cb (X) and every probability measure µ ∈ M(X), it can be shown that if T has an attractive probability
µ∗ , then µ∗ is a T -invariant probability, and T is uniquely ergodic (thus, T
has only one invariant probability, namely µ∗ ).
Barnsley and Elton [3] introduced a family of conditions that were named
average contractivity conditions in Barnsley, Demko, Elton, and Geronimo
[2], and can be stated as follows: given q ∈ R, q > 0, we say that (w, p)
satisfies the q-average contractivity condition if there exists α ∈ R, 0 ≤ α < 1,
k
pi dq (wi x, wi y) ≤ αdq (x, y) for every x ∈ X and y ∈ X. If
such that
i=1
(w, p) satisfies the 1-average contractivity condition, we will say that (w, p)
satisfies the standard average contractivity condition, or simply, the average
contractivity condition.
Given y ∈ X and z ∈ X, we say that z is a child of y, or that y is
a parent of z if there exist x ∈ X, n ∈ N ∪ {0}, and n + 1 nonnegative
integers i1 , i2 , . . . , in+1 , ih ∈ {1, 2, . . . , k} for every h = 1, 2, . . . , n + 1 such
that y = wi1 wi2 · · · win x (y = x if n = 0) and z = wi1 wi2 · · · win win+1 x.
We say that (w, p) satisfies the parent-child average contractivity condition if
there exists α ∈ R, 0 ≤ α < 1, such that
(1.1)
k
pi d(wi y, wi z) ≤ αd(y, z)
i=1
whenever z is a child of y, y ∈ X, z ∈ X.
Given x ∈ X, set


there exist n ∈ N and n integers i1 , i2 , . . . , in , 

∪ {x}.
O(x) = y ∈ X ih ∈ {1, 2, . . . , k} for every h = 1, 2, . . . , n, such


that y = wi1 wi2 · · · win x
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Radu Zaharopol
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We call O(x) the orbit of x under the action of T (or of (w, p)). The closure
O(x) of O(x) (in the metric topology of X) is called the orbit-closure of x.
Our first criterion states that if (w, p) satisfies the parent-child average
contractivity condition, then (w, p) has an attractive probability if and only
O(x) = ∅ if and only if O(x) ∩ O(y) = ∅ for every x ∈ X and y ∈ X
if
x∈X
(that is, the intersection of any two orbit-closures is nonempty).
Clearly, if (w, p) satisfies the 1-average contractivity condition, then
(w, p) satisfies the parent-child average contractivity condition, as well. We
will offer an example of an i.f.s. with constant probabilities (w, p) which satisfies the parent-child average contractivity condition, but for every q ∈ R,
q > 0, (w, p) fails to satisfy the q-average contractivity condition. Moreover,
(w, p) satisfies our criterion based on the parent-child average contractivity
condition, but does not satisfy the conditions of Theorem 1 of [3]. Finally,
note that while any q-average contractivity condition, q ∈ R, q > 0, is stated
in terms of two independent variables, the parent-child average contractivity
condition can be stated in terms of only one variable (the common ancestor
of the parent and the child). Therefore, our condition is easier to check.
If A is a nonempty subset of X, we denote by diam A or |A| the diameter
of A. That is, diam A = sup{d(x, y) | x ∈ A, y ∈ A}. The set A is said to be
bounded if diam A < ∞. If a ∈ X and r ∈ R, r > 0, we use the notation
B(a, r) for the open ball centered at a of radius r. That is, B(a, r) = {x ∈
X | d(a, x) < r}. Clearly, a subset A of X, A = ∅, is bounded if and only if
A ⊆ B(a, r) for some a ∈ X and r ∈ R, r > 0.
Let (w, p), w = (w1 , w2 , . . . , wk ), p = (p1 , p2 , . . . , pk ), be an i.f.s. with
probabilities defined on (X, d). We say that (w, p) is ELM-weakly hyperbolic
if the following two conditions are satisfied:
(ELM 1) For every bounded subset B of X, B = ∅, and for every ε ∈
R, ε > 0, there exists a positive integer nε,B that depends only on ε and B such
that |wi1 wi2 · · · win (B)| < ε for every n ≥ nε,B and every ij ∈ {1, 2, . . . , k},
j = 1, 2, . . . , n.
(ELM 2) For every nonempty compact subset K of X, there exists
a bounded subset B of X such that K ⊆ B and wi (B) ⊆ B for every i =
1, 2, . . . , k.
The notion of ELM (Edalat-Lasota-Myjak) weak hyperbolicity appears
in Edalat [5] in the case in which (X, d) is compact, and in Theorem 5.1 of
Lasota and Myjak [11] in the general case.
Our second criterion states that if (w, p) is an ELM-weakly hyperbolic
i.f.s. with constant probabilities, then (w, p) has an attractive probability. The
criterion strengthens and extends Corollary 3.3 of Edalat [5], and complements
Theorem 5.1 of Lasota and Myjak [11].
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Equicontinuity and existence of attractive probability measures
263
The two criteria are independent. That is, there exist i.f.s. that satisfy
the conditions of the first criterion but fail to satisfy the condition of the
second criterion, and there exist ELM-weakly hyperbolic i.f.s. with constant
probabilities that fail to satisfy the conditions of the first criterion. We will
discuss such examples in the last section.
In proving the two criteria we will use the same basic approach. Given
an i.f.s. (w, p) that satisfies the conditions of one of the criteria, we first
prove that (w, p) is equicontinuous (see the next section for the definition of
equicontinuous i.f.s.). Next, we use some of the results of [29] in order to prove
that (w, p) is uniquely ergodic, so (w, p) has a unique invariant probability,
say µ∗ . Finally, we prove that µ∗ is actually an attractive probability of (w, p).
The paper is organized as follows. In the next section (Section 2) we
review briefly the terminology and the notation used in [29]. In Section 3 we
prove the first criterion. Finally, in the last section (Section 4) we prove the
second criterion.
2. PRELIMINARIES ON MARKOV-FELLER OPERATORS
Our goal in this section is to review the results on Markov-Feller operators that are used in this paper. As is well-known these operators have been
studied extensively (see, for example, the monographs by Hernández-Lerma
and Lasserre [6], Meyn and Tweedie [16], Revuz [20], Rosenblatt [21], and the
papers by Lasota and Myjak [10], [11], [12], [13], [14], and by Lasota and Yorke
[15]). In our approach here we will follow [29].
Let (X, d) be a locally compact separable complete metric space, and let
M(X) be the Banach lattice of all real-valued signed Borel measures on X
(the norm on M(X) is the usual one, namely, the total variation norm).
A linear operator T : M(X) → M(X) is called a Markov operator if
T µ ≥ 0 and T µ = µ whenever µ ∈ M(X), µ ≥ 0. It is easy to see
that if T is a Markov operator, then T is a positive contraction of M(X) and
T = 1.
As already pointed out in Introduction, we will let Bb (X) be the space of
all real-valued Borel measurable bounded functions
on X. Given f ∈ Bb (X)
and µ ∈ M(X), we use the notation f, µ = f (x) dµ(x).
Let Cb (X) be the Banach lattice of all real-valued continuous bounded
functions defined on X (the norm on Cb (X) is the usual sup (uniform) norm
defined by f = sup |f (t)| for every f ∈ Cb (X)).
t∈X
Let S : Cb (X) → Cb (X) be a linear operator, and let T : M(X) → M(X)
be a Markov operator. The ordered pair (S, T ) is called a Markov-Feller pair if
(2.1)
Sf, µ = f, T µ
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Radu Zaharopol
6
for every f ∈ Cb (X) and µ ∈ M(X).
A Markov operator T : M(X) → M(X) is called a Markov-Feller operator if there exists a linear operator S : Cb (X) → Cb (X) such that (S, T ) is a
Markov-Feller pair.
A measure µ ∈ M(X), µ ≥ 0, µ = 1, is called an invariant probability
of T (or of (S, T )) if T µ = µ. We also say that µ is a T -invariant probability.
We say that T (or (S, T )) is uniquely ergodic if T has one and only one
invariant probability.
We say that a probability measure µ∗ , µ∗ ∈ M(X) is an attractive probability of T (or of (S, T )) if the sequence (f, T n µ)n∈N∪{0} converges to f, µ∗ for every f ∈ Cb (X) and every probability measure µ ∈ M(X). It is easy to
see that if T has an attractive probability µ∗ , then µ∗ is the unique invariant
probability of T , so T is uniquely ergodic.
If x ∈ X, we denote by δx the Dirac measure concentrated at x (that is,
δx is the probability measure in M(X) defined by δx ({x}) = 1).
As usual, the support of a signed measure ν, ν ∈ M(X) is denoted by
∞
supp (T n δx ). We call O(x) the orbit
supp ν. Let x ∈ X, and set O(x) =
n=0
of x under the action of T (or (S, T )). The closure O(X) of O(X) (in the
topology defined by the metric d on X) is called the orbit-closure of x under
the action of T (or (S, T )). An element z of X is called a universal element
O(X).
for (S, T ) (or for T ) if z ∈
x∈X
Naturally, we say that (S, T ) (or T ) has universal elements if
O(x) = ∅.
x∈X
Let (fn )n∈N be a sequence of real-valued functions defined on X. We say
that the sequence (fn )n∈N is equicontinuous if for every convergent sequence
(xk )k∈N of elements of X, and for every ε ∈ R, ε > 0, there exists kε ∈ N
such that |fn (xk ) − fn (x)| < ε for every k ≥ kε and every n ∈ N provided
that x = lim xk . We say that (fn )n∈N converges uniformly on the compact
k→∞
subsets of X if there exists a function f : X → R such that for every nonempty
compact subset K of X and for every ε ∈ R, ε > 0, there exists nK,ε ∈ N such
that |fn (x) − f (x)| < ε for every n ≥ nε and every x ∈ K. Naturally, in this
case, we say that f is the uniform limit of (fn )n∈N on the compact subsets of
X, or that (fn )n∈N converges uniformly to f on the compact subsets of X. We
say that the sequence (fn )n∈N is a uniformly Cauchy sequence on the compact
subsets of X if for every nonempty compact subset K of X, and for every
ε ∈ R, ε > 0, there exists nK,ε ∈ N such that |fn (x) − fm (x)| < ε for every
n ≥ nK,ε , m ≥ nK,ε, and x ∈ K. It can be shown (see Proposition 1.3.7
of [29]) that a sequence (fn )n∈N converges uniformly on the compact subsets
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Equicontinuity and existence of attractive probability measures
265
of X if and only if (fn )n∈N is a Cauchy sequence on the compact subsets of
X. Also, it can be shown (see Proposition 1.3.8 of [29]) that if (fn )n∈N is a
bounded sequence of elements of Cb (X), and if there exists f : X → R such
that (fn )n∈N converges uniformly on the compact subsets of X to f , then
f ∈ Cb (X), and the sequence (fn )n∈N is equicontinuous.
The Markov-Feller pair (S, T ) (or the operator S) is called equicontinuous
if the sequence (S n f )n∈N∪{0} is equicontinuous whenever f ∈ C0 (X). Note
that if S has the property that the sequence (S n f )n∈N∪{0} is a uniformly
Cauchy sequence on the compact subsets of X whenever f ∈ C0 (X), then S
is equicontinuous. Note also that the equicontinuity of (S, T ) as defined here
is called also C0 (X)-equicontinuity in [29].
Theorem 2.1. Let (S, T ) be an equicontinuous Markov-Feller pair, and
assume that (S, T ) has universal elements. Then either (S, T ) does not have
invariant probabilities, or (S, T ) is uniquely ergodic.
Proof. See the proof of Corollary 4.1.10 (a) of [29].
Theorem 2.2. If (S, T ) is an equicontinuous Markov-Feller pair defined
on (X, d), and if O(x) ∩ O(y) = ∅ for every x ∈ X and y ∈ Y , then either
(S, T ) does not have invariant probabilities, or (S, T ) is uniquely ergodic.
Proof. See the proof of Theorem 4.1.11 of [29].
Now, let (w, p), w = (w1 , w2 , . . . , wk ), p = (p1 , p2 , . . . , pk ) be an i.f.s.
with constant probabilities defined on (X, d), and let T be the operator defined
in Introduction. That is, let T : M(X) → M(X) be defined by T µ(A) =
k
pi µ wi−1 (A) for every µ ∈ M(X) and A ∈ B(X). Also let S : Cb (X) →
i=1
Cb (X) be defined by
(2.2)
Sf (x) =
k
pi f (wi (x))
i=1
for every f ∈ Cb (X) and x ∈ X. Note that S is well-defined because
k
i=1
pi f ◦wi
belongs to Cb (X) whenever f ∈ Cb (X). Observe that for every f ∈ Cb (X)
and x ∈ X we have
k
2
pi1 f ◦ wi1 (x) =
S f (x) = S(Sf )(x) = S
=
k
i1 =1
pi2
k
i1 =1
pi1 (f ◦ wi1 ◦ wi2 )
i1 =1
(x) =
k
i1 =1, i2 =1
pi1 pi2 f (wi1 wi2 (x)) ,
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Radu Zaharopol
8
and, in general,
n
S f (x) =
(2.3)
k
n pi1 pi2 · · · pin f (wi1 wi2 · · · win (x))
s=1 is =1
for every n ∈ N.
We now note that (S, T ) is a Markov-Feller pair. Since (as pointed out
in Introduction) T is a Markov operator, in order to show that (S, T ) is a
Markov-Feller pair we have to prove that S and T satisfy (2.1). To this
k
pi f ◦ wi is a real-valued bounded Borel measurable function
end, note that
i=1
whenever f ∈ Bb (X). Therefore, we can use (2.2) to extend S to an operator,
denoted again by S, such that the extended operator is defined from Bb (X) to
Bb (X). Using the extended operator S, it is easy to see that (2.1) is satisfied
whenever f = 1A , A ∈ B(X). So (2.1) is satisfied whenever f is a simple
function. Finally, if f ∈ Bb (X) (in particular, if f ∈ Cb (X)), then using the
fact that there exists a sequence of simple functions that converges uniformly
to f , we obtain that (2.1) is satisfied. We call (S, T ) the Markov-Feller pair
defined by (w, p).
Clearly, the notions of invariant probability, unique ergodicity, attractive
probability, and orbit as defined in Introduction agree with the corresponding notions defined in this section. For example, the orbit of an element
x ∈ X under the action of (w, p) as defined in Introduction is the same as
the orbit of x under the action of (S, T ) as defined in this section, since
supp (T δx ) = {w1 (x), w2 (x), . . . , wk (x)} for every x ∈ X whenever (S, T ) is
defined by (w, p).
Let (An )n∈N be a sequence of subsets of X and set


there exists xn ∈ An for every n ∈ N such 

.
Li An = x ∈ X that the sequence (xn )n∈N converges
n→∞


to x in the metric topology of (X, d)
The set Li An is called the (topological) lower limit of the sequence (An )n∈N
n→∞
(for details on topological lower limits, and related notions see Kuratowski [9]).
Given an i.f.s. (w, p) with constant probabilities defined on (X, d), let
(S, T ) be the Markov-Feller pair defined by (w, p), set σ(x) = Li supp (T n δx )
n→∞
σ(x).
for every x ∈ X, and set σ =
x∈X
Theorem 2.3. If the i.f.s. (w, p) has an attractive probability µ∗ , then
supp µ∗ = σ.
Proof. See the proof of Theorem 2.3 of [28].
9
Equicontinuity and existence of attractive probability measures
267
Finally, a few words about Lipschitz functions. As usual, a function
f : X → R is called a Lipschitz function if there exists λ ∈ R, λ ≥ 0 such
that |f (x) − f (y)| ≤ λd(x, y) for every x ∈ X and y ∈ X. The set of all
Lipschitz functions defined on X is denoted by Lip (X). We also use the
notation Lip1 (X) for the set of all Lipschitz functions f : X → R which have
the property that |f (x) − f (y)| ≤ d(x, y) for every x ∈ X and y ∈ X.
3. AVERAGE CONTRACTIVITY CONDITIONS
Our goal in this section is to prove the criterion that is based on the
parent-child average contractivity condition, and that was described in Introduction.
Let (X, d) be a locally compact separable complete metric space, let
(w, p), w = (w1 , w2 , . . . , wk ), p = (p1 , p2 , . . . , pk ), be an i.f.s. with constant
probabilities defined on (X, d), and let (S, T ) be the Markov-Feller pair defined
by (w, p).
Proposition 3.1. If (w, p) satisfies the parent-child average contractivity condition, then the sequence (S n f )n∈N∪{0} converges uniformly on the
compact subsets of X whenever f ∈ C0 (X).
Proof. In view of our discussion on uniform convergence on the compact
subsets of X in Section 2 (see also Proposition 1.3.7 of [29]), the proof of the
proposition will be complete if we show that (S n f )n∈N∪{0} is uniformly Cauchy
on the compact subsets of X for every f ∈ C0 (X). To this end, we will show
that (S n f )n∈N∪{0} is uniformly Cauchy on the compact subsets of X in the
following three cases:
(i) f ∈ Lip1 (X) ∩ C0 (X);
(ii) f ∈ Lip (X) ∩ C0 (X);
(iii) f ∈ C0 (X) (the general case).
(i) Since (w, p) satisfies the parent-child average contractivity condik
pi d(wi y, wi z) ≤
tion, we may and do pick α ∈ R, 0 ≤ α < 1, such that
i=1
αd(y, z) whenever y is a child of z, y ∈ X, z ∈ X.
For every nonempty compact subset K of X set MK = sup
Since the function φK : K → R defined by φK (x) =
k
i=1
k
x∈K i=1
pi d(wi x, x).
pi d(wi x, x) for every
x ∈ K is continuous, it follows that MK is finite. Thus, 0 ≤ MK < +∞ for
every nonempty compact subset K of X.
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Radu Zaharopol
10
Now, let f ∈ Lip1 (X) ∩ C0 (X), let K be a nonempty compact subset of
X, and let ε ∈ R, ε > 0. We have to prove that there exists nε,K ∈ N such
that |S n f (x) − S m f (x)| < ε for every n ≥ nε,K and m ≥ nε,K .
To this end, let nε,K be the first natural number n such that
1
MK < ε.
αn · 1−α
Using formula (2.1), the fact that f ∈ Lip1 (X), and keeping in mind that
for every n ∈ N, every n + 1 elements i1 , i2 , . . . , in+1 of the set {1, 2, . . . , k}, every h ∈ {1, 2, . . . , n}, and every x ∈ X we have that wih wih+1 · · · win win+1 x is a
child of wih wih+1 · · · win x (so we can apply the parent-child average contractivity condition to wih wih+1 · · · win win+1 x and wih wih+1 · · · win x), we obtain that
for every x ∈ K
n+1 k
n+1
n
f (x) − S f (x)| = pi1 pi2 · · · pin+1 f wi1 wi2 · · · win+1 (x) −
|S
s=1 is =1
−
n k
s=1 is =1
pi1 pi2 · · · pin f (wi1 wi2 · · · win (x)) =
n+1
k
pi1 pi2 · · · pin+1 f wi1 wi2 · · · win+1 (x) −
=
s=1 is =1


n k
k
pi1 pi2 · · · pin 
pin+1  f (wi1 wi2 · · · win (x)) ≤
−
s=1 is =1
≤
n+1
k
in+1 =1
pi1 pi2 · · · pin pin+1 f (wi1 wi2 · · · win win+1 (x))−f (wi1 wi2 · · · win (x)) ≤
s=1 is =1
≤
n+1
k
pi1 pi2 · · · pin pin+1 d wi1 wi2 · · · win win+1 (x), wi1 wi2 · · · win (x) ≤
s=1 is =1
≤α
n+1
k
pi2 pi3 · · · pin pin+1 d wi2 wi3 · · · win win+1 (x), wi2 wi3 · · · win (x) ≤
s=2 is =1
≤ α2
n+1
k
pi3 pi4 · · · pin pin+1 d wi3 wi4 · · · win win+1 (x), wi3 wi4 · · · win (x) ≤
s=3 is =1
≤ · · · ≤ αn−1
k
k
pin pin+1 d win win+1 (x), win (x) ≤
in+1 =1 in =1
≤ αn
k
in+1 =1
pin+1 d win+1 (x), x ≤ αn MK
11
Equicontinuity and existence of attractive probability measures
269
for every n ∈ N ∪ {0}.
Thus, for every m ∈ N, m ≥ nε,K , n ∈ N, n ≥ nε,K , m < n, and every
x ∈ K we have
|S n f (x) − S m f (x)| ≤ |S n f (x) − S n−1 f (x)| + |S n−1 f (x) − S n−2 f (x)|+
+|S n−2 f (x) − S n−3 f (x)| + · · · + |S m+1 f (x) − S m f (x)| ≤
≤ αn−1 MK + αn−2 MK + αn−3 MK + · · · + αm MK =
= αm MK (αn−m−1 + αn−m−2 + · · · + α + 1) ≤
≤ αm MK
Accordingly,
of X in this case.
∞
αi = αm MK ·
i=0
n
(S f )n∈N∪{0} is
1
< ε.
1−α
uniformly Cauchy on the compact subsets
(ii) If f ∈ Lip (X) ∩ C0 (X), then there exists λ ∈ R, λ > 0, such that
|f (x)−f (y)| ≤ λd(x, y) for every x ∈ X and y ∈ X; consequently, the function
g = f /λ belongs to Lip1 (X) ∩ C0 (X). Using our discussion of case (i), we
obtain that (S n g)n∈N∪{0} is uniformly Cauchy on the compact subsets of X.
Accordingly, the sequence (S n f )n∈N∪{0} is uniformly Cauchy on the compact
subsets of X, as well.
(iii) Let f ∈ C0 (X), let K be a compact subset of X, and let ε ∈ R, ε > 0.
We have to prove that there exists nε,K ∈ N such that |S n f (x) − S m f (x)| < ε
for every n ≥ nε,K and m ≥ nε,K .
Since Lip (X)∩C0 (X) is dense in C0 (X) (see, for example, Corollary 1.3.4
of [29]), there exists g ∈ Lip (X) ∩ C0 (X) such that f − g = sup |f (t) −
t∈X
g(t)| < ε/3. Using the discussion of case (ii), we obtain that there exists
nε/3,K (g) ∈ N such that |S n g(x) − S m g(x)| < ε/3 for every n ≥ nε/3,K (g),
m ≥ nε/3,K (g), and x ∈ K. Set nε,K = nε/3,K (g). Taking into consideration
that S n (f − g) ≤ f − g < ε/3 for every n ∈ N (because S is a positive
contraction of Cb (X)), we obtain that
|S n f (x)−S m f (x)| ≤ |S n f (x)−S n g(x)|+|S n g(x)−S m g(x)|+|S m g(x)−S m f (x)| <
ε ε ε
ε
+ f − g < + + = ε
3
3 3 3
for every n ≥ nε,K , m ≥ nε,K , and x ∈ K. < f − g +
Proposition 3.1 has the following consequence:
Corollary 3.2. If (w, p) satisfies the parent-child average contractivity
condition, then the Markov-Feller pair (S, T ) defined by (w, p) is equicontinuous.
270
Radu Zaharopol
12
Proof. By Proposition 3.1, the sequence (S n f )n∈N∪{0} converges uniformly on the compact subsets of X for every f ∈ C0 (X). Using the remarks
made on equicontinuity in Section 2 before Theorem 2.1, we obtain that (S, T )
is equicontinuous. Now, assume that the i.f.s. (with constant probabilities) (w, p) satisfies
the parent-child average contractivity condition. Then, by Proposition 3.1, for
every x ∈ X the sequence (S n f (x))n∈N∪{0} is convergent whenever f ∈ C0 (X).
Therefore, given x ∈ X, the map εx : C0 (X) → R, εx (f ) = lim S n f (x) for
n→∞
every f ∈ C0 (X) is well-defined. Clearly, εx is positive (that is, εx (f ) ≥ 0
whenever f ∈ C0 (X), f ≥ 0) and linear. Therefore, εx is also bounded
(continuous) for every x ∈ X. Since we identify the topological dual of C0 (X)
with M(X), we may and do think of the εx , x ∈ X, as elements of M(X).Thus,
it makes sense to ask if the maps εx , x ∈ X, are T -invariant (as elements of
M(X)). The next lemma answers this question.
Lemma 3.3. T εx = εx for every x ∈ X.
Proof. The proof follws from the discussion at the beginning of Section 2.1 of [29]. To be specific, for every Banach limit L and for every x ∈ X,
(L)
(L)
let εx : C0 (X) → R be defined by εx (f ) = L (S n f (x))n∈N∪{0} for every f ∈ C0 (X). Since (S n f (x))n∈N∪{0} is a convergent sequence for every
(L)
f ∈ C0 (X) and x ∈ X, we have εx = εx for every Banach limit L and x ∈ X.
By Theorem 2.1.1 of [29], εx is a T -invariant measure for every x ∈ X. Note that Lemma 3.3 does not tell us that the measures εx , x ∈ X,
are nonzero. Our next goal is to prove that these measures are not only
nonzero, but that they are all probability measures. To this end, we will use a
construction similar to the one given by Barnsley and Elton [3] (see also [27]).
Given the i.f.s. (w, p), w = (w1 , w2 , w3 , . . . , wk ), p = (p1 , p2 , p3 , . . . , pk ),
set Ωk = {1, 2, 3, . . . , k}, let P(Ωk ) be the σ-algebra of all subsets of Ωk , and
think of p = (p1 , p2 , p3 , . . . , pk ) as a probability measure on (Ωk , P(Ωk )) defined by p({i}) = pi for every i = 1, 2, 3, . . . , k. Note that p stands for the
k-uple (p1 , p2 , p3 , . . . , pk ), as well as for a probability measure on (Ωk , P(Ωk )).
This abuse of notation will not lead to any confusion throughout our discussion.
Let Ω = ΩN
k = {(in )n∈N | in ∈ Ωk for every n ∈ N}, and let F be the
product σ-algebra on Ω. Thus, F is generated by the rectangles
Rj1 j2 j3 ...jl = {(in )n∈N ∈ Ω | ih = jh for every h ∈ {1, 2, 3, . . . , l}},
l ∈ N, jh ∈ Ωk , for every h = 1, 2, 3, . . . , l.
It is well known (see, for example, Corollaire on p. 79 of Neveu’s book
[17]) that there exists a unique probability measure P on (Ω, F) such that
13
Equicontinuity and existence of attractive probability measures
271
P (Rj1 j2 j3 ...jl ) = pj1 pj2 pj3 · · · pjl for every l ∈ N and every l positive integers
j1 , j2 , j3 , . . . , jl such that jh ∈ {1, 2, 3, . . . , k} for every h = 1, 2, 3, . . . , l.
Lemma 3.4. Assume that the i.f.s. with constant probabilities (w, p),
w = (w1 , w2 , w3 , . . . , wk ), p = (p1 , p2 , p3 , . . . , pk ), satisfies the parent-child
average contractivity condition, and let x0 ∈ X. Then there exists an Fmeasurable subset Ω0 of Ω such that P (Ω0 ) = 1 and such that for every
(in )n∈N ∈ Ω0 the sequence (wi1 wi2 wi3 · · · win x0 )n∈N converges in the metric
topology of X.
Proof. For every n ∈ N, let hn : Ω → R be a function defined by
hn ((il )l∈N ) = d wi1 wi2 wi3 · · · win x0 , wi1 wi2 wi3 · · · win win+1 x0
for every (il )l∈N = (i1 , i2 , i3 , . . . , in , in+1 , . . .), (il )l∈N ∈ Ω.
Clearly, the functions hn , n ∈ N, are integrable (because hn is a realvalued simple measurable function for every n ∈ N).
Since (w, p) satisfies the parent-child average contractivity condition,
there exists α ∈ R, 0 ≤ α < 1, that satisfies inequality (1.1) whenever z is a
child of y, y ∈ X, z ∈ X.
Since wi1 wi2 wi3 · · · wil wil+1 x0 is a child of wi1 wi2 wi3 · · · wil x0 whenever
l ∈ N, and i1 , i2 , i3 , . . . , il , il+1 are l + 1 natural numbers such that ij ∈
{1, 2, 3, . . . , k} for every j = 1, 2, 3, . . . , l, l + 1, we obtain that
k
m+1
pi1 pi2 pi3 · · · pim pim+1 ·
hm dP =
s=1 is =1
·d wi1 wi2 wi3 · · · wim x0 , wi1 wi2 wi3 · · · wim wim+1 x0 =
=
m+1
k
pi2 pi3 · · · pim pim+1 ·
s=2 is =1
k
·
pi1 d wi1 wi2 · · · wim x0 , wi1 wi2 · · · wim wim+1 x0 ≤
i1 =1
≤α
m+1
k
pi2 pi3 · · · pim pim+1 d wi2 wi3 · · · wim x0 , wi2 wi3 · · · wim wim+1 x0 =
s=2 is =1
=α
hm−1 dP
for every m ∈ N, m ≥ 2. Thus,
2
m−1
hm−2 dP ≤ · · · ≤ α
h1 dP =
hm dP ≤ α hm−1 dP ≤ α
272
Radu Zaharopol
k
m−1
=α
14
pi1 pi2 d (wi1 x0 , wi1 wi2 x0 ) ≤ α
m
i1 =1,i2 =1
k
pi2 d (x0 , wi2 x0 ) ≤ αm c
i2 =1
for every m ∈ N, where c = max{d(x0 , wi (x0 )) | i = 1, 2, 3, . . . , k}.
∞ cα
< +∞.
hm dP ≤ c(α + α2 + · · · ) = 1−α
Accordingly,
m=1
Since the functions hm , m ∈ N, are positive (that is, hm ((il )l∈N ) ≥ 0 for
every m ∈ N and every (il )l∈N ∈ Ω), we can apply the Beppo-Levy theorem,
∞
hm is actually P -integrable
and obtain that the [0, ∞]-valued function
(that is,
m=1
∞
m=1
hm dP < +∞), so there exists an F-measurable subset Ω0 of
Ω such that P (Ω0 ) = 1 and
Since
∞
hm ((il )l∈N ) =
m=1
∞
m=1
hm ((il )l∈N ) < +∞ for every (il )l∈N ∈ Ω0 .
∞
d wi1 wi2 · · · wim x0 , wi1 wi2 · · · wim wim+1 x0 < +∞,
m=1
the sequence (wi1 wi2 · · · wim x0 )m∈N is convergent for every (il )l∈N ∈ Ω0 . Indeed, given (il )l∈N ∈ Ω0 , for every ε ∈ R, ε > 0, there exists mε ∈ N such
∞
d wi1 wi2 · · · wim x0 , wi1 wi2 · · · wim wim+1 x0 < ε. Therefore, for evthat
m=mε
ery l ∈ N, l ≥ mε , and j ∈ N, j ≥ mε , l < j, we have
d wi1 wi2 · · · wil x0 , wi1 wi2 · · · wij x0 ≤
≤ d wi1 wi2 · · · wil x0 , wi1 wi2 · · · wil wil+1 x0 +
+d wi1 wi2 · · · wil wil+1 x0 , wi1 wi2 · · · wil wil+1 wil+2 x0 + · · · +
+d wi1 wi2 · · · wil · · · wij−1 x0 , wi1 wi2 · · · wil · · · wij−1 wij x0 ≤
≤
∞
d wi1 wi2 · · · wim x0 , wi1 wi2 · · · wim wim+1 x0 < ε.
m=mε
Thus, the sequence (wi1 wi2 · · · wim x0 )m∈N is a Cauchy sequence. Hence
(wi1 wi2 · · · wim x0 )m∈N converges. Lemma 3.4 enables us to prove that the measures εx , x ∈ X, defined
before Lemma 3.3 are probability measures.
Proposition 3.5. Assume that the i.f.s. with constant probabilities (w, p),
w = (w1 , w2 , . . . , wk ), p = (p1 , p2 , . . . , pk ), satisfies the parent-child average
contractivity condition. Then εx0 is a probability measure for every x0 ∈ X.
15
Equicontinuity and existence of attractive probability measures
273
Proof. Let x0∈X, and let (Ω, F, P ) be the probability space defined before Lemma 3.4. Let Ω0 be an F-measurable subset of Ω such that P (Ω0 ) = 1
and such that the sequence (wi1 wi2 · · · win x0 )n∈N converges in the metric topology of X whenever (in )n∈N ∈ Ω0 (the existence of Ω0 is assured by Lemma 3.4).
Now, let φ : Ω → X be a function defined by
lim wi1 wi2 · · · wil x0 if (il )l∈N ∈ Ω0
l→∞
φ ((il )l∈N ) =
x0
if (il )l∈N ∈ Ω0 .
Taking into consideration the way in which Ω0 was chosen, φ is well-defined.
For every n ∈ N, let φn : Ω → X be defined by φn ((il )l∈N ) = wi1 wi2 · · ·
win x0 for every (il )l∈N ∈ Ω. Clearly, the functions φn , n ∈ N, are measurable
in the sense that φ−1
n (A) ∈ F for every n ∈ N and every A ∈ B(X). Obviously,
the sequence (φn ((il )l∈N ))n∈N converges to φ ((il )l∈N ) whenever (il )l∈N ∈ Ω0 .
Hence (φn )n∈N converges to φ P -a.e. Using a standard measure theoretical
result (see, for example, p. 58 of Doob [4]) we obtain that φ is measurable
(that is, φ−1 (B(X)) ⊆ F).
We now define a function µ̄ : B(X) → R by µ̄(A) = P (φ−1 (A)) for every
A ∈ B(X). Since P is a probability measure on (Ω, F), it is easy to see that µ̄
is a probability measure on (X, B(X)). Thus, in order to complete the proof
of the proposition, it is enough to show that µ̄ = εx0 .
For every f ∈ Bb (X), set fφ = f ◦ φ. Clearly, fφ is a measurable
function (from Ω to R) whenever f ∈ Bb (X). Using a well-known result in
measure-theoretical probability (see, for example,
Theorem 7, pp. 194–195 of
Shiryayev’s book [22]), we obtain that f dµ̄ = fφ dP for every f ∈ Bb (X).
Now, let f ∈ C0 (X), and set fφn = f ◦ φn for every n ∈ N. Since f is
continuous, and since (φn )n∈N converges to φ on Ω0 , the sequence (fφn )n∈N
converges to fφ on Ω0 . Therefore, (fφn )n∈N converges P -a.e. to fφ . Since
max {|fφn ((il )l∈N )| , |fφ ((il )l∈N )|} ≤ sup |f (x)| = f < +∞ for every n ∈ N
x∈X
and every (il )l∈N ∈ Ω, and since P is a probability measure, we can use
the
dominated convergence theorem
to obtain
that the sequence
Lebesgue
fφn dP n∈N converges, and that lim fφn dP = fφ dP = f dµ̄.
n→∞
Taking into consideration that
fφn dP =
n k
s=1 is =1
pi1 pi2 · · · pin f (wi1 wi2 · · · win x0 ) = S n f (x0 )
for every n ∈ N, we obtain that
fφn dP n∈N also converges to εx0 (f )
(= f, εx0 ).
We have therefore proved that f, µ̄ = f, εx0 for every f ∈ C0 (X).
Hence µ̄ = εx0 . 274
Radu Zaharopol
16
Note that if (w, p) is an i.f.s. with constant probabilities that satisfies the
parent-child average contractivity condition, and if (S, T ) is the Markov-Feller
pair defined by (w, p) then, by Proposition 3.1, the sequence (S n f (x))n∈N
converges whenever f ∈ C0 (X) and x ∈ X, so the maps εx : C0 (X) → R,
εx (f ) = lim S n f (x) for every f ∈ C0 (X), x ∈ X, are well-defined. The maps
n→∞
εx , x ∈ X can be thought of as elements of M(X). By Lemma 3.3, εx is
T -invariant for every x ∈ X, and εx , x ∈ X, are all probability measures by
Proposition 3.5. It follows that (w, p) has an attractive probability measure
if and only if εx = εy for every x ∈ X and y ∈ X. Indeed, if (w, p) has an
attractive probability, say µ∗ , then µ∗ is the unique invariant probability of T ,
so εx = µ∗ for every x ∈ X. Conversely, if εx = εy for every x ∈ X and y ∈ X,
then if we set µ∗ = εx for some x ∈ X, we obtain that (S n f (x))n∈N converges
to f, µ∗ for every f ∈ C0 (X) and x ∈ X. Therefore, by the Lebesgue dominated convergence theorem, the sequence (f, T n µ)n∈N converges to f, µ∗ whenever µ ∈ M(X) is a probability measure, and whenever f ∈ C0 (X) since
f, T n µ = S n f, µ for every n ∈ N. Thus, the criterion for the existence of
an attractive probability measure that is the main topic of this section and
that we are going to discuss now can be thought of as offering necessary and
sufficient conditions for the equalities εx = εy for every x ∈ X and y ∈ X to
hold true.
Theorem 3.6. Assume that the i.f.s. (w, p) has constant probabilities
and satisfies the parent-child average contractivity condition. Then the following assertions are equivalent.
(a) The i.f.s. (w, p) has an attractive probability measure.
O(x) = ∅.
(b)
x∈X
(c) O(x) ∩ O(y) = ∅ for every x ∈ X and y ∈ X.
Proof. Throughout the proof we let (S, T ) be the Markov-Feller pair
defined by (w, p).
(a)⇒(b) Let µ∗ be the attractive probability measure of (w, p). Clearly,
supp µ∗ = ∅. Therefore, in order tocomplete the proof of the implication, it
O(x).
is enough to prove that supp µ∗ ⊆
x∈X
To this end, let σ(x) be the topological lower limit of the sequence
σ(x) by Theorem 2.3, and
(supp (T n δx ))n∈N , x ∈ X. Since supp µ∗ =
x∈X
O(X).
since σ(x) ⊆ O(X) for every x ∈ X, we have supp µ∗ ⊆
x∈X
(b)⇒(c) is obvious.
17
Equicontinuity and existence of attractive probability measures
275
(c)⇒(a) We first note that (S, T ) has invariant probabilities. Indeed, in
view of our discussion preceding this theorem, the measures εx , x ∈ X, are
well-defined, are T -invariant, and are probability measures.
Since the Markov-Feller pair (S, T ) is equicontinuous by Corollary 3.2,
and since we assume that (c) holds, the conditions of Theorem 2.2 are satisfied,
so, (S, T ) is uniquely ergodic. Thus, εx = εy for every x ∈ X and y ∈ X. But
then, using the last remarks of the discussion preceding this theorem, we obtain
that (w, p) has an attractive probability. In general, the parent-child average contractivity condition alone does
not guarantee the existence of an attractive probability measure. That is,
there exist i.f.s. with constant probabilities that satisfy the parent-child average contractivity condition, but fail to have an attractive probability. The
following example illustrates this point.
Example 3.7. Let X = [0, 1]∪{2, 3}, and let d be the metric on X defined
by d(x, y) = |x − y| for every x ∈ X and y ∈ X (that is, d is the restriction
to X of the usual metric on R). Let (w, p) be the i.f.s. defined as follows:
w = (w1 , w2 ), where w1 : X → X is the identity map (that
x is, w1 (x) = x
2 if x ∈ [0, 1]
;
for all x ∈ X), and w2 : X → X is defined by w2 (x) =
x if x ∈ {2, 3}
p = (1/2, 1/2). Finally, let (S, T ) be the Markov-Feller pair defined by (w, p).
It is easy to see that δ0 , δ2 , and δ3 are T -invariant probability measures,
so (w, p) does not have an attractive probability measure (because, as pointed
out in Section 2, the existence of an attractive probability for (w, p) implies
the unique ergodicity of T ).
Since w1 w2 = w2 w1 = w2 , z is a child of y if and only if at least one of
the following conditions is satisfied:
(i) z = y
or
(ii) z = 12 y and y ∈ [0, 1].
If condition (i) is satisfied (that is, if z = y), then
1
1
d(w1 y, w1 z) + d(w2 y, w2 z) = 0.
2
2
If condition (ii) is satisfied, then
1
3
1
1
1
d(w1 y, w1 z) + d(w2 y, w2 z) = d(y, z) + d(y, z) = d(y, z).
2
2
2
4
4
Thus, (w, p) satisfies the parent-child average contractivity condition since
1
3
1
d(w1 y, w1 z) + d(w2 y, w2 z) ≤ d(y, z)
2
2
4
whenever z is a child of y. 276
Radu Zaharopol
18
As pointed out in Introduction, the q-average contractivity conditions,
q > 0 are stated in terms of two independent variables. By contrast, the
parent-child average contractivity condition can be stated in terms of only
one variable (a common ancestor of the parent and the child). Thus, in many
cases the parent-child average contractivity condition might be easier to check.
Clearly, if an i.f.s. (w, p) satisfies the 1-average contractivity condition,
then (w, p) satisfies the parent-child averaga contractivity condition. As can
be expected, there exist i.f.s. that satisfy the parent-child average contractivity
condition, but, for every q ∈ R, q > 0, they fail to satisfy the q-average
contractivity condition. Actually, as the next example illustrates, one can
find i.f.s. that satisfy the conditions of Theorem 3.6, that fail to satisfy the
q-average contractivity condition for every q ∈ R, q > 0, and fail to satisfy the
conditions of Theorem 1 of Barnsley and Elton [3].
Example 3.8. Let X = [0, 1/2], and let d be the metric on X defined
as the restriction to X of the usual metric on R (that is, d is defined by
d(x, y) = |x − y| for every x ∈ X and y ∈ X). Let (w, p) be the i.f.s. defined
as follows: w = (w1 , w2 ), where w1 : X → X is defined by w1 (x) = x2 for
every x ∈ X, and w2 : X → X is defined by w2 (x) = x for every x ∈ X (that
is, w2 is the identity map on X); p = (1/2, 1/2).
Since w1 w2 = w2 w1 = w1 , z is a child of y if and only if z = y 2 or z = y.
If z = y 2 , then y ∈ [0, 1/2] and z ∈ [0, 1/4]. Therefore,
1 2
1
|y − z 2 | + |y − z| =
2
2
7
1
= |y − z|(|y + z| + 1) ≤ |y − z|.
2
8
p1 d(w1 y, w1 z) + p2 d(w2 y, w2 z) =
Thus, inequality (1.1) is satisfied for α = 7/8. If y = z, then (1.1) is satisfied
for every α ∈ [0, 1]. In particular, (1.1) is satisfied for α = 7/8. Thus, (w, p)
satisfies theparent-child average contractivity condition. Since it is obvious
O(x), (w, p) satisfies the conditions of Theorem 3.6 (and it
that 0 ∈
x∈X
is easy to see that δ0 is an attractive probability of (w, p). Indeed, δ0 is an
invariant probability of (w, p); since, by Theorem 3.6, (w, p) has an attractive
probability measure, δ0 has to be the attractive probability of (w, p)).
Now, let q ∈ R, q > 0, and note that
sup
x∈[0, 12 ], y∈[0, 12 ], x=y
=
p1 dq (w1 x, w1 y) + p2 dq (w2 x, w2 y)
=
dq (x, y)
sup
x∈[0, 21 ], y∈[0, 12 ], x=y
1 2
2 |x
− y 2 |q + 12 |x − y|q
=
|x − y|q
19
Equicontinuity and existence of attractive probability measures
=
sup
x∈[0, 21 ], y∈[0, 12 ], x=y
1
1
|x + y|q +
2
2
277
= 1.
Accordingly, for every q ∈ R, q > 0 the i.f.s. (w, p) does not satisfy the
q-average contractivity condition. Consequently, (w, p) does not satisfy the
conditions of Theorem 1 of [3] since, by Lemma 1 of [3], if we assume that
(w, p) satisfies the conditions of Theorem 1 of [3], then (w, p) satisfies a qaverage contractivity condition for some q ∈ R, q > 0. 4. WEAKLY HYPERBOLIC ITERATED FUNCTION SYSTEMS
As mentioned in Introduction our goal in this section is to prove that an
ELM-weakly hyperbolic i.f.s. with constant probabilities defined on a locally
compact separable complete metric space has an attractive probability. However, since Edalat’s definition [5] of weak hyperbolicity in a compact space
looks somewhat different from the Lasota-Myjak definition that stems from
Theorem 5.1 of [11], we will first show that the two definitions agree on a
compact metric space.
Let (w, p), w = (w1 , w2 , . . . , wk ), p = (p1 , p2 , . . . , pk ), be an i.f.s. with
constant probabilities defined on a compact metric space (X, d). In order to
show that the two definitions agree, we will use the term E-weak hyperbolicity
for the weak hyperbolicity as defined by Edalat. Thus, (w, p) is called Eweakly hyperbolic if the sequence (|wi1 wi2 · · · win (X)|)n∈N converges to zero
whenever (in )n∈N is a sequence of elements of the set {1, 2, . . . , k}.
Proposition 4.1. The following assertions are equivalent.
(a) The i.f.s. (w, p) is ELM-weakly hyperbolic.
(b) The i.f.s. (w, p) is E-weakly hyperbolic.
Proof. (a)⇒(b) is obvious because the entire space X is a bounded set.
(b)⇒(a) Clearly, (w, p) satisfies (ELM 2) because the entire space X is
compact, so X is also bounded.
Now, assume that (w, p) is E-weakly hyperbolic. We have to prove
that (w, p) satisfies (ELM 1). Clearly, in order to prove that (w, p) satisfies
(ELM 1), it is enough to prove that for every ε ∈ R, ε > 0 there exists,
nε ∈ N such that for every n̄ ∈ N, n̄ ≥ nε , and for every n̄ positive integers
ī1 , ī2 , . . . , īn̄ , 1 ≤ īh ≤ k, h = 1, 2, . . . , n̄, we have |wī1 wī2 · · · wīn (X)| < ε.
To this end, set Ωk = {1, 2, . . . , k}, and assume that Ωk is endowed with
the discrete topology. Next, let Ω be the set of all Ωk -valued sequences. That
is, Ω = ΩN
k = {(in )n∈N | in ∈ Ωk for every n ∈ N}, and consider the product
topology on Ω. By Tychonoff’s theorem, Ω is a compact topological space.
As in the construction preceding Lemma 3.4, for every l ∈ N and every
l positive integers j1 , j2 , . . . , jl , 1 ≤ jh ≤ k, h = 1, 2, . . . , l, let Rj1 j2 j3 ...jl
278
Radu Zaharopol
20
be the rectangle in Ω consisting of all sequences (in )n∈N such that i1 = j1 ,
i2 = j2 , . . . , il = jl .
Now, let ε ∈ R, ε > 0, and let Rε be the collection of all rectangles
Rj1 j2 j3 ...jm , m ∈ N, jh ∈ {1, 2, 3, . . . , k} for every h = 1, 2, 3, . . . , m such that
|wj1 wj2 wj3 · · · wjm (X)| < ε.
Since we assume that (b) holds, we have
Rj1 j2 j3 ...jm .
Ω=
Rj1 j2 j3 ...jm ∈Rε
Indeed, if (in )n∈N ∈ Ω, then (by (b)) there exists nε ∈ N such that |wi1 wi2 · · ·
winε (X)| < ε. Therefore, (in )n∈N ∈ Ri1 i2 ...inε ∈ Rε .
Since the rectangles Rj1 j2 ...jm , m ∈ N, jh ∈ {1, 2, . . . , k} for every h =
1, 2, . . . , m are open subsets of Ω, and since Ω is compact, there exist l ∈ N
and l rectangles Rj (1) j (1) ...j (1) , Rj (2) j (2) ...j (2) , . . ., Rj (l) j (l) ...j (l) in Rε such that
Ω=
l
s=1
1
2
m1
1
m2
2
1
2
ml
Rj (s) j (s) ...j (s) . Set nε = max ms .
1
2
ms
1≤s≤l
Now, let n̄ ∈ N, n̄ ≥ nε , and let ī1 , ī2 , . . . , īn̄ be n̄ positive integers such
that īh ∈ {1, 2, . . . , k} for every h = 1, 2, . . . , n̄. Then Rī1 ī2 ...īn̄ ⊆ Rj (s) j (s) ...j (s)
1
2
ms
for some s ∈ {1, 2, . . . , l} because if (in )n∈N ∈ Rī1 ī2 ...īn̄ , then (in )n∈N ∈
Rj (s) j (s) ...j (s) for some s ∈ {1, 2, . . . , l} (since the l rectangles Rj (1) j (1) ...j (1) ,
1
2
ms
1
2
m1
(s)
Rj (2) j (2) ...j (2) ,. . .,Rj (l) j (l) ...j (l) form a (finite) covering of Ω), so i1 = ī1 = j1 ,
1
2
m2
(s)
1
2
ml
(s)
i2 = ī2 = j2 ,. . .,ims = īms = jms , and ims +1 = īms +1 , ims +2 = īms +2 , . . .,
in̄ = īn̄ . Hence Rī1 ī2 ...īn̄ ⊆ Rj (s) j (s) ...j (s) . (Note that we used here the obvious
ms
1
2
fact that if the intersection of two rectangles is nonempty, the rectangles are
equal, or one of the rectangles is a subset of the other.) By the definition of
the collection Rε and the fact that Rj (s) j (s) ...j (s) belongs to Rε , we obtain that
1
ms
2
|wī1 wī2 · · · wīn̄ (X)| = |wj (s) wj (s) . . . wj (s) (wīms +1 wīms +2 · · · wīn̄ (X))| ≤
1
2
ms
≤ |wj (s) wj (s) · · · wj (s) (X)| < ε.
1
2
ms
We now start to prepare the ground for the main result of this section.
Thus, let (w, p), w = (w1 , w2 , . . . , wk ), p = (p1 , p2 , . . . , pk ), be an i.f.s. with
constant probabilities defined on a locally compact separable complete metric
space (X, d), and let (S, T ) be the Markov-Feller pair defined by (w, p).
Proposition 4.2. If (w, p) is ELM-weakly hyperbolic, then (S, T ) is
equicontinuous.
Proof. We have to prove that for every convergent sequence (xm )m∈N
of elements of X, for every f ∈ C0 (X), and for every ε ∈ R, ε > 0, there
21
Equicontinuity and existence of attractive probability measures
279
exists mε ∈ N such that |S n f (xm ) − S n f (x)| < ε for every n ∈ N ∪ {0} and
every m ≥ mε , where x = lim xm . To this end, let (xm )m∈N be a convergent
m→∞
sequence of elements of X, let x = lim xm , and let ε ∈ R, ε > 0.
m→∞
Since we have to prove that for every f ∈ C0 (X) there exists mε ∈ N
such that |S n f (xm ) − S n f (x)| < ε for every m ≥ mε and every n ∈ N ∪ {0},
we will prove the existence of mε by assuming first that f ∈ Lip (X) ∩ C0 (X).
That is, we will prove the existence of mε in the following two cases:
(i) f ∈ Lip (X) ∩ C0 (X);
(ii) f ∈ C0 (X) (the general case).
(i) Let f ∈ Lip (X) ∩ C0 (X). Since f ∈ Lip (X), we may and do pick
L ∈ R, L > 0, such that |f (y) − f (z)| ≤ L d(y, z) for every y ∈ X and z ∈ X.
Set K = {xm | m ∈ N} ∪ {x}. Since K obviously is a compact subset of X, and
since (w, p) is ELM-weakly hyperbolic, there exists a bounded subset B of X
such that K ⊆ B and wi (B) ⊆ B for every i = 1, 2, . . . , k. Using condition
(ELM 1), we obtain that there exists nε ∈ N such that |wi1 wi2 · · · win (B)| <
ε/L for every n ∈ N, n ≥ nε , and every n elements i1 , i2 , . . . , in of the set
{1, 2, . . . , k}.
For every n ∈ N, n ≥ nε , and for every m ∈ N, using equation (2.3)
we obtain
|S n f (xm ) − S n f (x)| =
n k
pi1 pi2 · · · pin (f (wi1 wi2 · · · win xm ) − f (wi1 wi2 · · · win x)) ≤
= s=1 is =1
≤
k
n pi1 pi2 · · · pin |f (wi1 wi2 · · · win xm ) − f (wi1 wi2 · · · win x)| ≤
s=1 is =1
≤
n k
pi1 pi2 · · · pin Ld (wi1 wi2 · · · win xm , wi1 wi2 · · · win x) ≤
s=1 is =1
≤
k
n s=1 is =1
pi1 pi2 · · · pin L |wi1 wi2 · · · win (B)| <
k
n s=1 is =1
pi1 pi2 · · · pin L ·
ε
= ε.
L
Since f, Sf, S 2 f, . . . , S nε −1 f are continuous functions, there exists mε ∈
N such that |S n f (xm ) − S n f (x)| < ε for every m ≥ mε and every n =
0, 1, 2, . . . , nε − 1. Accordingly, |S n f (xm ) − S n f (x)| < ε for every m ∈ N,
m ≥ mε , and every n ∈ N ∪ {0}.
(ii) Now, let f ∈ C0 (X). Since Lip (X) ∩ C0 (X) is dense in C0 (X) (see,
for example, Corollary 1.3.4 of [29]), there exists g ∈ Lip (X)∩C0 (X) such that
280
Radu Zaharopol
22
f − g < ε/3. Using case (i), we obtain that there exists mε/3 (g) ∈ N such
that |S n g(xm ) − S n g(x)| < ε/3 for every m ≥ mε/3 (g) and every n ∈ N ∪ {0}.
Let mε = mε/3 (g). Since f − g < ε/3 and since S is a positive contraction of Cb (X), we have S n f − S n g < ε/3 for every n ∈ N ∪ {0}. Therefore,
|S n f (xm ) − S n f (x)| ≤ |S n f (xm ) − S n g(xm )| + |S n g(xm ) − S n g(x)| +
+ |S n g(x)−S n f (x)| ≤ S n f −S n g + |S n g(xm )−S n g(x)| + S n f −S n g <
ε ε ε
< + + =ε
3 3 3
for every m ∈ N, m ≥ mε and every n ∈ N ∪ {0}. Proposition 4.3. Assume that (w, p) is ELM-weakly hyperbolic. Then
the sequence (S n f )n∈N∪{0} converges pointwise whenever f ∈ C0 (X), that
is, for every f ∈ C0 (X) and every x ∈ X the sequence (of real numbers)
(S n f (x))n∈N∪{0} is convergent.
Proof. Obviously, we have to prove that (S n f (x))n∈N∪{0} is a Cauchy
sequence whenever f ∈ C0 (X) and x ∈ X.
As in the proof of Proposition 4.2, we will carry out the proof by dealing first with f ∈ Lip (X) ∩ C0 (X). That is, we will consider the following
two cases:
(i) f ∈ Lip (X) ∩ C0 (X);
(ii) f ∈ C0 (X) (the general case).
(i) Let f ∈ Lip (X) ∩ C0 (X). Since f is a Lipschitz function, we may and
do pick L ∈ R, L > 0, such that |f (x) − f (y)| ≤ Ld(x, y) for every x ∈ X and
y ∈ X.
We have to show that for every x ∈ X and ε ∈ R, ε > 0, there exists
nε ∈ N such that |S n f (x) − S m f (x)| < ε for every n ≥ nε and m ≥ nε . To this
end, let x ∈ X and let ε ∈ R, ε > 0. Since the set {x} is compact, and since
(w, p) satisfies condition (ELM 2), there exists a bounded subset B of X such
that x ∈ B and wi (B) ⊆ B for every i = 1, 2, . . . , k. Using condition (ELM 1)
we obtain that there exists nε ∈ N such that |wi1 wi2 · · · win (B)| < ε/L for
every n ≥ nε and every n elements i1 , i2 , . . . , in of the set {1, 2, . . . , k}.
Now, let m ∈ N and n ∈ N be such that nε ≤ m < n. Using equation
(2.2), we obtain
|S n f (x) − S m f (x)|
n k
pi1 pi2 · · · pim pim+1 · · · pin f wi1 wi2 · · · wim wim+1 · · · win x −
= s=1 is =1
−
k
m s=1 is =1
pi1 pi2 · · · pim f (wi1 wi2 · · · wim x) =
23
Equicontinuity and existence of attractive probability measures
281
n k
= pi1 pi2 · · · pim pim+1 · · · pin f wi1 wi2 · · · wim wim+1 · · · win x −
s=1 is =1
−
k
m 
pi1 pi2 · · · pim 
s=1 is =1
···
≤
k

pim+1  
im+1 =1
k
pin
in =1
n
k
k

pim+2  · · ·
im+2 =1
f (wi1 wi2 · · · wim x) ≤
pi1 pi2 · · · pim pim+1 · · · pin f wi1 wi2 · · · wim wim+1 · · · win x −
s=1 is =1
−f (wi1 wi2 · · · wim x) ≤
≤L
k
n pi1 pi2 · · · pim pim+1 · · · pin d wi1 wi2
s=1 is =1
· · · wim wim+1 · · · win x , wi1 wi2 · · · wim x
<
k
n since x∈B and
s=1 is =1
pi1 pi2 · · · pim pim+1 · · · pin L
ε
= ε.
L
wim wim+1 ···win x∈Bi ;
therefore,
ε
d(wi1 wi2 ···wim (wim+1 ···win x),wi1 wi2 ···wim x)< L
for every i1 ,i2 ,...,im ,im+1 ,...,in
By interchanging the roles of m and n in the above computation, we also
obtain that |S n f (x) − S m f (x)| < ε whenever m ∈ N and n ∈ N are such that
nε ≤ n < m.
(ii) Now, let f ∈ C0 (X), let x ∈ X, and let ε ∈ R, ε > 0. Since
Lip (X) ∩ C0 (X) is dense in C0 (X), there exists g ∈ Lip (X) ∩ C0 (X) such that
f − g < ε/3. Using case (i), we obtain that there exists nε ∈ N such that
|S n g(x) − S m g(x)| < ε/3 for every n ≥ nε and m ≥ nε .
Since S is a positive contraction of Cb (X), we have S l f − S l g < ε/3
for every l ∈ N ∪ {0}. Therefore,
|S n f (x)−S m f (x)| ≤ |S n f (x)−S n g(x)|+|S n g(x)−S m g(x)|+|S m g(x)−S m f (x)| <
ε
< S n f − S n g + + S m g − S m f < ε
3
for every n ≥ nε and every m ≥ nε . Thus, the sequence (S n f (x))n∈N∪{0} is a
Cauchy sequence. 282
Radu Zaharopol
24
Assume that (w, p) is an ELM-weakly hyperbolic i.f.s. Using Proposition 4.3, we obtain that for every x ∈ X the map εx : C0 (X) → R defined by
εx (f ) = lim S n f (x) for every f ∈ C0 (X) is well-defined. Clearly, the maps
n→∞
εx , x ∈ X, are positive linear functionals on C0 (X), so the εx , x ∈ X, are
also continuous. Thus, we may and do think of the εx , x ∈ X, as elements
of M(X). Note that we have already encountered the maps εx , x ∈ X, in
Section 3 (see the construction made before Lemma 3.3). Using the same arguments as in the proof of Lemma 3.3, it is easy to see that the measures εx ,
x ∈ X, are T -invariant (that is, T εx = εx for every x ∈ X).
As in the previous section, we have to prove that the measures εx , x ∈
X, are probability measures. However, for ELM-weakly hyperbolic i.f.s., the
arguments used are different. We need the following result.
Lemma 4.4. Assume that the i.f.s. (w, p) is ELM-weakly hyperbolic.
Then the sequence (wi1 wi2 · · · win x)n∈N converges for every x ∈ X and every sequence (in )n∈N of elements of {1, 2, . . . , k}.
Proof. Let x ∈ X, and let (in )n∈N be a sequence of elements of {1, 2, . . . , k}.
We have to prove that (wi1 wi2 · · · win x)n∈N is a Cauchy sequence of elements
of X. To this end, note that the set {x} is compact in X. So, taking into
consideration that (w, p) is ELM-weakly hyperbolic, we obtain that there exists a bounded subset B of X such that x ∈ B, and wi (B) ⊆ B for every
i = 1, 2, . . . , k.
Now, let ε ∈ R, ε > 0. Using again the fact that (w, p) is ELM-weakly
hyperbolic, we obtain that there exists nε ∈ N such that |wj1 wj2 · · · wjn (B)| < ε
for every n ≥ nε and every n elements j1 , j2 , . . . , jn of the set {1, 2, . . . , k}.
Let m ∈ N, m ≥ nε, and n ∈ N, n ≥ nε , and assume that n < m (if n > m,
we interchange the roles of m and n). Since both x and win+1 win+2 · · · wim x
belong to B, we have
d wi1 wi2 · · · win x, wi1 wi2 · · · win win+1 win+2 · · · wim x =
= d wi1 wi2 · · · win x, wi1 wi2 · · · win win+1 win+2 · · · wim x < ε.
Thus, (wi1 wi2 · · · win x)n∈N is a Cauchy sequence.
We now use Lemma 4.4 to prove that the measures εx , x ∈ X, are
probabilities.
Proposition 4.5. If (w, p) is an ELM-weakly hyperbolic i.f.s., then εx0
is a probability measure for every x0 ∈ X.
Proof. We will use the construction discussed in the proofs of Lemma 3.4
and Proposition 3.5.
To this end, let x0 ∈ X, let Ω be the set of all sequences (in )n∈N , in ∈
{1, 2, . . . , k} for every n ∈ N. That is, Ω = ΩN
k where Ωk = {1, 2, . . . , k}.
25
Equicontinuity and existence of attractive probability measures
283
Consider the σ-algebra P(Ωk ) on Ωk , and let F be the product σ-algebra on
Ω. Also, let P be the probability measure on (Ω, F) defined as in Section 3.
For every n ∈ N let φn : Ω → X be the function defined in the proof of
Proposition 3.5. That is, φn ((im )m∈N ) = wi1 wi2 · · · win x0 for every (im )n∈N ∈
Ω. Then, by Lemma 4.4, the sequence (φn )n∈N converges pointwise (everywhere). Let ψ be the pointwise limit of (φn )n∈N . Using a well-known result in
measure theory (see p. 58 of Doob [4]), we obtain that ψ is measurable (that
is, ψ −1 (B) ∈ F whenever B ∈ B(X)). Thus, it makes sense to define a map
ν̄ : B(X) → R by ν̄(A) = P (ψ −1 (A)) for every A ∈ B(X). It is easy to see
that ν̄ ∈ M(X), and that ν̄ is a probability measure.
Clearly, the proof of the proposition will be complete if we show that ν̄ is
actually equal to εx0 . However, the proof of the equality ν̄ = εx0 follows along
the same lines as the proof of the equality µ̄ = ε0 in Proposition 3.5. In our
case now, the sequence (f ◦ φn )n∈N converges pointwise
(everywhere) to f ◦ ψ
(X).
So,
on
one
hand
the
sequence
(
f◦φn dP )n∈N converges
whenever
f
∈
C
0
n
to f dν̄ while, on the other hand, using the fact
that f ◦ φn dP = S f (x0 )
for every n ∈ N, we obtain that the sequence ( f ◦ φn dP )n∈N also converges
to f, εx0 , f ∈ C0 (X). In order to prove the main result of this section, we need just one more
detail, which is spelled out below.
Proposition 4.6. Assume that (w, p) is an ELM-weakly hyperbolic i.f.s.
(a) For every x ∈ X, y ∈ X, and ε ∈ R, ε > 0, there exists nε,x,y ∈ N
such that d (wi1 wi2 · · · win x, wi1 wi2 · · · win y) < ε for every n ≥ nε,x,y and every
n elements i1 , i2 , . . . , in of {1, 2, . . . , k}.
(b) For every x ∈ X, y ∈ X, and every sequence (in )n∈N of elements
of {1, 2, . . . , k}, the sequences (wi1 wi2 · · · win x)n∈N and (wi1 wi2 · · · win y)n∈N
(which converge by Lemma 4.4) have the same limit.
Proof. (a) Let x ∈ X, y ∈ X, and ε ∈ R, ε > 0. Since (w, p) is ELMweakly hyperbolic, and since {x, y} is a compact subset of X, there exist a
bounded subset B of X, and a natural number nε,x,y such that {x, y} ⊆ B,
wi (B) ⊆ B for every i = 1, 2, . . . , k, and such that d(wi1 wi2 · · · win z1 , wi1 wi2
· · · win z2 ) < ε for every z1 ∈ B, z2 ∈ B, n ∈ N, n ≥ nε,x,y , and every n elements
i1 , i2 , . . . , in of {1, 2, . . . , k}. In particular, d(wi1 wi2 · · · win x, wi1 wi2 · · · win y) <
ε for every n ∈ N, n ≥ nε,x,y , and any n natural numbers i1 , i2 , . . . , in such
that ih ≤ k for every h = 1, 2, . . . , n.
(b) The proof of (b) is obvious in view of (a). We are now in a position to discuss the main result of this section.
Theorem 4.7. If (w, p) is an ELM-weakly hyperbolic i.f.s., then (w, p)
has an attractive probability.
284
Radu Zaharopol
26
Proof. We will actually offer two proofs of the theorem, one that uses
both Theorem 2.1 and Proposition 4.2, and one that uses neither Theorem 2.1,
nor Proposition 4.2. Naturally, the second proof is a bit longer.
First proof (using Theorem 2.1 and Proposition 4.2). By Proposition 4.2,
the Markov-Feller pair (S, T ) defined by (w, p) is equicontinuous. By Proposition 4.3, the measures εx , x ∈ X, are well-defined. As pointed out before
Lemma 4.4, the measures εx , x ∈ X, are T -invariant, and by Proposition 4.5
the are all probability measures. Thus, T has invariant probabilities. The pair
(S, T ) has universal elements. Indeed, if x ∈ X and (in )n∈N is a sequence of
elements of {1, 2, . . . , k}, then lim wi1 wi2 · · · win x exists by Lemma 4.4. By
n→∞
Proposition 4.6 (b), the limit is a universal element for (S, T ). Thus, we can
apply Theorem 2.1 in order to conclude that (S, T ) is uniquely ergodic.
Let µ∗ be the unique invariant probability of T . Then εx = µ∗ for every
x ∈ X and, using the Lebesgue dominated convergence theorem in the same
way as in the comments preceding Theorem 3.6, we obtain that µ∗ is actually
an attractive probability measure for (w, p).
Second proof (without using Theorem 2.1 and Proposition 4.2). By
Proposition 4.3, the maps εx , x ∈ X, defined before Lemma 4.4 are welldefined. Moreover, these maps are T -invariant measures. By Proposition 4.5,
the measures εx , x ∈ X, are probability measures. Using the comments preceding Theorem 3.6, we see that in order to prove that (w, p) has an attractive
probability measure, it is enough to show that εx = εy for every x ∈ X and
y ∈ X. Clearly, we are done if we prove that lim |S n f (x) − S n f (y)| = 0 for
n→∞
every f ∈ C0 (X), x ∈ X, and y ∈ X.
To this end, let f ∈ C0 (X), x ∈ X, y ∈ X, and ε ∈ R, ε > 0. Since f is
uniformly continuous, there exists δ ∈ R, δ > 0, such that |f (z1 ) − f (z2 )| < ε
whenever z1 ∈ X, z2 ∈ X, d(z1 , z2 ) < δ. By Proposition 4.6-(a), there exists
nδ,x,y ∈ N such that d (wi1 wi2 · · · win x, wi1 wi2 · · · win y) < ε for every n ≥ nδ,x,y
and every n elements i1 , i2 , . . . , in of {1, 2, . . . , k}. Using equation (2.2) we get
|S f (x)−S f (y)| ≤
n
n
k
n pi1 pi2 · · · pin f (wi1 wi2 · · · win x)−f (wi1 wi2 · · · win y)
s=1is =1
<ε
k
n pi1 pi2 · · · pin = ε
s=1 is =1
for every n ≥ nδ,x,y .
Since we proved that for every ε > 0 there exists nε ∈ N (where nε =
nδ,x,y ) such that |S n f (x) − S n f (y)| < ε for every n ≥ nε , we conclude that
lim |S n f (x) − S n f (y)| = 0. n→∞
27
Equicontinuity and existence of attractive probability measures
285
Note that Theorem 4.7 extends and strengthens Corollary 3.3 of Edalat
[5], and complements Theorem 5.1 of Lasota and Myjak [11].
A natural question that one may ask is whether or not the criteria for the
existence of an attractive probability measure discussed in Theorems 3.6 and
4.7 are independent (that is, whether or not one of the criteria is an extension
of the other). We will conclude the paper by showing that the two criteria are
independent. To this end, note that the i.f.s. of Example 3.8 does not satisfy
the conditions of Theorem 4.7 because the i.f.s. is not ELM-weakly hyperbolic
since d (w2n (x), w2n (y)) = d(x, y) for every x ∈ [0, 1/2], y ∈ [0, 1/2], and n ∈ N,
where w2 is the map defined in the example. On the other hand, we showed
that the i.f.s. satisfies the conditions of Theorem 3.6.
It remains to show that there exist i.f.s. that satisfy the conditions of
Theorem 4.6, but fail to satisfy the conditions of Theorem 3.6. In the next
example, we exhibit such an i.f.s. The example is a slight modification of an
example of Edalat [5].
Example 4.8. Let X = [0, 1], and let d be the usual metric on X defined
by the absolute value (d(x, y) = |x − y| for every x ∈ [0, 1] and y ∈ [0, 1]).
Let (w, p) be the i.f.s. with constant probabilities
as follows:
defined
5
w = (w1 , w2 ) where w1 : [0, 1] → [0, 1], w1 (x) = max 0, 2 x − 2 for every
x ∈ [0, 1], and w2 : [0, 1] → [0, 1], w2 (x) = max 0, 14
5 x − 2 for every x ∈ [0, 1];
p = (1/2, 1/2).
Since w1 (x) ≤ w2 (x) for every x ∈ [0, 1], since both w1 and w2 are increasing functions, and since w2n (1) = 0 for every n ≥ 3, the i.f.s. (w, p) is
ELM-weakly hyperbolic, so, clearly, (w, p) satisfies the conditions of Theorem 4.6.
However, (w, p) does not satisfy the parent-child average contractivity
condition. So, (w, p) does not satisfy the conditions of Theorem 3.6. Indeed,
let y = 1 and let z = w2 (1) = 45 . Then z is a child of y, and
1
53
> = d(y, z). p1 d(w1 y, w1 z) + p2 d(w2 y, w2 z) =
100
5
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Received 29 June 2006
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