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Self-similar measures for iterated function systems driven by weak contractions.

1. Introduction and main result.

Hutchinson [Hu81] showed the following result: Let N [greater than or equal to] 2. Let X be a complete metric space. Let [p.sub.1], ..., [p.sub.N] [member of] (0,1) such that [[summation].sup.N.sub.i=1] [p.sub.i] = 1. Let [f.sub.1], ..., [f.sub.N] be contractions on X. Then, there exist a unique compact set K and a unique probability measure [mu] on K such that K = [[union].sup.N.sub.i=1] [f.sub.i](K) and

[mu](A) = [N.summation over (i=1)] [p.sub.i] [mu] ([f.sup.-1.sub.i] (A))

for any Borel subset A of K.

In this paper we consider the case that [f.sub.1], ..., [f.sub.N] are weak contractions. Iterated function systems driven by weak contractions are considered in [AF04, Ha85-1, Ha85-2, L04], for example. There are several different definitions of weak contractions, here we adopt the following definition.

Definition 1.1 (Weak contractions in the sense of Browder [Br68], cf. [J97]). Let (X, d) be a metric space and f : X [right arrow] X be a map. Then, we say that f is a weak contraction in the sense of Browder if there exists an increasing right-continuous function [phi] : [0, +[infinity]) [right arrow] [0, +[infinity]) such that

[phi](t) < t, t > 0, d(f(x), f(y)) [less than or equal to] [phi](d(x,y)), x,y [member of] X.

Hata [Ha85-1,Ha85-2] extended the result of [Hu81] and showed that if each [f.sub.i] is a weak contradiction on X, then there exists a unique compact subset K of X such that K = [[union.sup.N.sub.i=1] [f.sub.i] (K). Hata's definition is different from the Browder's one, but it follows that they are equivalent.

In this paper we show that

Theorem 1.2. Let (X, d) be a complete metric space and [f.sub.1], ..., [f.sub.N] be weak contractions. Let K be the unique compact subset of X such that K = [[union].sup.N.sub.i=1] [f.sub.i](K). Let [p.sub.1], ..., [p.sub.N] [member of] (0, 1) such that [[summation].sup.N.sub.i=1] [p.sub.i] = 1. Then, there exists a unique probability measure [mu] on K such that

(1) [mu](A) = [N.summation over (i=1)] [p.sub.i] [mu] ([f.sup.-1.sub.i](A))

for any Borel subset A of K.

Barnsley [Ba05, Ba06] considered an inhomogeneous version of this result, specifically, he showed that there exists a unique Borel probability measure [mu] on a topological space X such that

[mu](A) = p[[mu].sub.0](A) + [N.summation over (i=1)] [p.sub.i] [mu]([f.sup.-1.sub.i](A)), [for all]A: Borel subset of X,

where each [f.sub.i] is a continuous transformation on X, p + [[summation].sup.N.sub.i=1] [p.sub.i] = 1, p > 0, [p.sub.i] [greater than or equal to] 0 for each i, and, [[mu].sub.0] is a probability measure on X. This framework is general, however, the assumption that p > 0 is essential.

Our second result is a collage theorem.

Theorem 1.3. Let (X, d) be a complete metric space and [f.sub.1], ..., [f.sub.N] be weak contractions. Let K be the unique compact subset of X such that K = [[union].sup.N.sub.i=1][f.sub.i](K). Let [d.sub.Haus] be the Hausdorff distance between compact subsets of X. Then, for any M > [epsilon] > 0, there exists [delta] > 0 such that if a compact subset L of X satisfies that

(2) [d.sub.Haus](L, [[union].sup.N.sub.i=1] [f.sub.i](L)) [less than or equal to] [delta], and

(3) [d.sub.Haus](K,L) [less than or equal to] M, then,

[d.sub.Haus](K, L) [less than or equal to] [epsilon].

If [f.sub.1], ..., [f.sub.N] are contractions, then, the collage theorem is shown by [BEHL86]. Since we add (3), the above result is not an extension of [BEHL86]. However, we believe that (3) is not a large constraint. If (X, d) is compact, there exists M such that (3) is satisfied for any compact subset L of X.

Finally we state a collage theorem for probability measures. Let (X, d) be a complete metric space and [f.sub.1], ..., [f.sub.N] be weak contractions. Let K be the unique compact subset of X such that K = [[union].sup.N.sub.i=1] [f.sub.i](K). Let P(K) be the set of probability measures on K. For f : K [right arrow] R, let Lip(f) be the Lipschitz constant for f. For [mu], v [member of] P(K), let

[mathematical expression not reproducible].

This is called the Monge-Kantorovich metric. (P(K),D) is a compact metric space. See [Ba06, Theorem 2.4.15 and Definition 2.4.16] for details.

Theorem 1.4. Let (X,d) be a complete metric space and [f.sub.1], ..., [f.sub.N] be weak contractions. Let K be the unique compact subset of X such that K = [[union].sup.N.sub.i=1] [f.sub.i](K) and m be the solution for (1). Let [p.sub.1], ..., [p.sub.N] [member of] (0,1) such that [[summation].sup.N.sub.i=1] [p.sub.i] = 1. Then, for any [epsilon] > 0, there exists [delta] > 0 such that if a probability measure v on K satisfies that

(4) D(v, [N.summation over (i=1)] [p.sub.i] v [omicron] [f.sup.-1.sub.i]) [less than or equal to] [delta],

then,

D(v, [mu]) [less than or equal to] [epsilon].

Before we proceed to proof, we give an example.

Example 1.5. Let X =[0,1], N = 2, [p.sub.1] = [p.sub.2] = 1/2, [f.sub.1](x) = x/(x + 1), and [f.sub.2](x) = 1/(2 - x). Then, the distribution function of the solution m of (1) is the Minkowski question-mark function [M1905]. In this particular case, it is shown in Kessebohmer-Stratmann [KeSt08] that the Hausdorff dimension for [mu] is strictly smaller than one.

2. Proofs.

Definition 2.1 (Hata's definition of weak contractions [Ha85-2, Definition 2.1]). Let (X, d) be a metric space and f : X [right arrow] X be a map. Then, we say that f is a weak contraction in the sense of Hata if for any t > 0

[mathematical expression not reproducible].

Lemma 2.2 (Cf. [J97, Theorem 1]). Let (X, d) be a metric space and f : X [right arrow] X be a map. Then, f is a weak contraction in the sense of Hata if and only if f is a weak contraction in the sense of Browder.

Proof. If f is a weak contraction in the sense of Browder, that is, [J97, Condition (a) of Theorem 1] holds, then it is obvious that f is a weak contraction in the sense of Hata. Conversely, assume that f is a weak contraction in the sense of Hata. Then, [J97, Condition (f) of Theorem 1] holds for

[mathematical expression not reproducible]

Then, by [J97, Theorem 1], f is a weak contraction in the sense of Browder.

[W91, Proposition A4.5] also discusses several conditions for Hata's definition of weak contractions.

Now we proceed to the proof of Theorem 1.2.

If f : X [right arrow] X is a weak contraction and not a contraction on a metric space X, then, Lip(g [omicron] f) = Lip(g) may occur for a function g on X, and it would be difficult to give an upper bound for

[mathematical expression not reproducible].

Therefore, it seems that the proof of [Hu81] does not work well in a direct manner. Our idea is that we first show the metric D is identical with the first Wasserstein metric on P(K) thanks to the duality theorem of Kantorovich-Rubinstein [KR58] (see also Villani's book [V09, Particular Case 5.16]), and then use several definitions for weak contractions which are equivalent to Browder's definition. Their equivalences are established by [J97, Theorem 1].

Proof. By the fixed point theorem of Browder, it suffices to show that for any t > 0,

[mathematical expression not reproducible].

Since

[mathematical expression not reproducible],

it suffices to show that for each i,

(5) [mathematical expression not reproducible].

For [mu], v [member of] P(K), let [PI]([mu], v) be the set of probability measures on X x X whose marginal distributions to the first and second coordinates are [mu] and v respectively. By the duality theorem of [KR58],

[mathematical expression not reproducible]. (7)

If [mathematical expression not reproducible],

[mathematical expression not reproducible].

Since [f.sub.i] is a weak contraction, by the condition of Krasnoselskii-Stetsenko [KrSt69], whose equivalence with Browder's definition is established by Jachymski [J97, Theorem 1 (d)], there exists a continuous function [[psi].sub.i] : [0, +[infinity]) [right arrow] [0, +[infinity]) such that [[psi].sub.i](t) > 0 if t > 0, and,

(6) d([f.sub.i](x), [f.sub.i](y)) [less than or equal to] d(x,y) - [[psi].sub.i](d(x,y)), x, y [member of] K.

We show that a contradiction occurs if we take a sufficiently small [delta] > 0.

Since K is compact, there exists M such that [sup.sub.x,y[member of]K] d(x, y) [less than or equal to] M. Take sufficiently small [epsilon] [member of] (0, 1) so that 4[epsilon]t [less than or equal to] M. Then,

[mathematical expression not reproducible].

Hence,

[mathematical expression not reproducible].

Since [[psi].sub.i] is positive and continuous,

[mathematical expression not reproducible].

Therefore,

[mathematical expression not reproducible].

Hence,

[mathematical expression not reproducible].

By taking infimum with respect to [gamma],

(7) [mathematical expression not reproducible].

By (6), [[psi].sub.i](u) [less than or equal to] u for any u [less than or equal to] M, and hence,

[mathematical expression not reproducible].

Hence,

[mathematical expression not reproducible].

Hence,

[mathematical expression not reproducible].

Thus (5) follows.

Now we show the collage theorem.

Proof of Theorem 1.3. Assume that there exist M > [epsilon] > 0 such that for any [delta] > 0 there exists a compact subset L of X satisfying (2), (3) and

(8) [d.sub.Haus] (K,L) > [epsilon].

Since K = [[union].sup.N.sub.i=1] [f.sub.i](K),

[mathematical expression not reproducible]

Hence,

(9) [mathematical expression not reproducible].

Since [f.sub.i] is a weak contraction, there exists a continuous function [[psi].sub.i] : [0, +[infinity]) [right arrow] [0, +[infinity]) such that [[psi].sub.i](t) > 0 if t > 0, and, (6) holds. It follows that

[mathematical expression not reproducible].

Since [[psi].sub.i](t) [greater than or equal to] 0 for any t [greater than or equal to] 0,

[mathematical expression not reproducible].

By (3),

[mathematical expression not reproducible].

By this and (9),

[mathematical expression not reproducible].

We remark that by the continuity and positivity for [[psi].sub.i],

[mathematical expression not reproducible].

Hence if we take

[mathematical expression not reproducible]

and an associated L, then, by (8), a contradiction occurs.

Remark 2.3. (i) We are not sure whether we can drop (3) or not. It is added because we do not know about the long-time behavior of [[psi].sub.i](t) appearing in the above proof. If [lim.sub.t[right arrow][infinity]] [[psi].sub.i](t) > 0, we can remove (3). If [f.sub.i] is contractive, then, we can take [[psi].sub.i] (t) := (1 - Lip([f.sub.i]))t.

(ii) [AF04, Proposition 4.3] considers a weak contractivity for the Barnsley-Hutchinson operator.

However, their definition of weak contractions [AF04, Definition 3.1], which is also adopted by [R01], is stronger than the one we adopt. If [AF04, Definition 3.1] is adopted, we can drop (3).

Finally we show Theorem 1.4.

Proof. The outline is the same as in the proof of Theorem 1.3, so we give a sketch only. Assume that there exists [epsilon] > 0 such that for any [delta] [member of] (0, [epsilon]) there exists a probability measure v on K satisfying (4), and

(10) D([mu], v) > [epsilon].

We have that for some i,

(11) D([mu], v) - [delta] [less than or equal to] D([mu] [omicron] [f.sup.-1.sub.i], v [omicron] [f.sup.-1.sub.i]).

Let M such that [sup.sub.x,y[member of]K] d(x, y) [less than or equal to] M. Then, we can show that by replacing [epsilon]t with D([mu], v)/4 in the proof of Theorem 1.2, and by recalling (7),

[mathematical expression not reproducible].

By this, (10) and (11),

(12) [mathematical expression not reproducible]

(13) [mathematical expression not reproducible].

Hence a contradiction occurs if

[mathematical expression not reproducible].

doi: 10.3792/pjaa.94.31

Acknowledgments. The author was supported by Grant-in-Aid for JSPS Fellows (16J04213) and also by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.

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[Br68] F. E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 27 35.

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[Hu81] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713 747.

[J97] J. R. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc. 125 (1997), no. 8, 2327 2335.

[KR58] L. V. Kantorovic and G. S. Rubinstein, On a space of completely additive functions, Vestnik Leningrad. Univ. 13 (1958), no. 7, 52 59.

[KeSt08] M. Kessebohmer and B. O. Stratmann, Fractal analysis for sets of non-differentiability of Minkowski's question mark function, J. Number Theory 128 (2008), no. 9, 2663 2686.

[KrSt69] M. A. Krasnosel'skii and V. Ja. Stecenko, On the theory of concave operator equations, Sibirsk. Mat. Z. 10 (1969), 565 572.

[L04] K. Lesniak, Infinite iterated function systems: a multivalued approach, Bull. Pol. Acad. Sci. Math. 52 (2004), no. 1, 1 8.

[M1905] H. Minkowski, Zur Geometrie der Zahlen, in Verhandlungen des dritten Internationalen Mathematiker-Kongresses (Heidelberg, 1904), 164 1 73, Druck und Verlag von B. G. Teubner, Leipzig, 1905; available at https://faculty.math.illinois.edu/ ~reznick/Minkowski.pdf.

[R01] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), no. 4, 2683 2693.

[V09] C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.

[W91] K. R. Wicks, Fractals and hyperspaces, Lecture Notes in Mathematics, 1492, Springer-Verlag, Berlin, 1991.

By Kazuki OKAMURA

School of General Education, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan

(Communicated by Masaki Kashiwara, m.j.a., March 12, 2018)

2010 Mathematics Subject Classification. Primary 28A80, 47H09.
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