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F. self-similar structure. It gives a sufficient condition for K\VQ to be connected. 9. Let (K,S,{Fi}ies) nite self-similar structure. Assume that, for any p,q G Vb, there exists a homeomorphism g : K —* K such that g(Vo) = Vb and g(p) = q. Then K\Vo is connected. If a connected p. c. f. self-similar structure satisfies the assumption of the above proposition, we say that the self-similar structure is weakly symmetric. To prove the above proposition, we need the following lemmas. 10. 9. Let C be a connected component of K\VQ.

Assume that v(K) < oo and v(I) = 0. // there exists c > 0 such that μ^yj) < cv(Kw) for any w G W*, then μ(A) < cis(A) for any A G M^μ) n M{y). In particular, //(I) = 0. Proof. Let U be an open subset of K. Set W(U) = {w G W* : Kw C U}. For w,v G W(U), we define w > v if and only if Hw D Ev. Then > is a partial order on W(U). If W~*~(U) is the collection of maximal elements in 28 Geometry of Self-Similar Sets W(U) with respect to this order, then U = Uwew+(U)Kw and KWC\KV c X for w ^ v e W+(U). 2, for any A G A4(/x) D M(v), there exists a decreasing sequence of open sets {Ok}k>1 such that A C Ok for any k, μ(^k>lOk) = μ{A) and u(nk>iOk) = v(A).

See, for example, [124] and [158]. 2. Let (X, d) be a metric space and let μ be a Borel regular measure on (X,M). Assume that μ{X) < oo. 3 (Bernoulli measure). Let S be a finite set. M P ), where E = 5 N , p or an w that satisfies μ {Tlw) = pWlpW2 .. -Pwm f V — ^1^2 • • • w m G W*. p This measure μ is called the Bernoulli measure on E with weight p. Remark. In this book, all the measures we will encounter are supposed to be complete unless otherwise stated. Also the Bernoulli measure with weight p is characterized as the unique Borel regular probability measure on E that satisfies μ(A) = Y^piμ(ar\A)) ies for any Borel set A C E.