By M.A. Armstrong

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9) holds. 1. We now proceed with the proof of the second statement. It follows directly from the first statement that Cape (uzi) ::; Cape (uzi) = i=l i=l = m~ {CapeZi} l:$•:$n ~~~n {capezi}::; Cape (Qzi). 1. This completes the proof of the theorem. • Let X and X' be sets endowed with C-structures r = (:F,e,TJ,'l/J) and r' = (:F' I e' I TJ1 I cp') satisfying Conditions Al, A2, and A3'. The following statement shows that the lower and upper Caratheodory capacities are invariant under a bijective map x: X -+ X' which preserves the C-structures.

One well-known class of sets for which the coincidence usually takes place IS the class of limit sets for some geometric constructions (see Chapter 5). For subsets which are invariant under a dynanucal system one can pose another problem of the coincidence of the Hausdorff dimension and box dimension of invariant measures. In order to explain this let us consider a map f: U -+ Rm, where U c IR"' is an open domain. '(Z) = 1). The stochastic properties of the map /IZ are closely related to the topolog~cal structure of the set Z that, in many uphysically" interesting situations, resembles a Cantor-like set.

On X, denote by AI' the set of points for which Condition A5 holds. ,a(x) ~ dc,p(x). Denote also by M(Z) the set of Borel measures p. ). 2. (x). pEM(Z) zEA~ {3. Proof. It follows from the definition of (3 that for any e > 0 one can find a measure p. -almost every x E Z. 3 implies now that dime Z ~dime p. ~ {3- e and the desired result follows. • We still assume that {(U) = 1 for any U E F. The previous results give rise to the following notion. Given a ~ 0, define Da = {x E Ap: dc,p(x) =a}. The function jp(a) = dime Da is called the Caratheodory dimension spectrum specified by the measure p..