Hyperbolic IFSs, attractors and fractal tops in .NET Development barcode code39 in .NET Hyperbolic IFSs, attractors and fractal tops

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Hyperbolic IFSs, attractors and fractal tops generate, create code 39 none with .net projects Console application Figure 4.39 The to Visual Studio .NET bar code 39 ps dynamical system and the branches of its inverse, which de ne a restricted IFS.

. The idea of a dire cted IFS is new to fractal geometry. But a more general structure, of iterated set maps, has been investigated in [69] and is quite closely related..

4.17 The top of a directed IFS The following will tell us that most of the theory of transformations between fractal tops goes through for directed IFSs. Hence lies our success in colour-stealing, using directed IFSs in place of hyperbolic IFS, as illustrated in Figure 4.37.

The main difference, in general, is that there is no tops dynamical system. We present the main ideas somewhat concisely, and in such a way that they can be used in 5 in connection with superfractals. These key ideas are the same as in Section 4.

14, and the proofs, which we omit, are entirely analogous. I want to stress here that although these ideas are very simple, they may look complicated because they require quite a few symbols for their expression. Let F = {X; f 1 , f 2 , .

. . , f N } be a hyperbolic IFS.

Let AF denote its attractor and let F : {1,2,...

,N } AF denote the associated addressing function. Let {1,2,..

.,N } be closed and de ne A(F,. = F ( ).. When is shift inva riant, (F, ) is of course a directed IFS and A(F, ) is a deterministic fractal, but we want to discuss sets of the form of A(F, ) quite generally. Indeed, in 5 we will represent certain V -variable and random fractal sets in just this way. Now let (F, ) : A(F, ) be de ned by (F, ) (x) = max{ : F ( ) = x} for all x A(F, ) .

. 4.17 The top of a 3 of 9 for .NET directed IFS Notice that ( F (F, ) )(x) = x We de ne the restricted tops code space.

(F, ). for all x A(F, ) .. (F, ) ). to be the set := (F, (F,. A(F,. = (F, ) ( F ( )) A(F,. (F, ) .. and we de ne (F, ). to be the restrict ion of F to the closure of (F, ) continuously onto A(F, ) .. The latter maps the closure of is 1 D e f i n i t i o n 4.17.1 The set of sets Q (F, ) := (F, ) (x) : x A(F, called a restricted code structure of the set A(F, ) .

. When (F, ) is a di .net framework Code 39 Full ASCII rected IFS we call Q (F, ) the restricted code structure of the directed IFS (F, ). We remark that clearly on the one hand not all sets possess restricted code structures and on the other hand a set may possess many restricted code structures.

Moreover, we may consider projective restricted code structures and M bius restricted code structures , with obvious meanings; then we discover o that projective restricted code structure is a property of projective geometry, and so on, along the lines discussed at the end of 3. D e f i n i t i o n 4.17.

2 We say that two restricted code structures Q (F, ) and Q (G, ) are homeomorphic iff there is a homeomorphism : (F, ) (G, ) that respects the code structures, that is, such that q Q (F, ) (q) Q (G, ) . T h e o r e m 4.17.

3 Let the two restricted code structures Q (F, ) and Q (G, ) be homeomorphic. Then A(F, ) and A(G, ) are homeomorphic. That is, there exists a homeomorphism H : A(F, ) B(G, ) such that H A(F, If the homeomorphism : (G, ) then.

(F, ) ). = A(G, ) .. (G, ). has the property that (. (F, ) ). (G,. H = (F, ) .. P r o o f This fol barcode 39 for .NET lows similar lines to the discussion in Section 4.14 and is therefore omitted.

Analogous results to those concerning fractal transformations apply to directed IFSs and, as we will see in 5, in connection with superfractals. This extends our ability to construct homeomorphisms between pictures and between objects. One immediate application is to the construction of synthetic imagery.

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