Let (C ) denote Cantor 's discontinuum which is obtained if we remove recursively the open middle third of any remaining interval 0,1 leadsto left 0, frac 1 3 ight cup left frac 2 3 , 1 ight leadsto left 0, frac 1 9 ight cup left frac 2 9 , frac 1 3 ight cup left frac 2 3 , frac 7 9 ight cup left f...

To show that C f1C cup f2C where f1x frac13x and f2x frac13x frac23 we need to show two things 1 C s ...

Let (C) denote Cantor's discontinuum which is obtained if we remove recursively the open middle third of

Question:

Let \(C\) denote Cantor's discontinuum which is obtained if we remove recursively the open middle third of any remaining interval:

\[[0,1] \leadsto\left[0, \frac{1}{3}\right] \cup\left[\frac{2}{3}, 1\right] \leadsto\left[0, \frac{1}{9}\right] \cup\left[\frac{2}{9}, \frac{1}{3}\right] \cup\left[\frac{2}{3}, \frac{7}{9}\right] \cup\left[\frac{8}{9}, 1\right] \leadsto \ldots\]

Show that \(C=f_{1}(C) \cup f_{2}(C)\) with \(f_{1}(x)=\frac{1}{3} x\) and \(f_{2}(x)=\frac{1}{3} x+\frac{2}{3}\) and \(\operatorname{dim} C=\log 2 / \log 3\).

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