Let (C) denote Cantor's discontinuum which is obtained if we remove recursively the open middle third of
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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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Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher
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