Cantor function: Difference between revisions

Revision as of 01:39, 21 September 2020

Revision as of 12:35, 25 September 2020

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:<math>\gamma_D\circ c= c\circ \gamma_C</math>

The dyadic monoid itself has several interesting properties. It can be viewed as a finite number of left-right moves down an infinite [[binary tree]]; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set. In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additonaladditional examples can be found in the article on [[de Rham curve]]s. Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the [[modular group]] <math>SL(2,\mathbb{Z}).</math>

Note that the Cantor function bears more than a passing resemblance to [[Minkowski's question-mark function]]. In particular, it obeys the exact same symmetry relations, although in an altered form.