Talk:Minkowski's question-mark function: Difference between revisions

Revision as of 03:52, 29 March 2006

Revision as of 02:31, 30 March 2006

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The fractal and self-similar nature of the function is unclear. Exactly how does the modular group describe the self-similarity? [[User:AxelBoldt|AxelBoldt]] 03:52, 29 March 2006 (UTC)

:Yes, its a sloppy statement; it is only the "period-doubling monoid" inside SL(2,Z) (and not the modular group PSL(2,Z)) that applies to period-doubling fractals. (Different people seem to call this [[monoid]] different names). Consider the operators R and S:

::<math>[S?](x) = ?(x/1+x) = ?(x)/2</math>

:Note that the above is a self-symmetry for x in [0,1]: the question mark on the interval [0,1/2] is a half-size of the whole thing. Let R be a reflection:

::<math>[R?](x) = ?(1-x) = 1-?(x)</math>

:Then R and S generate the monoid: that is, any string of the form

::<math>S^m R S^n R S^p ...</math>

:for positive integers m,n,p ... is a self-symmetry of the question mark. The requirement that m,n,p be poistive is what makes it a monoid,nt a group. [[User:Linas|linas]] 02:31, 30 March 2006 (UTC)