Some modular functions, such as the Riemann zeta one, display a fractal behaviour.
Does also the curve of the j-function display power law properties, or a scale free structure?
It it feasible to calculate the j-function's power law slope, or its Lyapunov exponent? Is the j-function somehow correlated with the Feigenbaum constant of logistic plots?
Dear Arturo,
The Klein's J- invariant is a modular function.
It transforms some regions into beautiful geometric shapes but not necessarily fractals. It reflects some regions ( different fundamental regions of the complex plane) into same graphs or ( homeomorphic images) due to periodicity (and modularity).
On the other side, a fractal is an abstract object that has a self - similar pattern.i.e., (Topologically) the image of a local neighborhood of the (fractal) at some point has the same image as the whole fractal itself. Both J-invariant and Fractals generate amazing images, but with different nature.
Regards
Dear Issam,
therefore you think that the j-function is not scale-free. Thanks a lot for your comment.
@ Arturo,
Hope I get your point " is not scale-free."
Topologically, the transformations of the domains are scale-free, but can't consider as fractals.
The transformations of different fundamental domains are scale-free, but neighborhoods never generate the original image.
As an example, see the attached file.
Best regards
Despite the previous answers, I have plotted the modular discriminant (which is a modular form), and the result seems to be a self-similar fractal.
@ Fabio,
It seems a very nice example. Did you mean the discriminant of the (J-function)? In general, some of the modular forms are Fractals.
But not all of them.
Best regards
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