### relationship between Gronwall functions and the Hausdorff measurement..?

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#### hgjf2

• Fractal Friar
• Posts: 111

#### relationship between Gronwall functions and the Hausdorff measurement..?

« on: March 16, 2019, 10:09:41 AM »
relationship between Gronwall functions and the area of Julia sets the Hausdorff measurement of Julia sets?

The table of Hausdorff measurement of Julia sets is few simmilar with the table of Areas of Julia sets, but the table of Hausdorf measurements overcome the interior of Mandelbrot set associated the Julia sets.
Also the Hausdorff dimension [ln(Ne)/ln(1/r)] is more more diffcult to calculate at Julia fractals, due Julia fractals don't have exacthly self simmilarity how has Cantor dust and Syerpinsky gasket, and the dimension Hausdorff was been rendered only on PC on programming languages (C#/C++/JAVA/VBASIC) with approximative methods.
The Hausdorff dimension has formula lim{n->0}[ln(Ne)/ln(1/r)] where (Ne) is minimal number of the topological spheres with ray (r) whick covers the fractal.
The my researches continuing. I yet checking if exist a relatioship between Gronwall functions like W(z^grad(P))=P(W(z)) and the Hausdorf dimension of the Julia set based by the polynom P, if exists formula for area of Julia set about I told at a my previous topic.
For Julia set based by P=z^2+1/4 the Hausdorff dimension is 1,0812
For Julia set based by P=z^2+i the Hausdorff dimension is 1,2
For Julia set based by P=z^2-1 the Hausdorff dimension is 1,2683
For Julia set based by P=z^2-0,123+0,745*i the Hausdorff dimension is 1,3934
etc.
Also the Hausdorff dimension was helped cartographer for to measured the Coast of Norway and the length of Odessa Catacombs.

#### hgjf2

• Fractal Friar
• Posts: 111

#### Re: relationship between Gronwall functions and the Hausdorff measurement..?

« Reply #1 on: July 05, 2019, 10:15:09 AM »
Also the saddle points of graph H(a,b) = Haussdorf measurement of Julia sets based by fc(z)=z^3+k*z^2+c where c=a+bi are maybe kompassbrot kind (2 and above) coordinate locations, for each (k)
Saddle point mean point where partial derivative of H at any direction is 0 in this point when H has negative surphace.

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