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Could you please provide me with the graph of the fractal set produced by the given following IFS?

Consider an IFS $\{\phi_i, i=1,...,9\}$ on ${X}=[0, \infty) \times[0,1]$ defined as follows $$ \phi_i(x, y)=\left(\begin{array}{cc} \overline s_{i} & 0 \\\\ 0 & \underline s_{i} \end{array}\right)\left(\begin{array}{l} x \\\\ y \end{array}\right)+\left(\begin{array}{c} i-1 \\ \displaystyle\sum_{k=1}^{i-1} \underline s_{k} \end{array}\right), $$ where $$ \underline s_{i}=\frac{2^{i-1}}{3^i}\quad\text{and}\quad \overline s_{i}=\underline s_{i}+\frac{i}{10} $$ for $i=1,2, \ldots, 9$. Now, we have that $$ \underline s_{i}~d\Big((x,y),(x^*, y^*)\Big)\leq d(\phi_i(x, y),\phi_i(x^*, y^*))\leq \overline s_{i}~d\Big((x,y),(x^*, y^*)\Big). $$ I need the graphical representation of the attractor generated by the IFS $\{\phi_i, i=1,...,9\}.$

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  • $\begingroup$ What is $\lambda$? What is $\phi$? Is there a reason for using under- and overlines on the $s$ values? It's really not clear what you're asking here. Maybe an example would help? $\endgroup$
    – Chris Lewis
    Commented Jun 20, 2023 at 10:42
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    $\begingroup$ Thank you for your reply. I correct it now. $\endgroup$
    – B-S
    Commented Jun 20, 2023 at 11:15

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Sure, here is the IFS, but I am not sure you gave the correct bounds for it, as it seem to only occupy $0\leq x \leq 10$ but be much larger in the $y-$direction. I plotted it for $0\leq x \leq 10$, $0 \leq y \leq 20$, with 280.000 pts.

The Mathematica code for this is provided below. See if I interpreted your formulation correctly.

What is the motivation for studying this particular IFS?

IFS picture

st[i_]:=2.0^(i-1)/3^i;
sb[i_]:=st[i]+i/10.0;
IFS[{x_,y_}]:=With[{i=RandomInteger[{1,9}]},
{x st[i]+i-1,y sb[i]+Sum[sb[k],{k,1,i-1}]}
];
pts=NestList[IFS,{0,0},280000];

view={{0,10},{0,20}};
ListPlot[pts,
AspectRatio->1/2,
PlotRange->1.1view,
PlotStyle->{Black,PointSize[0.002]},
Axes->False]
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  • $\begingroup$ Thank you very much for your help. we study the strictly bounds of fractal dimensions of measures generated by bi-Lipschitz IFS. $\endgroup$
    – B-S
    Commented Jun 20, 2023 at 16:54
  • $\begingroup$ @G.Bilel I see, you need a figure for a paper then? $\endgroup$
    – Per Alexandersson
    Commented Jun 20, 2023 at 19:49
  • $\begingroup$ Yes, thank you very much! $\endgroup$
    – B-S
    Commented Jun 20, 2023 at 20:17
  • $\begingroup$ Please, what software do you use? $\endgroup$
    – B-S
    Commented Jun 21, 2023 at 7:14
  • $\begingroup$ I used Mathematica, with the code above. $\endgroup$
    – Per Alexandersson
    Commented Jun 21, 2023 at 10:55

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