Circle Inversion Fractals

Inversion Limit Sets

Convergence to the Limit Set

Suppose the circles C₁, ... , C_N bound discs D₁, ... , D_N having disjoint interiors.

Starting with a point z₀ outside the discs, we pick a circle C_i1 at random and invert z₀, obtaining a new point

z₁ = I_i1(z₀).

Then we pick another circle C_i2 at random, and invert z₁, obtaining a new point

z₂ = I_i2(z₁) = I_i2(I_i1(z₀)).

Continue in this way,

z_k = I_ik(z_k-1) = I_ik(I_ik-1(...I_i1(z₀)...)).

with the restriction that no i_j equals i_j+1.

Sketch

The limit set as the nested intersection, recalling the Cantor set construction, and the standard deterministic IFS for the gasket.

Because the inverting circles are disjoint, the radii converge to 0.

A standard adderss argument shows the limit set of any orbit is identical with the intersection of this nested sequence.

Return to circle inversion fractals.