Cyclic Driven IFS

2-Cycle Addresses

Coordinates The limiting points in the last example have coordinates (1/3, 0) and (2/3, 0).

To see this, say (x₁, y₁) is the point with address (12)^infinity and (x₂, y₂) is the point with address (21)^infinity. Then notice

T2(x1, y1) has address 2(12)infinity =

2(12)(12)(12)(12)... = (21)(21)(21)(21)... = (21)infinity

Because T₂(x₁, y₁) and (x₂, y₂) have the same (infinite) address,

T₂(x₁, y₁) = (x₂, y₂)

By a similar argument,

T₁(x₂, y₂) = (x₁, y₁).

combining these two, we see

T1T2(x1, y1) = (x1, y1)

and

T2T1(x2, y2) = (x2, y2).

From the first we obtain

(x₁, y₁) = T₁T₂(x₁, y₁) = T₁(x₁/2 + 1/2, y₁/2) = (x₁/4 + 1/4, y₁/4),

so

x₁ = x₁/4 + 1/4 and y₁ = y₁/4

Solving for x₁ and y₁, we obtain (x₁, y₁) = (1/3, 0). A similar argument gives (x₂, y₂) = (2/3, 0).

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