Cyclic Driven IFS

Constant Cycles and Fixed Points

The sequence of points generated by applying T₁ repeatedly converges to the point with address 1111... = 1^infinity.

We show this is the fixed point of T₁, and find its coordinates.

Fixed point. Say (x*, y*) is the point with address 1^infinity. Then

T₁(x*, y*) has address 1(1^infinity) = 1^infinity.

Because T₁(x*, y*) and (x*, y*) have the same (infinite) address, they must be the same point. That is,

T₁(x*, y*) = (x*, y*)

and (x*, y*) is the fixed point of T₁.

Coordinates. We see

(x*, y*) = T₁(x*, y*) = (x*/2, y*/2),

and so (x*, y*) = (0, 0).

Similar arguments show 2^infinity, 3^infinity, and 4^infinity are the fixed points of T₂, T₃, and T₄, respectively.

These points have coordinates (1, 0), (0, 1), and (1, 1), respectively. For example,

(x*, y*) = T₂(x*, y*) = ((x* + 1)/2, y*/2).

So x* = x*/2 + 1/2 and y* = y*/2, so x* = 1 and y* = 0.

Return to Cyclic IFS.