| The sequence of points generated by applying T1 repeatedly converges
to the point with address 1111... = 1∞. |
| We show this is the fixed point of T1, and find its coordinates. |
| Fixed point. Say (x*, y*) is the point
with address 1∞. Then |
| T1(x*, y*) has address
1(1∞) = 1∞. |
| 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∞, 3∞, and 4∞
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. |