| 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)∞ and
(x2, y2) is the
point with address (21)∞. Then notice |
| T2(x1, y1) has address
2(12)∞ = |
| 2(12)(12)(12)(12)... =
(21)(21)(21)(21)... = (21)∞ |
|
| 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). |