WOLFRAM|DEMONSTRATIONS PROJECT

A Rational Function that Iterates to the Identity

​
x
1
curve parameter a
2
zoom
6
-6
-4
-2
2
4
6
-6
-4
-2
2
4
6

The function
f
a
(x)=
ax-(1+a+a
2
)
x-(1+a)
has the property that
f(f(f(x)))=x
for all
x
in the extended reals (i.e.,
x
can be infinite as well as finite). In the graph,
x
can be taken to be any one of the three points on the
x
axis. The sides of the polygon trace out the steps of the triple composition.