A Rational Function that Iterates to the Identity

x

1

curve parameter a

2

zoom

6

The function

f

a

(x)=

ax-(1+a+

2

a

)

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.