Now let us take it a bit further for a general real base .

Proposition. The only real base for which the complex function has a fixed point with is and in this case the only such fixed point is .

Simalarly to the previous post we assume a fixed point being given by

then for parabolicity we have to prove

and we have the equation system:

(1) and (2)

Both equation squared and added yields

and hence

For the case : and and so

.

Now we put this into equation (1) with :

We substitute ,

. But the functions on both sides are well known. The right side is a circle with radius 1 and the left side is always above or below the right side in the region . So equality happens exactly at .

This implies however only satisfies equation (1). Then is the only real base for which has a parabolic fixed point and this fixed point is given by and and is hence the real number .

Proposition. The only real base for which the complex function has a fixed point with is and in this case the only such fixed point is .

Simalarly to the previous post we assume a fixed point being given by

then for parabolicity we have to prove

and we have the equation system:

(1) and (2)

Both equation squared and added yields

and hence

For the case : and and so

.

Now we put this into equation (1) with :

We substitute ,

. But the functions on both sides are well known. The right side is a circle with radius 1 and the left side is always above or below the right side in the region . So equality happens exactly at .

This implies however only satisfies equation (1). Then is the only real base for which has a parabolic fixed point and this fixed point is given by and and is hence the real number .