What is a Parabolic Fixed Point?

Question

I know the definitions of hyperbolic and elliptic fixed point (or equilibrium). However, when I google I find references to 'parabolic elliptic points' but not a proper definition.

Answer 1

Especially for Möbius transformations, you have three types of transformations:

• hyperbolic: one fixed point is stable, one is unstable.
• parabolic: there is one fixed point which is not stable nor unstable.
• elliptic: the fixed point is indifferent.

In general, parabolic fixed points are such that all trajectories converge to the fixed point, but points in a neighborhood eventually quit the neighborhood.

For example, the map $f(z) = \frac{(2+i)x + 1}{x+(2-i)}$ fixes the unit complex circle. It has a fixed point $i$, and it moves all points but $i$ counter-clockwise. Therefore any point of the unit circle is eventually moved close to $i$, but points that start close to $i$ get far away before getting close. That's why it's not a stable fixed point.

Answer 2

In case of discrete dynamical system fixed point is rationally indifferent or parabolic if it's multiplier of map at fixed point is a root of unity, see :

• Milnor's book "Dynamic in one complex variable"
• wiki page

HTH