# Is “indeterminate” a better name than “indifferent” for neutral fixed-points?

let $f(t)=t$ be an analytic function which has a fixed-point at $t$. The multiplier $\lambda (t) = f'(t)$ is just the 1st derivative evaluated at the fixed-point. The standard nomenclature is that when $\| \lambda (t) \|=1$ the fixed-point is said to be indifferent, or neutral. However, it does not necessarily mean that trajectories can't be attracted or repelled by this point, only that its 1st derivative equals 1.

Wouldn't "indeterminate" be a better name than "indifferent" ?

The function $(1 / ( 1 - \| \lambda (t) \| ) )$ is ill-defined when $\lambda(t) = 1$

On the other hand, it also common to use "parabolic" (and in my opinion much better) in opposition to "hyperbolic". The latter (usually) means that $|f'(t)|\ne1$.
• I agree with your interpretation of indeterminate vs indifferent. I think that parabolic is typically reserved for the case where $f'(t)$ is a root of unity, though, which is a stronger assumption that $|f'(t)|=1$. As such, parabolic points are definitely not indeterminate, as the Leau-Fatou flower theorem describes their dynamics quite precisely. – Mark McClure Mar 4 '17 at 23:32