Fix $N>1$.  Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R})  &\rightarrow C^{\infty}(\mathbb{R},\mathbb{R})
\\
g & \mapsto g \circ f,
\end{aligned}
$$
is a bounded operator.  Here the topology on $C^{\infty}(\mathbb{R},\mathbb{R})$ is the [Whitney topology][1].

Let $C^f$ denote the adjoint operator $C_f$.  Is there a criterion on $f$ to verify when $C^f$ has at most $N$ linearly independent eigenvectors?


  [1]: https://ncatlab.org/nlab/show/C-infinity+topology