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

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?