Equality of the dynamical sets of two commuting transcendental entire functions

In this paper, we study the dynamics of commuting transcendental entire functions $f$ and $g$, where $g$ is of the form $af^p + b$ with $a,b \in \C$, $p \in \N$, and $a \neq 0,1$.

We establish that the escaping sets, filled Julia sets, and bungee sets of $f$ and $g$ all coincide.

As an immediate consequence, we obtain in particular that the Julia sets of $f$ and $g$ are identical.

Our theorem extends the 1998 result of Poon and Yang.

Furthermore, following Wang and Yang, we consider a non-constant polynomial $Q$ and permutable entire functions $f$ and $g$ satisfying the relation $Q(g)=aQ(f)+b$, where $a(\neq 0,1), b \in \C$.

In this more general setting, we also prove that the escaping sets, the filled Julia sets, and the bungee sets of $f$ and $g$ are equal.