Universal properties of the isotropic Laplace operator on homogeneous trees

Let $P$ be the isotropic nearest neighbor transition operator on a homogeneous tree. We consider the $\lambda$-eigenfunctions of $P$ for $\lambda$ outside its $\ell^2$ spectrum, i.e., the eigenfunctions with eigenvalue $\gamma=\lambda - 1$ of the Laplace operator $Delta=P- \mathbb I$, and also the $\lambda-$polyharmonic functions, that is, the union of the kernels of $(Delta-\gamma \mathbb I)^n$ for $n\geqslant 0$. We prove that, on a suitable Banach space generated by the $\lambda-$polyharmonic functions, the operator $e^{Delta-\gamma \mathbb I}$ is hypercyclic, although $Delta-\gamma \mathbb I$ is not.