Composition Operators on the Little Lipschitz space of a rooted tree

In this work, we study the composition operators on the little Lipschitz space ${\mathcal L}_0$ of a rooted tree $T$, defined as the subspace of the Lipschitz space ${\mathcal L}$ consisting of the complex-valued functions $f$ on $T$ such that $$\lim_{|v|\to\infty}|f(v)-f(v^-)|=0,$$ where $v^-$ is the vertex adjacent to the vertex $v$ in the path from the root to $v$ and $|v|$ denotes the number of edges from the root to $v$. Specifically, we give a complete characterization of the self-maps $\phi$ of $T$ for which the composition operator $C_\phi$ is bounded and we estimate its operator norm. In addition, we study the spectrum of $C_\phi$ and the hypercyclicity of the operators $\lambda C_\phi$ for $\lambda \in {\mathbb C}$.