Invariance principle for local time by quasi-compactness

The objective of this paper is to prove a functional weak invariance principle for a local time of a process of the form $X_{n}=\varphi\circ T^{n}$ where $\left(X,\mathcal{B},T,m\right)$ is a measure preserving system with a transfer operator acting quasi-compactly on a large enough Banach space of functions and $\varphi\in L^{2}\left(m\right)$ is an aperiodic observable.