Weak invariance principle for the local times of Gibbs-Markov processes

The subject of this paper is to prove a functional weak invariance principle for the local time of a process generated by a Gibbs-Markov map. More precisely, let $\left(X,\mathcal{B},m,T,\alpha\right)$ is a mixing, probability preserving Gibbs-Markov{\normalsize{}. and let $\varphi\in L^{2}\left(m\right)$ be an aperiodic function with mean $0$. Set $S_{n}=\sum_{k=0}^{n}X_{k}$ and define the hitting time process $L_{n}\left(x\right)$ be the number of times $S_{k}$ hits $x\in\mathbb {Z}$ up to step $n.$ The normalized local time process $l_{n}\left(x\right)$ is defined by $l_{n}\left(t\right)=\frac{L_{n}\left(\left\lfloor \sqrt{n}x\right\rfloor \right)}{\sqrt{n}},\,\, x\in\mathbb{R}$. We prove under that $l_{n}\left(x\right)$ converges in distribution to the local time of the Brownian Motion. The proof also applies to the more classical setting of local times derived from a subshift of finite type endowed with a Gibbs measure.