A Large deviation and an escape rate result for special semi-flows

In this paper we consider a smooth flow $(\Lambda,\Phi^t)$ builded from suspending over a (non-invertible topologically mixing) subshift of finite type, and we equip it with an equilibrium measure $\nu$ on $\Lambda.$ The two main theorems are a large deviation and an escape rate result. The first theorem gives an explicit formula for $X>0$ and $Y$ such that $$\nu\left\{x\in\Lambda: \left|\int F\circ \Phi^s (x) ds-\int F d\mu\right|>\epsilon\right\}\leq \exp(-Xt+\log t+Y)$$ for $t\gg>1\gg\epsilon>0,$ where $F:\Lambda\to\mathbb{R}$ is smooth. The second theorem gives an explicit lower bound for the asymptotic behaviour of the escape rate of $\nu$ through a small hole.