If $T:[0,1] \rightarrow [0,1]$ preserves Lebesgue, then $\liminf_n(n|T^n(x)-x|) \leq 1$

Question

Let $T:[0,1] \rightarrow [0,1]$ be a measurable function such that $T$ preserves Lebesgue, then for almost all point:

$$\liminf_n(n|T^n(x)-x|) \leq 1$$