This seems like a really basic question, and yet I haven't managed to find the answer!

Let $(X,\Sigma,\mu,T)$ be a measure-preserving dynamical system. Does there necessarily exist at least one sequence $(x_n)_{n \in \mathbb{N}}$ of points in $X$ such that $T(x_{n+1})=x_n$ for all $n \in \mathbb{N}$?

If not, what about in the particular case that $(X,\Sigma)$ is a standard measurable space?