Linear fractional recurrences are given as $z_{n+k}=A\left(z\right)/B\left(z\right)$, where $A\left(z\right)$ and $B\left(z\right)$ are linear functions of $z_n,z_{n+1},\cdots ,z_{n+k-1}$. In this article we consider two questions about these recurrences: (1) Find $A\left(z\right)$ and $B\left(z\right)$ such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each $k$ there are $k$-step linear fractional recurrences...