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...

We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets $U_i$ which are isomorphic to closed smooth hypersurfaces in ${ℂ}^{n+1}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety $X\subset {ℂ}^m$ there is a generically-finite (even quasi-finite) polynomial mapping $f:{ℂ}^n\to {ℂ}^m$ such that $X\subset S_f$. This gives (together with [3]) a full characterization of irreducible components of the set $S_f$ for generically-finite polynomial mappings $f:{ℂ}^n\to {ℂ}^m$.

Suppose that $R\subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$, then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R\to R_1$ and $S\to S_1$ along $V$ such that $R_1\to S_1$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.

In this paper we study a notion of local volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study an invariant for normal isolated singularities, generalizing a volume defined by J. Wahl for surfaces. We also compare this generalization to a different one arising in recent work of T. de Fernex, S. Boucksom, and C. Favre.