Search

We prove several results for the arithmetic dynamics of monomial maps, including Silverman’s conjectures on height growth, dynamical Mordell–Lang conjecture, and dynamical Manin–Mumford conjecture. These results were originally known for monomial maps on algebraic tori. We extend them to arbitrary toric varieties.

Let $$f:S\dashrightarrow S$$ be a rational self-map of a smooth complex projective surface $$S$$ and $$\eta $$ be a meromorphic two-form on $$S$$ satisfying $$f^*\eta = \delta \eta $$ for some $$\delta \in \mathbf {C}^*$$ . We show that under a mild topological assumption on $$f$$ , there is a birational change of domain $$\psi : X\dashrightarrow S$$ such that $$\psi ^*\eta $$ has no zeros. In this context, we investigate the notion of algebraic stability for $$f$$ , proving that $$f$$ can be made algebraically stable if and only if it acts nicely on the poles of $$\eta $$ . We illustrate this last result in the case $$\eta = \frac{dx\wedge dy}{xy}$$ , where we translate our stability result into a condition on whether a circle homeomorphism associated to $$f$$ has rational rotation number.

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$, we can find a toric model with at worst quotient singularities where $f$ is $k$-stable. If $f$ is replaced by an iterate one can find a $k$-stable model as soon as the dynamical degrees $\lambda _k$ of $f$ satisfy $\lambda _k^2>\lambda _{k-1}\lambda _{k+1}$. On the other hand, we give examples of monomial maps $f$, where this condition is not satisfied and where the degree sequences $\deg _k(f^n)$ do not satisfy any linear recurrence. It follows that such an $f$ is not $k$-stable on any toric model with at worst quotient singularities.

For a finite dimensional vector space equipped with a C \mathbb C -algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps.

For a one-dimensional local domain C M constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of C M is described using these residue maps.

Given a rational self-map f : X X on an n-dimensional Kahler manifold X, one can define a pullback map f ∗ : H(X) → H(X) for 0 ≤ p ≤ n. In general, the pullback does not commute with iteration; that is, (f ∗)k = (f k)∗. Following Sibony ([Si]; see also [FoSi]), we call the map f (algebraically) stable if the action on the cohomology of X is compatible with iterations. More precisely, f is called p-stable if the pullback on H(X) satisfies (f ∗)k = (f k)∗ for all k ∈N. If f is not p-stable on X, one might try to find a birational change of coordinate h : X ′ X such that f = h−1 f h is p-stable. This is not always possible even for p = 1, as shown by Favre [Fa]. However, for p = 1 and n = 2, one can find such a stable model (with at worst quotient singularities) for quite a few classes of surface maps [DFa; FaJ]. Also for p = 1, such a model can be obtained for certain monomial maps [Fa; JW; L]. For an n × n integer matrix A = (ai,j ), the associated monomial map fA : (C∗)n → (C∗)n is defined by fA(x1, . . . , xn) = (∏

The height function measures the arithmetic complexity of a point on a variety over [Formula: see text]. The canonical height function measures the asymptotic height growth (relative to the degree growth) of a point under a dominant rational map. One property desired for the canonical height function is the Northcott finiteness property, which states that there are only finitely many points for a bounded degree and a bounded height. We show that the canonical height function for dominant rational maps does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has the desired nice properties.

We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution., Comment: 13 pages

We show that an XML DTD (Document Type Definition) can be viewed as the fixed point of a parametric content model. Based on the parametric content model, we develop a model of modular transformation of XML documents. A fold operator is used to capture a class of functions that consume valid XML document trees in a bottom-up matter. Similarly, an unfold operator is used to generate valid XML document trees in a top-down fashion. We then show that DTD-aware XML document transformation, which consumes a document of one DTD and generates a document of another DTD, can be thought as both a fold operation and an unfold operation.This leads us to model certain DTD-aware document transformations by mappings from the source content models to the target content models. From these mappings, we derive DTD-aware XML document transformational programs. Benefits of such derived programs include automatic validation of the target documents (no invalid document will be generated) and modular property in the composition of these programs (intermediate results from successive transformations can be eliminated).