Related papers: A Local-Global Criterion for Dynamics on P^1

Let $V$ be a vector space over a finite field $k$. We give a condition on a subset $A \subset V$ that allows for a local criterion for checking when a function $f:A \to k$ is a restriction of a polynomial function of degree $<m$ on $V$. In…

Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to…

For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case.…

We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for…

For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a…

A global optimization problem is studied where the objective function $f(x)$ is a multidimensional black-box function and its gradient $f'(x)$ satisfies the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant $K$.…

Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial. In this paper, we show that a point P in P^1(Kbar) has f-canonical height zero if and only if P is…

Let F: P^N_K --> P^N_K be a morphism of degree d > 1 defined over a field K that is algebraically closed and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function G_F associated to F is Holder…

In this paper, the study of the global orbit pattern (gop) formed by all the periodic orbits of discrete dynamical systems on a finite set $X$ allows us to describe precisely the behaviour of such systems. We can predict by means of closed…

We compare different local-global principles for torsors under a reductive group G defined over a semiglobal field F. In particular if the F-group G s a retract rational F-variety, we prove that the local global principle holds for the…

We give a generalization to higher dimensions of Silverman's result on finiteness of integer points in orbits. Assuming Vojta's conjecture, we prove a sufficient condition for morphisms on P^N so that (S,D)-integral points in each orbit are…

In this paper we define and study the global Hilbert-Kunz multiplicity and the global F-signature of prime characteristic rings which are not necessarily local. Our techniques are made meaningful by extending many known theorems about…

Given a polynomial or a rational map f we associate to it a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with…

We study the rationality of the Artin-Mazur zeta function of a dynamical system defined by a polynomial self-map of A^1(k), where k is the algebraic closure of the finite field F_p. The zeta functions of the maps f(x)=x^m for (p,m)=1 and…

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$…

In [GTZ08, GTZ12], the following result was established: given polynomials $f,g\in\mathbb{C}[x]$ of degrees larger than $1$, if there exist $\alpha,\beta\in\mathbb{C}$ such that their corresponding orbits $\mathcal{O}_f(\alpha)$ and…

It was recently argued that one-dimensional systems of several strongly interacting fermions of different mass undergo critical transitions between different spatial orderings when the external confinement adiabatically changes its shape.…

For some Maltsev conditions $\Sigma$ it is enough to check if a finite algebra $\mathbf A$ satisfies $\Sigma$ locally on subsets of bounded size, in order to decide, whether $\mathbf A$ satisfies $\Sigma$ (globally). This local-global…

An n dimensional monomial dynamical system over a finite field K is a nonlinear deterministic time discrete dynamical system with the property that each of the n component functions is a monic nonzero monomial function in n variables. In…

In the first part of this paper we show that a set $E$ has locally finite $s$-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and…