Good reduction of periodic points on projective varieties

Good Reduction of Periodic Points on Projective Varieties

We consider dynamical systems arising from iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. Our goal is to obtain information about the dynamical system over the field of fractions by studying the dynamical system over the residue field. In particular, we aim to bound the possible primitive periods for a periodic point. This is discu...

Periodic Points for Good Reduction Maps on Curves *

The periodic points of a morphism of good reduction for a smooth projective curve with good reduction over Q p form a discrete set. This is used to give an interpretation of the morphic height in terms of asymptotic properties of periodic points, and a morphic analogue of Jensen's formula.

Existence of Rational Points on Smooth Projective Varieties

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of ...

Good reduction of abelian varieties

On recurrence equations associated with invariant varieties of periodic points

A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. Some of them had been known for some years, while some others have been found recently. In this contribution we would like to show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic points of higher dimensional integra...