Arithmetic Dynamics - Dynamically Defined Points Lying On Subvarieties

Dynamically Defined Points Lying On Subvarieties

There are general conjectures due to Shouwu Zhang and others concerning subvarieties that contain infinitely many periodic points or that intersect an orbit in infinitely many points. These are dynamical analogues of, respectively, the Manin–Mumford conjecture, proven by Raynaud, and the Mordell–Lang conjecture, proven by Faltings. The following conjectures illustrate the general theory in the case that the subvariety is a curve.

Conjecture Let F : PN → PN be a morphism and let CPN be an irreducible algebraic curve. Suppose that either of the following is true:
(a) C contains infinitely many points that are periodic points of F.
(b) There is a point PPN such that C contains infinitely many points in the orbit OF( P).
Then C is periodic for F in the sense that there is some iterate F(k) of F that maps C to itself.

Read more about this topic:  Arithmetic Dynamics

Famous quotes containing the words defined, points and/or lying:

    An alcoholic has been lightly defined as a man who drinks more than his own doctor.
    Alvan L. Barach (1895–1977)

    We only part to meet again.
    Change, as ye list, ye winds: my heart shall be
    The faithful compass that still points to thee.
    John Gay (1685–1732)

    It is impossible to calculate the moral mischief, if I may so express it, that mental lying has produced in society. When a man has so far corrupted and prostituted the chastity of his mind as to subscribe his professional belief to things he does not believe he has prepared himself for the commission of every other crime.
    Thomas Paine (1737–1809)