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 C ⊂ PN 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 P ∈ PN 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:
“Long before Einstein told us that matter is energy, Machiavelli and Hobbes and other modern political philosophers defined man as a lump of matter whose most politically relevant attribute is a form of energy called “self-interestedness.” This was not a portrait of man “warts and all.” It was all wart.”
—George F. Will (b. 1941)
“When our relatives are at home, we have to think of all their good points or it would be impossible to endure them. But when they are away, we console ourselves for their absence by dwelling on their vices.”
—George Bernard Shaw (1856–1950)
“Into her lying down head
His enemies entered bed....”
—Dylan Thomas (1914–1953)