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:
“Now defined as art, the totem has lost cult, taboo, and custom.”
—Mason Cooley (b. 1927)
“In writing biography, fact and fiction shouldn’t be mixed. And if they are, the fictional points should be printed in red ink, the facts printed in black ink.”
—Catherine Drinker Bowen (1897–1973)
“Into her lying down head
His enemies entered bed....”
—Dylan Thomas (1914–1953)