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:
“The depth and strength of a human character are defined by its moral reserves. People reveal themselves completely only when they are thrown out of the customary conditions of their life, for only then do they have to fall back on their reserves.”
—Leon Trotsky (1879–1940)
“If I were in the unenviable position of having to study my work my points of departure would be the “Naught is more real ...” and the “Ubi nihil vales ...” both already in Murphy and neither very rational.”
—Samuel Beckett (1906–1989)
“Every modern male has, lying at the bottom of his psyche, a large, primitive being covered with hair down to his feet. Making contact with this Wild Man is the step the Eighties male or the Nineties male has yet to take. That bucketing-out process has yet to begin in our contemporary culture.”
—Robert Bly (b. 1926)