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:
“There is no such thing as a language, not if a language is anything like what many philosophers and linguists have supposed. There is therefore no such thing to be learned, mastered, or born with. We must give up the idea of a clearly defined shared structure which language-users acquire and then apply to cases.”
—Donald Davidson (b. 1917)
“Wi’ joy unfeigned brothers and sisters meet,
An’ each for other’s weelfare kindly spiers:
The social hours, swift-winged, unnoticed fleet;
Each tells the uncos that he sees or hears;
The parents, partial, eye their hopeful years;
Anticipation forward points the view:”
—Robert Burns (1759–1796)
““Do not be afraid; for see -I am bringing you good news of great joy for all the people: to you is born this day in the city of David a Savior, who is the Messiah, the Lord. This will be a sign for you: you will find a child wrapped in bands of cloth and lying in a manger.””
—Bible: New Testament, Luke 2:10 -12.
Angels to the Shepherds.