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:
“Coming to terms with the rhythms of women’s lives means coming to terms with life itself, accepting the imperatives of the body rather than the imperatives of an artificial, man-made, perhaps transcendentally beautiful civilization. Emphasis on the male work-rhythm is an emphasis on infinite possibilities; emphasis on the female rhythms is an emphasis on a defined pattern, on limitation.”
—Margaret Mead (1901–1978)
“The dominant metaphor of conceptual relativism, that of differing points of view, seems to betray an underlying paradox. Different points of view make sense, but only if there is a common co-ordinate system on which to plot them; yet the existence of a common system belies the claim of dramatic incomparability.”
—Donald Davidson (b. 1917)
“To dine, drink champagne, raise a racket and make speeches about the people’s consciousness, the people’s conscience, freedom and so forth while servants in tails are scurrying around your table, just like serfs, and out in the severe cold on the street await coachmen—this is the same as lying to the holy spirit.”
—Anton Pavlovich Chekhov (1860–1904)