Dynamical Mordell-Lang results via Euclidean uniformizationsjhs/JMMSF2010/Scanlon.pdfGiven an...

Dynamical Mordell-Lang results via Euclidean

uniformizations

Thomas Scanlon

University of California, Berkeley

AMS

San Francisco

15 January 2010

Thomas Scanlon University of California, Berkeley

Dynamical Mordell-Lang results via Euclidean uniformizations

Basic dynamical Mordell-Lang problem

Problem

Given an algebraic dynamical systems f : X → X over a �eld K , a

point a ∈ X (K ) and a subvariety Y ⊆ X , describe

{n ∈ N : f ◦n(a) ∈ Y (K )}



Why is this called a Mordell-Lang problem?

Let

X be an abelian variety,

f : X → X be given by translation by some element

γ ∈ X (K ), and

a be the identity element of X ,

then Of (a) := {f ◦n(a) : n ∈ N} is the monoid generated by γ.

The usual Mordell-Lang conjecture (when the characteristic of K is

zero) asserts in this case that Y (K ) ∩ Of (a) is a �nite union of

translates of submonoids of Of (a). In particular,

{n ∈ N : f ◦n(a) ∈ Y (K )} is a �nite union of points and arithmetic

progressions.



Why is this called a Mordell-Lang problem?

Let

X be an abelian variety,

f : X → X be given by translation by some element

γ ∈ X (K ), and

a be the identity element of X ,

then Of (a) := {f ◦n(a) : n ∈ N} is the monoid generated by γ.

The usual Mordell-Lang conjecture (when the characteristic of K is

zero) asserts in this case that Y (K ) ∩ Of (a) is a �nite union of

translates of submonoids of Of (a). In particular,

{n ∈ N : f ◦n(a) ∈ Y (K )} is a �nite union of points and arithmetic

progressions.

This rank one problem is a very special case of the Mordell-Lang

conjecture and was already addressed in the 1930s and 1940s by

Chaubuty, Skolem, Lech and Mahler.Thomas Scanlon University of California, Berkeley


Skolem's method

Theorem

Let X be a commutative algebraic group over a �eld K of

characteristic zero, γ ∈ X (K ), and Y ⊆ X a closed subvariety.

Then the set {n ∈ Z : [n]X (γ) ∈ Y (K )} is a �nite union of points

and arithmetic progressions.



Proof of theorem

Proof.

Choosing equations for X , γ, and Y over some �nitely

generated subring of K , we may assume that K is a p-adic �eld

and that all of the named objects have good integral models.

Using the theory of p-adic Lie groups, we can �nd a p-adic

analytic exponential map and thereby produce a p-adic

analytic function E : Zp → X (Qp) satisfying E (n) = [n]X (γ)for n ∈ N (after, possibly, replacing γ with a multiple).

The set {x ∈ Zp : E (x) ∈ Y (Qp)} is then the zero set of a

p-adic analytic function on Zp and is therefore a �nite union of

points and cosets of pnZp.



Skolem's method for general dynamical systems

Ghioca and Tucker have adapted Skolem's method to p-adic

dynamical systems with attracting �xed points at which the

di�erential is diagonalizable. More generally, variants hold

even when the dynamical systems are uniformized around

indi�erent �xed points.

In general, it is still an open problem whether every instance of

the dynamical Mordell-Lang problem for tuples of rational

functions in one variable is susceptible to this method.

Benedetto, Ghioca and Tucker have given examples of

polynomials f1, . . . , fn, p-adic number a1, . . . , an, and a p-adic

analytic function G (x1, . . . , xn) for which lim f ◦mi (ai ) exists for

each i ≤ m, but {m ∈ N : G (f ◦m1 (a1), . . . , f ◦mn (an)) = 0} isin�nite but does not contain an arithmetic progression.



A real analytic version of Skolem's method?

It would seem that Skolem's method does not apply in the

Euclidean topology for even if we had an analytic function E

satisfying E (n) = f ◦n(a) for all n ∈ N, since the integers are

discrete in R, from an analytic equation F (E (n)) = 0 for in�nitely

many n ∈ N, we could not reach a useful conclusion.



Analytic uniformization

If f : ∆→ ∆ is a complex analytic function from the disc

∆ := {x ∈ C : |x | < 1} back to itself for which f (0) = 0 but

λ := f ′(0) 6= 0 and |λ| < 1, then f is analytically conjugate

the map x 7→ λx .

More generally, if f : ∆n → ∆n is complex analytic, f (0) = 0,

and the eigenvalues of df0 are nonresonant, then f is

analytically conjugate to its linearization.

If f : ∆→ ∆ is a nonconstant complex analytic map and

f (0) = f ′(0) = 0, then f is conjugate to a power map x 7→ xN .

Of course, for a general analytic function in several complex

variables even with a �xed point there is no standard form.



Real analytic orbits

Proposition

If f is a real analytic function near the origin, a is close to the

origin, and limn→∞ f ◦n(a) = 0, then there is a function F de�nable

in Ran,exp for which F (n) = f ◦n(a) for n ∈ 2N.




Proposition




Ran,exp is the �eld of real numbers considered as a �rst-order

structure in the language of ordered �elds augmented by a function

symbol for the real exponential function and by function symbols

for the restrictions of real analytic functions to sets of the form

[−1, 1]n.




Proposition




Ran,exp is the �eld of real numbers considered as a �rst-order

structure in the language of ordered �elds augmented by a function

symbol for the real exponential function and by function symbols

for the restrictions of real analytic functions to sets of the form

[−1, 1]n.

Theorem (van den Dries,Macintyre, Marker)

Ran,exp is o-minimal



O-minimality

De�nition

A �rst-order structure (R, <, · · · ) is o-minimal if every de�nable

subset of R is a �nite union of points and intervals.



O-minimality

De�nition



Theorem

Let f1, . . . , fn be a �nite sequence of real analytic functions each

de�ned on some interval. Let a1, . . . , an be real numbers for which

limm→∞ f ◦mi (ai ) exists for each i . Then if X ⊆ Rn is a subanalytic

set, the set {m ∈ Z+ : (f ◦m1 (a1), . . . , f ◦mn (an)) ∈ X} is a �nite

union of points and arithmetic progressions.



O-minimality

De�nition



Theorem

Let f1, . . . , fn be a �nite sequence of real analytic functions each

de�ned on some interval. Let a1, . . . , an be real numbers for which

limm→∞ f ◦mi (ai ) exists for each i . Then if X ⊆ Rn is a subanalytic

set, the set {m ∈ Z+ : (f ◦m1 (a1), . . . , f ◦mn (an)) ∈ X} is a �nite

union of points and arithmetic progressions.

This holds with (f1, . . . , fn) replaced by any real analytic function

g : U → U on some open ball and a ∈ U for which a := lim g◦n(a)exists provided that there is an Ran,exp de�nable function

G : [1,∞)→ U satisfying G (n) = g◦n(a) for n ∈ Z+.Thomas Scanlon University of California, Berkeley


Cautionary remark about complex dynamics

Extending Skolem's method to complex analytic dynamics leads

very quickly to notorious issues in diophantine approximation. It

seems plausible that the the p-adic counterexamples may be

adapted to the complex analytic setting.



Cautionary remark about complex dynamics

Extending Skolem's method to complex analytic dynamics leads

very quickly to notorious issues in diophantine approximation. It

seems plausible that the the p-adic counterexamples may be

adapted to the complex analytic setting.

On the other hand, this method is not completely useless for

complex dynamics. For instance, if under the uniformization map,

the point under consideration takes on a root of unity as its angular

component, then the Mordell-Lang conclusion follows.



Higher rank dynamical Mordell-Lang problem

The di�cult cases of the usual Mordell-Lang conjecture concern

�nitely generated groups of rank greater than one. While the

dynamical Mordell-Lang problem for a single map is already

nontrivial, the higher rank case is much more complicated.








From point of view of understanding the induced structure on

dynamical orbits, the simplest higher rank problem is to understand

algebraic relations between di�erent points of single orbit.











Problem

Given an algebraic dynamical systems fi : Xi → Xi (for i ≤ n) over

a �eld K and a points ai ∈ Xi (K ) describe for Y ⊆∏n

i=1 Xi an

algebraic subvariety, describe

{(m1, . . . ,mn) ∈ Nn : (f ◦m1

1 (a1), . . . , f ◦mn

n (an)) ∈ Y (K )}











Problem

Given an algebraic dynamical systems fi : Xi → Xi (for i ≤ n) over

a �eld K and a points ai ∈ Xi (K ) describe for Y ⊆∏n

i=1 Xi an

algebraic subvariety, describe

{(m1, . . . ,mn) ∈ Nn : (f ◦m1

1 (a1), . . . , f ◦mn

n (an)) ∈ Y (K )}

Specializing further, we might want to take Xi = X , fi = f , and

ai = a to be the same for all i ≤ n.Thomas Scanlon University of California, Berkeley


Tropical analysis

Given

f (x) ∼ λx with 0 < |λ| < 1 or f (x) ∼ xN with N ≥ 2,

a close enough to zero so that limn→∞ f ◦n(a) = 0, and

F (x1, . . . , xn) =∑

Fαxα a convergent power series de�ning an

analytic hypersurface.



Tropical analysis

Given

f (x) ∼ λx with 0 < |λ| < 1 or f (x) ∼ xN with N ≥ 2,

a close enough to zero so that limn→∞ f ◦n(a) = 0, and

F (x1, . . . , xn) =∑

Fαxα a convergent power series de�ning an

analytic hypersurface.

One might hope to conclude from the equation

F (f ◦m1(a), . . . , f ◦mn(a)) = 0 that∑γimi = γ0 or∑γ1N

mi = γ0

for some integers γi .



Nonzrchimedian analytic relations

Replacing R or C by a valued �eld, one can pass from an analytic

equation to a linear relation amongst the exponents of the iteration.






In the case of f ′(0) 6= 0, the linear equation corresponds to an

analytic transformation of an algebraic torus.






In the case of f ′(0) 6= 0, the linear equation corresponds to an

analytic transformation of an algebraic torus.

In the super-attracting case of f ′(0) = 0, one has more information

from the Mann property isolated by van den Dries and Günayd�n.



Super-attracting case continued with the Mann property

De�nition

Let L be a �eld. The group Γ ≤ L× has the Mann property if for

any linear function L(x1, . . . , xn) =∑n

i=1 aixi ∈ L[x1, . . . , xn] thereare only �nitely many non-degenerate solutions to

L(γ1, . . . , γn) = 1 with γi ∈ Γ where a solution is degenerate if∑i∈I aiγi = 0 for some I ( {1, . . . , n}




De�nition





It follows that every system of (inhomogeneous) linear equations in

Γ is reducible to a system de�ned by equations of the form xi = γxjor xk = δ for γ and δ from Γ




De�nition







Returning to the dynamical problem, these valuations equations

correspond to the analytic equations f ◦`(xj) = xi or xk = f ◦`(a).




De�nition







Returning to the dynamical problem, these valuations equations

correspond to the analytic equations f ◦`(xj) = xi or xk = f ◦`(a).

It follows that all analytic relations on Of (a) are �nite unions of

relations de�ned by conjunctions of such equations.Thomas Scanlon University of California, Berkeley


Euclidean analytic relations

Even though for the Euclidean norm it is not true that∑

aαxα = 0

implies that |aαxα| = |aβxβ| 6= 0 for some α 6= β, via an

ultraproduct construction the conclusion of the nonarchimedian

theorem follows in the Euclidean topology as well.



Concluding speculation: Pila-Wilkie

Theorem (Pila-Wilkie)

Let X ⊆ Rn be a set de�nable in some o-minimal structure on R.

We de�ne X alg to be the union of all the in�nite connected

semialgebraic subsets of X . Then for every ε > 0 there is a

constant C such that #{a ∈ (X r X alg) ∩ Zn : |a| ≤ T} ≤ CT ε



Concluding speculation: Pila-Wilkie

Theorem (Pila-Wilkie)

Let X ⊆ Rn be a set de�nable in some o-minimal structure on R.

We de�ne X alg to be the union of all the in�nite connected

semialgebraic subsets of X . Then for every ε > 0 there is a

constant C such that #{a ∈ (X r X alg) ∩ Zn : |a| ≤ T} ≤ CT ε

Applying the Pila-Wilkie theorem to the o-minimal uniformizations

of algebraic dynamical systems, we would have bounds on the size

of the intersections of the monoids of the intersections of products

of orbits with algebraic varieties.



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