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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 )}
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
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
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
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
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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 )}
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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 )}
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
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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 .
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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 super-attracting case of f ′(0) = 0, one has more information
from the Mann property isolated by van den Dries and Günayd�n.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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}
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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}
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
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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}
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
Returning to the dynamical problem, these valuations equations
correspond to the analytic equations f ◦`(xj) = xi or xk = f ◦`(a).
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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}
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
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
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations
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.
Thomas Scanlon University of California, Berkeley
Dynamical Mordell-Lang results via Euclidean uniformizations