Algebraic dynamics - Indian Statistical Institutedst.model/slides/scanlon.pdf · Discrete dynamical systems A dynamical system is a pair (X ;f ) consisting of a set X and a self-map

Algebraic dynamics

Thomas Scanlon

UC Berkeley

2 January 2013

Thomas Scanlon (UC Berkeley) Algebraic dynamics 2 January 2013 1 / 24

Some acknowledgments

Some the work I will describe today, I have done on my own, but mostly, itis joint with such people as R. Benedetto, D. Ghioca, B. Hutz, P. Kurlberg,C. G. Lee, A. Medvedev, T. Tucker, and Y. Yasufuku supported by an NSFFocused Research Group grant on algebraic dynamics.


Discrete dynamical systems

A dynamical system is a pair (X , f ) consisting of a set X and a self-mapf : X → X .

The dynamics of this system come from the associated action

N× X → X

(n, a) 7→ f ◦n(a)


Discrete dynamical systems

A dynamical system is a pair (X , f ) consisting of a set X and a self-mapf : X → X .

The dynamics of this system come from the associated action

N× X → X

(n, a) 7→ f ◦n(a)


Some sets associated to a dynamical system

For a dynamical system (X , f ) we de�ne the following associated sets.

The orbit of a point: For a ∈ X we write Of (a) := {f ◦n(a) : n ∈ N}.Return sets: For a ∈ X and Y ⊆ X

E (a, f ,Y ) := {n ∈ N : f ◦n(a) ∈ Y }

(Pre-)periodic points:

Per(f ) := {a ∈ X : f ◦n(a) = a for some n ∈ Z+}

PrePer(f ) := {a ∈ X : Of (a) is �nite }





E (a, f ,Y ) := {n ∈ N : f ◦n(a) ∈ Y }








E (a, f ,Y ) := {n ∈ N : f ◦n(a) ∈ Y }








E (a, f ,Y ) := {n ∈ N : f ◦n(a) ∈ Y }





Algebraic dynamics

By an algebraic dynamical system we mean a dynamical system (X , f ) forwhich X is an algebraic variety and f : X → X is a (possibly just rational)map of varieties.

For purposes of this lecture, it su�ces to think of such a dynamicalsystem as g : Kn → Kn where K is some �eld and g is given by ann-tuple of polynomials.

Generalizing somewhat, it may have been better to de�ne a dynamicalsystem in a category to be an arrow f : X → X whose source andtarget are the same object. An algebraic dynamical systems is then adynamical system in the category of algebraic varieties.


Algebraic dynamics





Algebraic dynamics





Special points interpreted dynamically

If (X , f ) is an algebraic variety de�ned over a �eld K and a ∈ X (K ) isa K -rational point, then Of (a) ⊆ X (K ) consists entirely of K -rationalpoints.

If X is a semi-abelian variety (for example, if X (C) = (C×)g is someCartesian power of the multiplicative group), and f : X → X is givenby multiplication by n > 1 (in the sense of the group;(x1, . . . , xg ) 7→ (xn

1 , . . . , xng ) in the example), then PrePer(f ) is the set

of torsion points on X and Per(f ) is the set of torsion points of orderprime to n.


Special points interpreted dynamically

If (X , f ) is an algebraic variety de�ned over a �eld K and a ∈ X (K ) isa K -rational point, then Of (a) ⊆ X (K ) consists entirely of K -rationalpoints.

If X is a semi-abelian variety (for example, if X (C) = (C×)g is someCartesian power of the multiplicative group), and f : X → X is givenby multiplication by n > 1 (in the sense of the group;(x1, . . . , xg ) 7→ (xn

1 , . . . , xng ) in the example), then PrePer(f ) is the set

of torsion points on X and Per(f ) is the set of torsion points of orderprime to n.


Zhang's dynamical analogues of Diophantine conjectures

Dynamical Mordell-Lang: If f : X → X is an algebraic dynamicalsystem over C, a ∈ X (C) is any point, and Y ⊆ X is a closedsubvariety, then the return set E (a, f ,Y ) is a �nite union of pointsand arithmetic progressions.

Dynamical Manin-Mumford: If f : X → X is an algebraic dynamicalsystem over C, Y ⊆ X is a closed irreducible subvariety andY (C) ∩ PrePer is Zariski dense in Y , then Y is a preperiodicsubvariety.

Dense orbit conjecture: If f : X → X is an (irreducible) algebraicdynamical system de�ned over the algebraic numbers, then there is analgebraic point a ∈ X (Qalg ) for which Of (a) is Zariski dense in X .












Some quali�cations

Zhang works with a stronger notion of an algebraic dynamical system.A polarized algebraic dynamical system is a triple (X , f ,L ) where Lis an ample line bundle on X , f : X → X is a self-morphism, andf ∗L ≈ L ⊗q for some q > 1.

The dynamical Mordell-Lang conjecture may still hold with our weakernotion of algebraic dynamical system.

The dynamical Manin-Mumford conjecture (even with Zhang'sstronger hypotheses) is known to be false by work of Ghioca andTucker.

In general, for the dense orbit conjecture one needs the hypothesesthat f : X → X is dominant and that there is no dominant rationalmap g : X → Y with dim(Y ) > 0 and g ◦ f = g . This follows frompolarizability.




















From geometry to arithmetic, a logical caution

While the data de�ning an algebraic dynamical system are �nitary andcomputable (assuming that the ring operations on our underlying �eld arecomputable), it is not obvious that the sets of special points we areconsidering (eg Of (a), Per(f ), Of (a) ∩ Y (C), etc.) are computable.

The dynamical Mordell-Lang conjecture implies, in particular, that the setsof the form Of (a) ∩ Y (C) are computable (though it is not a formalconsequence of this conjecture that one may compute the set from apresentation of the problem).


From geometry to arithmetic, a logical caution

While the data de�ning an algebraic dynamical system are �nitary andcomputable (assuming that the ring operations on our underlying �eld arecomputable), it is not obvious that the sets of special points we areconsidering (eg Of (a), Per(f ), Of (a) ∩ Y (C), etc.) are computable.

The dynamical Mordell-Lang conjecture implies, in particular, that the setsof the form Of (a) ∩ Y (C) are computable (though it is not a formalconsequence of this conjecture that one may compute the set from apresentation of the problem).


Dense orbit conjecture and geometry

Recall that the dense orbit conjecture says that if (X , f ) is an (irreducible,dominant) algebraic dynamical system over an algebraically closed �eld K

which does not dominate a constant dynamical system, then there is apoint a ∈ X (K ) whose orbit is Zariski dense.

Considering the contrapositive, if the conjecture were false witnessed by(X , f ), then for each algebraic point a ∈ X (K ), the Zariski closure ofOf (a) would be a proper subvariety of X . Clearly, f (Of (a)) ⊆ Of (a) sothat f Of (a) ⊆ Of (a).

The dense orbit conjecture may be reformulated as asserting that there issome algebraic point which does not lie on a proper preperiodic subvariety.




















Invariant varieties

To say that a lies on a preperiodic variety is the same as saying that f ◦n(a)lies on an f ◦n-invariant variety for some n ∈ Z+.

The real problem for which the dense orbit conjecture is merely a testquestion for its solution is:

Problem

Qualitatively describe the invariant varieties for algebraic dynamicalsystems.


Invariant varieties

To say that a lies on a preperiodic variety is the same as saying that f ◦n(a)lies on an f ◦n-invariant variety for some n ∈ Z+.

The real problem for which the dense orbit conjecture is merely a testquestion for its solution is:

Problem

Qualitatively describe the invariant varieties for algebraic dynamicalsystems.


Invariant varieties and de�nable sets

The class of algebraic dynamical systems and their invariant subvarietiesmay be analyzed as special cases of de�nable sets in di�erence closed �elds.As such, they admit a tame geometry.


Model theory as tame geometry

Being de�nable in general has no consequences, but if we require that X bede�nable in a fairly inexpressive language, then we may deduce that Xenjoys certain tameness properties and avoids the pathologies of generalmathematical objects.

For the dynamical problems, we consider sets de�nable in di�erence closed�elds.

A di�erence �eld is a pair (K , σ) consisting of a �eld K and anendomorphism σ : K → K . It is di�erence closed if every �nite system ofpolynomial di�erence equations and inequations with coe�cients from K

which has a solution in some di�erence �eld extension already has asolution in K .














De�nable sets from algebraic dynamics

If (X , f ) is an algebraic dynamical system de�ned over the �eld K , then byregarding K as a di�erence �eld with the identity map σ = id : K → K , thede�nable set (X , f )] de�ned by a ∈ X and σ(a) = f (a) encodes thedynamical system.

When evaluated on (K , id), the de�nable set (X , f )] contains only the �xedpoints of f , but when evaluated on a di�erence closed �eld (U, σ), the setis Zariski dense in X (assuming that f is dominant and X is absolutelyirreducible) and the invariant subvarieties of f correspond to de�nablesubset of (X , f )].


De�nable sets from algebraic dynamics

If (X , f ) is an algebraic dynamical system de�ned over the �eld K , then byregarding K as a di�erence �eld with the identity map σ = id : K → K , thede�nable set (X , f )] de�ned by a ∈ X and σ(a) = f (a) encodes thedynamical system.

When evaluated on (K , id), the de�nable set (X , f )] contains only the �xedpoints of f , but when evaluated on a di�erence closed �eld (U, σ), the setis Zariski dense in X (assuming that f is dominant and X is absolutelyirreducible) and the invariant subvarieties of f correspond to de�nablesubset of (X , f )].


Structure theory for de�nable sets in di�erence closed �elds

The de�nable sets in di�erence closed �elds admit dimension theoriesmore re�ned than the usual algebraic dimensions. Generally, if (X , f )is a dynamical system, then dim(X , f )] ≤ dim(X ) with strictinequality possible. [Here dim(X , f )] could be taken to be Lascar rankand dim(X ) is the usual dimension from algebraic geometry.]

The one dimensional sets fall into three classes: �eld-like (essentiallyalgebraic curves over the �xed �eld), group-like (in correspondencewith de�nable groups have the property that every de�nable subset ofany Cartesian power is essentially a translate of a group), anddisintegrated (all de�nable relations on the set are essentially binary).

The higher dimensional de�nable sets may be �bred byone-dimensional sets and the �brations are mediated by de�nablegroups.












Invariant varieties for polynomial dynamics

Theorem (Medvedev)

If f (x) is a one-variable nonconstant rational function, then either the set

de�ned by σ(x) = f (x) is disintegrated or there is a one-dimensional

algebraic group G, isogeny φ : G → Gσ and nonconstant rational function

π : G → P1 for which f ◦ π = π ◦ φ.

Theorem (Medvedev, S.)

If f1, . . . , fn is a sequence of one variable nonlinear polynomials over C none

of which is special, then every invariant subvariety of the algebraic

dynamical system given by (x1, . . . , xn) 7→ (f1(x1), . . . , fn(xn)) comes from

invariant curves for (xi , xj) 7→ (fi (xi ), fj(xj)).

Here, �special� means linearly conjugate to a monomial or a scalarmultiple of a Chebyshev polynomial.The invariant curves for (fi , fj) have a very simple form. In the specialcase of (f , f ), they are horizontal, vertical or of the form y = g(x) forsome g commuting with f .



Theorem (Medvedev)













Theorem (Medvedev)













Theorem (Medvedev)












Corollaries for algebraic dynamics

Zhang's dense orbit conjecture is true for dynamical systems of theform (x1, . . . , xn) 7→ (f1(x1), . . . , fn(xn)) where each fi is anonconstant polynomial and at most one fi is linear.

The dynamical Manin-Mumford conjecture is true for such dynamicalsystems (for periodic points) provided that each fi lifts the Frobeniusin the sense that each is de�ned over Z and fi (x) ≡ xp (mod p).

In work of Baker and DeMarco, this characterization is a key step inthe proof of a dynamical Pink-Zilber theorem.












Skolem's method

Skolem introduced a p-adic analytic method for proving some instances ofthe Mordell conjecture in the 1930s.

In some cases, his method generalizes to dynamical problems: given analgebraic dynamical system (X , f ) over a �eld K of characteristic zero, apoint a ∈ X (K ), and a subvariety Y ⊆ X one may choose a prime p andregard (X , f ) and Y as being de�ned over Qp.

Sometimes, one can �nd a p-adic analytic function Φ : Zp → X (Qp) whichinterpolates the map n 7→ f ◦n(a). In this case, the set{α ∈ Zp : Φ(α) ∈ Y (Qp)} is the zero set of a one variable p-adic analyticfunction which by the identity principle must be a �nite union of points andsets of the form a + pnZp. The dynamical Mordell-Lang conjecture followsin this case.


Skolem's method





Skolem's method





Euclidean variant?

If instead of a p-adic analytic function one had a real analytic functionΦ : (0,∞)→ X (C) satisfying Φ(n) = f ◦n(a), then it would seem that wecould deduce nothing about the set {n ∈ N : f ◦n(a) ∈ Y (C)} from thefact that it is the intersection of N with the zero set of a real analyticfunction.

However, if we could arrange for Φ to be o-minimally de�nable, then thedynamical Mordell-Lang conjecture would follow.


Euclidean variant?

If instead of a p-adic analytic function one had a real analytic functionΦ : (0,∞)→ X (C) satisfying Φ(n) = f ◦n(a), then it would seem that wecould deduce nothing about the set {n ∈ N : f ◦n(a) ∈ Y (C)} from thefact that it is the intersection of N with the zero set of a real analyticfunction.

However, if we could arrange for Φ to be o-minimally de�nable, then thedynamical Mordell-Lang conjecture would follow.


O-minimality

A structure (R, <, . . .) is o-minimal if < is a total order and every de�nablesubset of R is a �nite union of points and intervals.

It follows from a theorem of Tarski that (R, <,+, ·, 0, 1) is o-minimal. Adeep theorem of Wilkie shows that (R, <,+, ·, exp, 0, 1) is also o-minimal.By work of van den Dries and Miller, the real numbers given together withall polynomials, the exponential function, and all real analytic functions (inany number of variables) restricted to boxes [0, 1]n is o-minimal.


O-minimality

A structure (R, <, . . .) is o-minimal if < is a total order and every de�nablesubset of R is a �nite union of points and intervals.

It follows from a theorem of Tarski that (R, <,+, ·, 0, 1) is o-minimal. Adeep theorem of Wilkie shows that (R, <,+, ·, exp, 0, 1) is also o-minimal.By work of van den Dries and Miller, the real numbers given together withall polynomials, the exponential function, and all real analytic functions (inany number of variables) restricted to boxes [0, 1]n is o-minimal.


O-minimal Skolem's method

There are algebraic dynamical systems to which the o-minimal Skolemmethod applies though the p-adic method does not. (Alas, not everyalgebraic dynamical system is susceptible to this method.)

The de�nable sets in higher dimension in o-minimal structures are alsotame. If one considers a dynamical system (X , f ) and point a forwhich there is an o-minimally de�nable interpolating mapΦ : (0,∞)→ X (C) for the orbit of a, then for any subvarietyY ⊆ X n, one may interpret{(`1, . . . , `n) ∈ Nn : (f ◦`1(a), . . . , f ◦`n(a)) ∈ Y (C)} as the integerpoints on a de�nable set. A counting theorem of Pila and Wilkie limitsthe number of such points not coming from �obvious� relations tobeing subpolynomial in the size of the integer.


O-minimal Skolem's method

There are algebraic dynamical systems to which the o-minimal Skolemmethod applies though the p-adic method does not. (Alas, not everyalgebraic dynamical system is susceptible to this method.)

The de�nable sets in higher dimension in o-minimal structures are alsotame. If one considers a dynamical system (X , f ) and point a forwhich there is an o-minimally de�nable interpolating mapΦ : (0,∞)→ X (C) for the orbit of a, then for any subvarietyY ⊆ X n, one may interpret{(`1, . . . , `n) ∈ Nn : (f ◦`1(a), . . . , f ◦`n(a)) ∈ Y (C)} as the integerpoints on a de�nable set. A counting theorem of Pila and Wilkie limitsthe number of such points not coming from �obvious� relations tobeing subpolynomial in the size of the integer.


Higher rank problem

Let X be an algebraic variety over some �eld K , f1, . . . , fn a sequence ofself-maps fi : X → X which commute with each other, a ∈ X (K ) a point,and Y ⊆ X a subvariety. What can one say aboutE := {(`1, . . . , `n) ∈ Nn : f ◦`11 ◦ · · · ◦ f ◦`nn (a) ∈ Y (K )}?

In general, not much. Consider X (C) = Cn,fi (x1, . . . , xn) = (x1, . . . , xi−1, xi + 1, xi+1, . . . , xn), Y de�ned over Z, anda = (0, . . . , 0). Then E = Y (Z) which is notoriously complicated.


Higher rank problem

Let X be an algebraic variety over some �eld K , f1, . . . , fn a sequence ofself-maps fi : X → X which commute with each other, a ∈ X (K ) a point,and Y ⊆ X a subvariety. What can one say aboutE := {(`1, . . . , `n) ∈ Nn : f ◦`11 ◦ · · · ◦ f ◦`nn (a) ∈ Y (K )}?

In general, not much. Consider X (C) = Cn,fi (x1, . . . , xn) = (x1, . . . , xi−1, xi + 1, xi+1, . . . , xn), Y de�ned over Z, anda = (0, . . . , 0). Then E = Y (Z) which is notoriously complicated.


Higher rank with simple geometry

One might hope that if the geometry of X is simple, then E should havesimple form, too, but Ghioca, Tucker and Zieve constructed an example ofa pair of commuting endomorphisms of the algebraic group (C×)3 forwhich E is the set of natural number points on a quadratic curve.

With Y. Yasufuku, I showed that, in fact, every exponential-polynomial set,a set of n-tuples of natural numbers de�ned by the vanishing of a function

of the form (`1, . . . , `n) 7→ P(`1, . . . , `n; {α`ji }) where P is a polynomialwith coe�cients from the ring of all algebraic integers and each αi is analgebraic integer, may be realized as a return set for a sequence of ncommuting endomorphisms of an algebraic torus.


Higher rank with simple geometry

One might hope that if the geometry of X is simple, then E should havesimple form, too, but Ghioca, Tucker and Zieve constructed an example ofa pair of commuting endomorphisms of the algebraic group (C×)3 forwhich E is the set of natural number points on a quadratic curve.

With Y. Yasufuku, I showed that, in fact, every exponential-polynomial set,a set of n-tuples of natural numbers de�ned by the vanishing of a function

of the form (`1, . . . , `n) 7→ P(`1, . . . , `n; {α`ji }) where P is a polynomialwith coe�cients from the ring of all algebraic integers and each αi is analgebraic integer, may be realized as a return set for a sequence of ncommuting endomorphisms of an algebraic torus.


Conclusion: room for chaos?

The model theoretic methods suggest that by averting to properties ofde�nable sets we may tame algebraic dynamics.

However, in joint work with several collaborators (Benedetto, Ghioca, Hutz,Kurlberg and Tucker), we showed that while the initial step of Skolem'smethod may be made to work in low dimension, probabilistically, it shouldalmost always fail in dimensions four and higher.

Thus, I would expect that the chaos inherent in Gödel's incompletenesstheorem will show itself even in the algebraic dynamics of a single mapf : X → X .












Algebraic dynamics - Indian Statistical Institutedst.model/slides/scanlon.pdf · Discrete dynamical systems A dynamical system is a pair (X ;f ) consisting of a set X and a self-map

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