Preperiodic Points and
Unlikely Intersections
joint work with Laura DeMarco
Matthew Baker
Georgia Institute of Technology
AMS Southeastern Section Meeting
November 6-7, 2010
Preperiodic points
• Let f(z) be a polynomial with complex
coefficients.
• A complex number z0 is called
preperiodic for f if the orbit
{z0, f(z0), f(2)(z0),…} is finite.
Iteration of z2 + c
• Given a complex number a, let Sa be
the set of complex parameters c such
that a is preperiodic for z2 + c.
• When a=0, the points of S0 belong to
the Mandelbrot set M and the closure of
S0 contains the entire boundary of M.
The Mandelbrot Set
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Unlikely intersections
Lemma: Sa is always a (countably) infinite set.
• Now fix two complex numbers a and b.
• What can we say about the set Sa  Sb? In
other words, for how many parameters c can
a and b be simultaneously preperiodic?
• Note that if a2 = b2 then Sa = Sb so Sa  Sb is
infinite.
Are a and b preperiodic?
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Main theorem
Theorem (B., DeMarco): If a2 ≠ b2 then
Sa  Sb is finite.
Although the statement is purely about
complex dynamics, the proof uses ideas
from number theory and involves
History of the problem
At an AIM workshop in January 2008, Umberto
Zannier (motivated by questions of David
Masser) asked if S0  S1 is finite.
Our theorem shows that the answer is yes…
though we don’t know what S0  S1 is!
Some experimental data
z 2:
0 --> 0, 1 --> 1
z2 - 1: 0 --> -1 --> 0, 1 --> 0 --> -1 --> 0
z2 - 2: 0 --> -2 --> 2 --> 2, 1 --> -1 --> -1
Conjecture (B., Connelly):
S0  S1 = {0,-1,-2}.
The theorem of Masser &
Zannier
Theorem (Masser-Zannier): The set of
complex numbers  such that both A()
and B() are torsion in the Legendre
curve
is finite. Here we take
An illustration of the MasserZannier theorem
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The “local-global principle”
Lemma: If f is a polynomial with rational
coefficients, then a rational number z0 is
preperiodic for f if and only if the orbit
{z0, f(z0), f(2)(z0),…}
is bounded in the real numbers and also in the
p-adic numbers for all prime numbers p.
Example: For f(z) = z2 - 1/2, the orbit of 0 is
{0, -1/2, -1/4, -7/16,…}
which is bounded in the usual topology on the
rationals but not in the 2-adic topology.
Boundedness loci
Given a complex number a, let Ma be
the set of complex parameters c such
that a stays bounded under iteration of
z2 + c. We call Ma the generalized
Mandelbrot set associated to a.
a=0
a=1
M0 and M1 superimposed
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Complex potential theory
• Given a compact subset E of the complex
plane and a probability measure  supported
on E, define the energy of  to be
• If there is a probability measure of finite
energy supported on E, then there is a unique
such measure E having minimal energy,
called the equilibrium measure for E.
Capacity
The capacity of E is defined to be
(or zero if there is no probability measure
of finite energy supported on E.)
Lemma:
Proof of the main theorem:
algebraic case
Sketch of the proof:
Assume a,b are algebraic and that
Sa  Sb = {c1,c2,c3,…} is infinite.
Easy lemma: All cn’s must be algebraic.
Let n be the discrete probability measure on
the complex plane supported equally on all
Galois conjugates of cn.
Proof of the main theorem:
algebraic case (cont’d)
• Using the fact that the cn’s belong to Sa, a
suitable arithmetic equidistribution theorem
shows that the sequence {n} converges
weakly to the equilibrium measure a for Ma.
• By symmetry, {n} also converges to b.
Hence a = b.
• This implies (taking supports and ‘filling in’)
that Ma = Mb.
• An argument from univalent function theory
now shows that a2 = b2.
What if a and b are
transcendental?
• If a is transcendental, then so is b and a,b,
and all cn’s belong to the algebraic closure of
• We think of k as the function field of an
algebraic curve.
• Can we mimic the above proof with algebraic
numbers replaced by algebraic functions?
The product formula
• The proof of the ‘suitable equidistribution
theorem’ uses the product formula for
algebraic numbers, which for a nonzero
rational number  just says that
• There is also a product formula for algebraic
functions, which says that a rational function
on the projective line has the same number of
zeros as poles, counting multiplicities.
The product formula for
algebraic functions
• Let k be the field of rational functions over
some algebraically closed constant field.
• The “places” (equivalence classes of absolute
values) of k are in 1-1 correspondence with
points of the projective line over k.
• If a place v corresponds to a point p of P1(k),
we define |f|_v = exp(-ord_p(f)) for any nonzero rational function f in k. Then
Absolute values on fields
• In the algebraic function case, all absolute
values appearing in the product formula are
non-Archimedean.
• So in order to mimic our proof from the
algebraic case, we need to use some sort of
non-Archimedean potential theory.
• Luckily, such a theory exists! But one needs
to use Berkovich’s theory of nonArchimedean analytic spaces.
The Berkovich affine line
• Let K be a complete and algebraically
closed non-Archimedean field.
• The Berkovich affine line
over K
is a locally compact and path-connected
space containing K (which is not locally
compact and is totally disconnected) as
a dense subspace.
The Berkovich affine line
The Berkovich affine line over K has the
structure of an infinitely branched real
tree:
Definition of
• As a set, the Berkovich affine line consists of
all real-valued multiplicative semi-norms on
K[T] which extend the usual absolute value
on K. It is endowed with the weakest
topology making the absolute value of a
polynomial f in K[T] continuous.
• Each element of K gives a semi-norm (by
evaluation), but there are lots of other seminorms, e.g. the supremum of |f(x)| as x
ranges over some closed disk D in K. (This is
multiplicative by `Gauss’ Lemma’.)
Potential theory on the
Berkovich line
The extra points allow one to `connect up’ the
totally disconnected space K in a natural way.
The tree structure on the Berkovich line allows
one to do potential theory.
As in the complex case, for compact subsets
of the Berkovich line one can define
capacities, equilibrium measures, Green’s
functions, etc. (B.-Rumely “Potential Theory
and Dynamics on the Berkovich Projective
Proof of the main theorem
(transcendental case)
Sketch of the proof:
Assume a,b are transcendental and that
Sa  Sb = {c1,c2,c3,…} is infinite.
Fix a place v of
Let n be the discrete probability measure
on the v-adic Berkovich affine line which
is supported equally on all
conjugates of cn.
Conclusion of the proof
• By equidistribution, the v-adic generalized
Mandelbrot sets Ma,v and Mb,v coincide for all
places v of k.
• By a local-global principle for function fields
due to Benedetto (2005), it follows that a is
preperiodic for z2+c iff b is, i.e. Sa=Sb.
• By Montel’s theorem, Sa (resp. Sb)
determines Ma (resp. Mb) inside
• Hence Ma = Mb and we conclude as in the
algebraic case!
Variant of the main theorem
Similar methods allow us to prove the following
result:
Theorem (B.,DeMarco): If f,g are rational
functions with complex coefficients then
Preper(f)  Preper(g) is infinite iff
Preper(f) = Preper(g).
Remark: This was recently proved
independently (and generalized to higher
dimensions) by X. Yuan and S. Zhang. When
f and g have algebraic coefficients, our result
follows from a theorem of A. Mimar.