Algebraic numbers
What is the smallest integer ! > 1?
ISLANDS ON
ALGEBRAIC SURFACES
Curtis T McMullen
Harvard University
Lehmer’s number
M(!) = product of conjugates with |!i| > 1
λ = 1.1762808182599175065 . . .
p(x) = 1 + x − x3 − x4 − x5 − x6 − x7 + x9 + x10
Answer: Lehmer’s Number?
1
0.5
!
-1
-0.5
0.5
1
-0.5
-1
! = 1.1762808182599175...
P(x) = x10+x9"x7"x6"x5"x4"x3+x+1
The (-2,3,7) pretzel knot
The (-2,3,7) pretzel knot
The (3,4,7) triangle group
Coxeter element
Coxeter Groups
e1
e2
e3
e4
e10
e5
e6
e7
e8
e9
<ei,ei> = 2
<ei,ek> = 0 or -1
W
O(n,Z) : generated by reflections si in ei
Coxeter element w = s1 s2 s3 s4 ... s10
The 38 minimal hyperbolic Coxeter diagrams
Coxeter system
Lehmer’s polynomial = det(xI-w) for E10
Theorem. The spectral radius of any w in any
Coxeter group satisfies r(w) = 1 or
r(w) ! !Lehmer > 1.
The 38
minimal
Coxeter
systems
λ(W, S)
det(xI − w)
Ah4
2.36921
(2.26844)
1 − x − 3x2 − x3 + x4
Ah5
2.08102
(1 + x)(1 − x − 2x2 − x3 + x4 )
Ah6
1.98779
(1.96355)
1 − 2x2 − 3x3 − 2x4 + x6
Ah7
1.88320
(1 + x)(1 + x + x2 )(1 − 2x + x2 − 2x3 + x4 )
Ah8
1.83488
(1.82515)
1 − x2 − 2x3 − 3x4 − 2x5 − x6 + x8
Bh5
1.72208
(1 + x)(1 − x − x2 − x3 + x4 )
Bh6
1.58235
1 − x2 − 2x3 − x4 + x6
Bh7
1.50614
(1 + x)(1 − x − x3 − x5 + x6 )
Bh8
1.45799
1 − x2 − x3 − x5 − x6 + x8
Bh9
1.42501
(1 + x)(1 − x − x3 + x4 − x5 − x7 + x8 )
Dh6
1.72208
(1 + x)2 (1 − x − x2 − x3 + x4 )
Dh7
1.58235
(1 + x)(1 − x2 − 2x3 − x4 + x6 )
Dh8
1.50614
(1 + x)2 (1 − x − x3 − x5 + x6 )
Dh9
1.45799
(1 + x)(1 − x2 − x3 − x5 − x6 + x8 )
Dh10
1.42501
(1 + x)2 (1 − x − x3 + x4 − x5 − x7 + x8 )
Eh8
1.40127
(1 + x + x2 )(1 − x2 − x3 − x4 + x6 )
Eh9
1.28064
(1 + x)(1 − x3 − x4 − x5 + x8 )
Eh10
1.17628
1 + x − x3 − x4 − x5 − x6 − x7 + x9 + x10
Coxeter system
λ(W, S)
det(xI − w)
K343
2.08102
(1 + x)(1 − x − 2x2 − x3 + x4 )
K3433
1.88320
(1 + x)2 (1 − 2x + x2 − 2x3 + x4 )
K44
K53
K533
5
5
x)2 (1
2.61803
(1 +
2.15372
(1 + x)2 (2 − 3x −
− 3x +
1.91650
√
(1 + x)(2 − x −
x2
2x3
x4
√
5x + 2x2 )
5x − x3 −
5x3 + 2x4 )
x6
1.58235
1−
L34333
1.40127
1 − x2 − x3 − x4 + x6
1.84960
2+x−
L4343
1.88320
(1 + x)(1 − 2x + x2 − 2x3 + x4 )
L443
2.08102
1 − x − 2x2 − x3 + x4
1.36000
(1 + x)(2 − x −
1.91650
√
√
2 − x − 5x − x3 − 5x3 + 2x4
2.15372
(1 + x)(2 − 3x −
1.72208
1 − x − x2 − x3 + x4
1.63557
(1 + x)(1 + x + x2 − 4x cos2 π/7)
3.09066
(2.89005)
(1 + x)(1 − 2x −
5
L353
L5333
L534
L54
L633
L73
Q3
5
5
5
6
7
−
+
√
L33433
−
Dynamics
x2 )
√
√
√
5x − 2 5x2 + x3 − 5x3 + 2x4
x2
√
5x + 2x2 − x3 −
√
√
√ 3
5x + 2x4 )
5x + 2x2 )
2x + x2 )
√
− 2 2x2 − x3 + x4
Q4
2.57747
1−x−
Q5
2.43750
(2.3963)
(1 + x)(1 − 2x + x2 −
X5
2.61803
(1 + x)3 (1 − 3x + x2 )
X6
2.61803
(1 + x)4 (1 − 3x + x2 )
√
2x2 − 2x3 + x4 )
f : X #X holomorphic diffeomorphism
of a compact complex manifold
What is the simplest interesting dynamical system?
Bowties
Complex Surfaces
Theorem (Cantat) A surface X admits an automorphism
f : X #X with positive entropy only if X is birational to:
•
•
•
the projective plane P2
a complex torus C2/$, or
a K3 surface.
Elliptic islands
Stochastic Sea
A=2
A=2.5
(1+x2)(1+y2)(1+z2)+Axyz = 2
Tame blowup
Ergodicity
A=8
Complex Orbit
A family of K3 surfaces
Islands Theorem
Synthesis
There exists a K3-surface automorphism f : X#X with a complex
invariant island -- a Siegel disk.
Analysis
Hodge theory:
study f* on H2(X) = H2,0
Lefschetz:
Tr(f*)= -1
Atiyah-Bott:
f* determines rotation DfP on TPX
f has a unique fixed point P
Transcendence: DfP not resonant
Siegel:
f
H1,1 H0,2
linear rotation near P
Number theory: P(t) = det(tI-f*|H2(X))
Gross-M:
P(t)
Torelli:
f*
f* acting on II3,19 = H2(X,Z)
[X and f:X#X]
Key ingredient: Degree 22 Salem number of trace -1
P(t) = 1+t-t3-2t4-3t5-3t6-2t7+2t9+4t10+5t11
+4t12+2t13-2t15-3t16-3t17-2t18-t19+t21+t22
X is not projective!
Rational Surfaces
Realization Theorem
X = blowup of P2 at n points
H2(X,Z)
Z1,n
KX
The Coxeter element of Wn can always be realized
by an automorphism Fn : Xn#Xn
of P2 blown up at n special points.
[En lattice]
Example: F3(x,y) = (y,y/x)
Theorem (Nagata) Every automorphism of X lies in the
Weyl group Wn O(Z1,n).
(x,y)#(y,y/x)#(y/x,1/x)#(1/x,1/y)#(1/y,x/y)#(x/y,x)#(x,y)
3
3
!
P2
KX = (-3,1,1,...,1)
1
2
1
2
4
Lehmer’s automorphism
10 points on a cuspidal cubic
{a,b} {0.499497, -0.0837358}
F10 : X10#X10
First case where h(Fn) > 0
Theorem. The map F10 has minimal positive entropy
among all surface automorphisms, namely
h(F10) = log(!Lehmer).
(x,y)# (y,y/x) + (a,b)
X3
12 points on 3 lines
{a,b} {-2.26123, 1.79288}
Synthesis
X = blowup of n points on a cuspidal cubic C in P2
[En lattice]
!
Pic0(Xn) # Pic0(C)
Coxeter element w
!
!
C
Eigenvalue ! of w
positions of n points on C
{a,b} {-0.926076, 0.61173}
11 points on a conic + line
Speed of convergence
Island on a rational surface
{a,b} {0.0444317 - 0.442239 I, -0.0444317 - 0.442239 I}