Algebraic integers Entropy, What is the smallest integer λ > 1? Algebraic Integers, and moduli of surfaces Curtis T McMullen Harvard University Lehmer’s number 1 Introduction Mahler measure Gross, Hironaka In this paper we use the gluing theory of lattices to constructK3surface λ =1.1762808182599175065 ... M(λ) = product of conjugates with |λi| > 1 automorphisms with small entropy. 3 4 5 6 7 9 10 Algebraic integers. A Salem number λ > 1 is an algebraic integer which p(x)=1+x − x − x − x − x − x + x + x is conjugate to 1/λ, and whose remaining conjugates lie on S1. There is a unique minimum Salem number λd of degree d for each even d. The smallest Lehmer’s Number known Salem number is Lehmer’s number, λ10. These numbers and their 1 Smallest Salem Numbers, by Degree minimal polynomials Pd(x), for d ≤ 14, are shown in Table 1. Pd(x) 0.5 2 λ2 2.61803398 x − 3x +1 λ10 4 3 2 λ4 1.72208380 x − x − x − x +1 -1 -0.5 0.5 1 6 4 3 2 λ6 1.40126836 x − x − x − x +1 8 5 4 3 λ8 1.28063815 x − x − x − x +1 -0.5 10 9 7 6 5 4 3 λ10 1.17628081 x + x − x − x − x − x − x + x +1 12 11 10 9 6 3 2 λ12 1.24072642 x − x + x − x − x − x + x − x +1 14 11 10 7 4 3 -1 λ14 1.20002652 x − x − x + x − x − x +1 λ Table 1. The smallest Salem numbers by degree, and their minimal 10 = 1.176280... Conjecture (Lehmer) λ10 = inf M(α) over 10 9 7 6 5 4 3 polynomials. P10(x) = x +x !x !x !x !x !x +x+1 all algebraic integers with M(α)>0. Surface dynamics. Now let F : X → X be an automorphism of a compact complex surface. It is known that the topological entropy h(F ) is determined by the spectral radius of F ∗ acting on H∗(X). More precisely, we have ∗ h(F ) = log ρ(F |H2(X)), (1.1) and if h(F ) > 0, then a minimal model for X is either a K3 surface, an Enriques surface, a complex torus or a rational surface [Ca]. The lower bound h(F ) ≥ log λ10 (1.2) holds for all surface automorphisms of positive entropy, by [Mc3]. In this paper, we will show that the lower bound (1.2) can be achieved on a K3 surface. Theorem 1.1 There exists an automorphism of a K3 surface with entropy h(F ) = log λ10. 1 Lehmer’s polynomial = det(xI-w) for E10 Dynamics [Compare 12, 18, 30, ∞] Theorem. The spectral radius of any w in any Coxeter group satisfies r(w) = 1 or r(w) ≥ λ10 > 1. f : X →X Proof: uses Hilbert metric on the Tits cone. What is the simplest interesting dynamical system? Entropy Torus examples X = torus Rn/Zn Entropy of English = h = about log 3 (or less) f : X →X linear map induced by A in GLn(Z) Schneier, Applied Cryptography, 1996 Number of possible English books with N characters h(f) = log (product of eigenvalues of A with |λ| > 1) N N is about 3 (not 26 ) = log [spectral radius of f* | H*(X) ] X compact, f : X →X continuous h(f) = log λ ⇔ Lehmer’s conjecture ⇔ N |{orbit patterns of length N}| ∼ λ . h(f) ≥ log λ10 if h(f)>0. Entropy on Complex Surfaces Entropy: Kähler case Theorem (Gromov, Yomdin) The entropy of an automorphism f : X →X of a compact Kähler manifold is X = compact complex manifold, dim = 2, given by: f : X → X holomorphic h(f) = log [spectral radius of f|H*(X)] . What small values can h(f) assume? Cor. For surfaces, h(f) = log [a Salem number] = log ρ(f|H2(X)) . (dim=1 ⇒ zero entropy) Complex Surfaces Complex torus C2/Λ Theorem (Cantat) A surface X admits an automorphism Theorem. There exists a 2D complex torus automorphism f : X →X with positive entropy only if X is birational to: with h(f) = log(λ6). (minimum possible for 2D torus) 6 • a complex torus C2/Λ, * 22 • a K3 surface*, or ( or Enriques) Complement. For a projective torus, one can achieve ∞ • the projective plane P2. h(f) = log(λ4) and this is optimal. (1.722 > 1.401) Synthesis Problem: Salem number ⇒ surface + map Synthesis: f|H1(X,Z) ⇒ Λ ⊂ C2 ⇒ X Rational Surfaces En spherical ⇔ Fn periodic ⇔ n ≤ 8 2 X = blowup of P at n points (Kantor, 1890s) 2 1,n ⊥ H (X,Z) ≃ Z ⊃ KX ≃ [En lattice] Example: F3(x,y) = (y,y/x) KX = (-3,1,1,...,1) (x,y)→(y,y/x)→(y/x,1/x)→(1/x,1/y)→(1/y,x/y)→(x/y,x)→(x,y) Aut(X) ⊂ W(En) (Nagata) 3 3 Theorem. The Coxeter automorphism of En can be 2 2 X3 realized by an automorphism Fn : Xn→Xn of P blown up P at n suitable points. 1 2 1 2 4 Lehmer’s automorphism Rational Surfaces: Synthesis F10 : X10→X10 X = blowup of n points on a cuspidal cubic C in P2 0 0 [En lattice] ≃ Pic (Xn) → Pic (C) ≃ C First case where h(Fn) > 0 Coxeter element w Eigenvalue λ of w Theorem. The map F10 has minimal positive entropy among all surface automorphisms, namely h(F10) = log(λ10). λ eigenvector of w ⇒ positions of n points on C 10 points{a,b} {0.499497, on a -0.0837358} cuspidal cubic Blowup at 11 points (x,y)→ (y,y/x) + (a,b) Bedford-Kim K3 surfaces/R K3 surfaces/R 3 X ⊂ R defined by (1 + x2)(1 + y2)(1 + z2)+Axyz =2 f : X→X defined by f = I I I x ◦ y ◦ z The map f is area-preserving! Complex Orbit K3 surfaces: Glue results Theorem. There exists a K3 surface automorphism with h(f) = log(λ10), and this is optimal. (minimum possible) (Oguiso - λ14 = 1.2002) Complement. For projective K3 surfaces, one can achieve h(f) = log(λ6). (1.401 > 1.176) Synthesis of a K3 automorphism with Assembly of a projective K3 entropy log λ10 automorphism with entropy log λ6 Coxeter automorphism of E10(a) Positive automorphism of A2 A2 2 ⊕ Salem factor F2 Coxeter automorphism of A2(2) (3, 7) 2 (0, 4) F3 (1, 5) (0, 2) 2,0 0,2 H +H 6 F F3 +transcendental cycles 3 Identity factor E8 Coxeter automorphism of A12(a) F13 Identity factor (0, 8) (2, 10) (0, 2) E8 NS(X) H2,0+H0,2 X Signature (0,12) +transcendental cycles determinant 9 NS(X): signature (1,9), determinant 9477 = 36·13 A2 A2 blows down to 3 points Problem: Find this example in projective space! The exceptional triangular billiard tables Moduli space Entropy on topological surfaces = Mg of Riemann surfaces X of genus g 1/4 EE Modg = {f : ∑g → ∑g}/isotopy = π1( g ) 6 M = { } 5/12 1/3 -- a complex variety, dimension 3g-3 h(f) = min {h(g) : g isotopic to f} 2/9 EE 7 = length of loop on moduli Teichmüller metric: every holomorphic map space represented by [f]. 2 4/9 1/3 f : H g → M * 1 1/5 ≥ log spectral radius of f on H (∑g) is distance-decreasing. EE 8 Stringlike Synthesis: f ⇒ a loop of Riemann surfaces 7/15 1/3 Conjectures on finite covers Kazhdan’s Theorem F ∑ → ∑ h h H ↓ ↓ sup log spectral radius of * 1 ↓ h(f) = F on H (∑h), ∑g → ∑g h f over all finite covers. Y → Jac(Y) = C /Λ ⇔ ↓ X The super period map is an isometry. Mg → Ah The hyperbolic metric on X is the limit of the C metrics inherited from the Jacobians of finite covers of X. (from the Teichmüller metric to the Kobayashi metric) Counterexamples Surface maps with genus →∞ (after Lanneau-Thiffeault, E. Hironaka, Farb-Leininger-Margalit) Theorem. The super period map Let δg = exp (length of shortest geodesic on . ) Mg g h 21 M → A Known: δ1= (3+√5)/2 ⇔ matrix SL2(Z) is not an isometry in the directionsC coming from 11 ∈ quadratic differentials with odd order zeros. ⟩ = λ2 = smallest Salem number of degree 2 δ2 = λ4 = smallest Salem number of degree 4 Corollary. The entropy of most mapping classes δg = 1 + O(1/g) (not Salem for g >> 0!) f : ∑g → ∑g Spectral Gap ⟨ g cannot be detected homologically, Conjecture: lim (δ ) = δ = (3+√5)/2 = λ even after passing to finite covers. g 1 2 A link L from the shortest loop Other fibrations M3 → S1 in the moduli space of genus one Teichmüller polynomial H1(M) 3 3 M = S -L -1 21 SL2(Z) 11 ∈ 1 -1 1 = Mod1 ∼= Mod0,4 -1 ⇔ Braids on 3 strands X2-3X+1 2n n -1 M3 = S3-L fibers over S1 Tn(X) = X - X (X+1+X ) + 1 Teichmüller polynomial: Lehmer’s number for topologists special values 4 3 2 T2(X) = X - X - X - X + 1 ⇒ genus 2 example with h(f) = log λ4 12 7 6 5 T6(X) = X - X - X - X + 1 = (X2-X+1)(X10+X9-X7-X6 -X5-X4 -X3+X+1) ⇒ genus 5 example with h(f) = log λ10 ψ = τA τB in the mapping- 1/g h(ψ) = log λ10 ⇒ genus g examples with h(f) ∼ log λ2 , class group for genus 5 Tn(X) in agreement with conjecture 2 1+√5 21 2 ⇒ 11 ⇒ Lehmers number in other fibrations ⇒ ⇒ genus 5 Lehmer’s number is implicit in the golden mean.