<latexit sha1_base64="4vblUilF8p5XKyKsKdrkZr/n/Ro=">ACYHicbVBbaxQxGP126mUdL93VN35aHERxGmL/oi1Bv4olTstoXNsmSy2TY0lzHJSIdh/ov/hof1Td/iZndVrT1QOBwLiQ5RamkD4R87yVrV65eu96/kd68dfvO+mB4d9/bynEx5lZd1gwL5Q0YhxkUOKwdILpQomD4uRV5x98Fs5La/ZCXYqpZkdGLiRnIUqzwRvEj/gc6TsWGjLKX2Sj0iL9MX71ymeYe8kI9wGym3Hmkptz5RSkaYt5ils8EmycgS+IfkF8nmzgZ9/AUAdmfD3hqdW15pYQJXzPtJTsowbZgLkivRprTyomT8hB2JSaSGaeGnzfK/LT6MyhwX1sVjAi7VvxsN097XuohJzcKxv+h14v+8SRUWz6aNGUVhOGrixaVwmCxGw/n0gkeVB0J407GtyI/Zo7xECdO6bLY6GhZn3GrtTXtuXpaR3pa27LbvWFKtWk326WRLpP97SwnWf4h7vcSVujDA9iAR5DU9iBt7ALY+DwFb7BD/jZ+5X0k/VkuIomvbPOPfgHyf3fF6xtw=</latexit> <latexit sha1_base64="+nLDbSRYPfNRVtMDhGQfdVOkmag=">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</latexit> <latexit sha1_base64="+nLDbSRYPfNRVtMDhGQfdVOkmag=">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</latexit> <latexit sha1_base64="IfnjtXF0P3l/79v2IGQAT87U0=">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</latexit> The Hecke groups G q G q = <S,T>, where Billiards and the arithmetic of ! ! 0 1 1 2 cos π /q S = T = . and non-arithmetic groups − 1 0 0 1 Curtis T McMullen allen q = 3 , 4 , 6 und 1 , ... ist so etwas wie ein Abgesehen von den vier F¨ Harvard University Koe ffi zientengesetz der Matrizen zu G q noch nicht bekannt. 2 —Leutbecher, 1974. 2 Aside from the four cases q = 3 , 4 , 6 and ∞ ... no rule for the coe ffi cients of the matrices in G q is yet known. Hecke, Rosen, Leutbecher, Cohen-Wolfart, Davis-Lelievre, ... Matrix entries in G 5 The golden Hecke group G 5 =(2,5, ∞ ) γ = (1+ √ 5)/2 z → z+ γ triangle Theorem group Let x = A + B γ be a matrix entry in G 5 . Then AB ≥ 0. z → -1/z Cor. i |x ′ | ≤ |x|. ζ 10 ↺ Complement. For any g in G 5 , we have |Tr g ′ | ≤ |Tr g| provided the latter is > 2. - γ /2 γ /2 0
Proofs: Holomorphic pentagon-to-star map G 5 = SL(X, ω ). Use the golden table! G 5 ′ 1 G 5 F γ 1 γ Cusps of the golden Hecke group Quadratic trace field Theorem Consider a form (X, ω ) of any genus g. The cusps of G 5 coincide with K = Q ( √ 5). Leutbecher, M Theorem If SL(X, ω ) is a lattice with quadratic trace field K, then the cross-ratios of its cusps coincide with K ∪ { ∞ }. Example: g=3, the regular 12-gon; handles G 12 . γ 1/ γ 1 0
Simple slopes, long trajectories Effective methods Theorem Let s = (a+b γ )/(c+d γ ) with |a|,|b|,|c|,|d| < M. Then the trajectory at slope s in the golden L has length at most s=1 s=7/3 s=5/3 exp(c(log M) 2 ). 4 segs 80 segs 1152 segs s=89/55 Sharp for s a Fibonacci number. 10 13 segs The arithmetic of G 5 Long periodic trajectories are uniformly distributed...aren’t they? Let R = {a ′ /a : a ≠ 0 is a matrix entry in G 5 }. Theorem I. R is contained in [-1,1]. II. The closure S of R is countable. III. S is homeomorphic to ω ω +1. IV. S is equal to the semigroup generated by R. Cor There is no simple characterization of the matrix coefficients of G 5 .
Long periodic trajectories are uniformly distributed...aren’t they? Non-optimality of optimal billiards Twists Q. of Davis-Lelievre Let m(s) = probability measure on a periodic trajectory of slope s in Q ( √ 5). Let M = {all lim m(s n ) for s n → 0.} Theorem (M) M is homeomorphic to ω ω +1.
<latexit sha1_base64="QzgJDznIFa9Bx+3WD1jahphWN6w=">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</latexit> <latexit sha1_base64="QzgJDznIFa9Bx+3WD1jahphWN6w=">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</latexit> <latexit sha1_base64="QzgJDznIFa9Bx+3WD1jahphWN6w=">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</latexit> <latexit sha1_base64="QzgJDznIFa9Bx+3WD1jahphWN6w=">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</latexit> Coxeter diagrams and Teichmüller curves Every Teichmüller curve V can be specified by a curve system (A i ), (B j ). `New’ proof of Veech dichotomy Gives h D τ A , D τ B i = Γ ⇢ SL ( X, ω ) . Usually of infinite index! Modular symbols organize all curves systems attached to V.
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