Dynamcs of contnued fractons n Q( 2) Robert Hnes January 2, 2017 At the end of the ntroducton to [7], Schmdt remarks that there s an ergodc theory for hs contnued fractons over Q( 2) smlar to that over Q() n [5]. Here s the verson of that n terms of nversons n eght crcles (wth cubc tangency) and sx dual crcles (wth octahedral tangency), generatng the group of reflectons n the deal, rght-angled cubeoctahedron. Consder the followng 14 moebus transformatons defned over Z[ 2] s 1 = (1 + 2 2) z 4 2 z 1 + 2 2, s 2 = z 2 z 1, s 3 = z + 2, s 4 = z, s 5 = (1 + 2 2) z 2 4 z 1 + 2 2, s 6 = (3 + 2 2) z 4 4 z 3 + 2 2, t 1 = z + 2 2, t 2 = z, t 3 = (5 2 2) z + 4 2 4 2 z + 5 + 2 2, t 4 = (3 2 2) z + 2 2 4 2 z + 3 + 2 2, z t 5 = 2 2 z + 1, t 6 = (3 2 2) z + 4 2 2 2 z + 3 + 2 2, t 7 = (1 2 2) z + 2 2 2 2 z + 1 + 2 2, t 8 = 3 z + 2 2 2 2 z + 3, nversons n the crcles whose nterors are A and B (see fgures below). These generate a dscrete group of sometres of hyperbolc three-space (of fnte covolume), reflectons n the sdes of an deal, rght-angled cubeoctahedron. The rght-angled part s what allows the constructon below (drectly analagous to the constructon for Q() and the rght-angled deal octahedron). We defne two dynamcal systems, T A, T B : C C (off some measure zero exceptonal sets) s w w A T A (w) = t w w A t z z B T B (z) = s z z B and use these to coordnatze the plane n two dfferent ways usng the alphabet s j, t j }. We wrte w = w = =1 w f TA (w) = w T 1 A w and smlarly z = z = =1 z f TB (z) = z T 1 B z. Wth these coordnates, T A, T B are the shft maps. [Talk about normal form, Farey sets, ratonal approxmaton, etc.] We now defne an nvertble extenson T of T A and T B (T wll extend T B and T 1 wll extend T A ), defned on a subset of the space of geodescs of hyperbolc 3-space P 1 (C) P 1 (C)\dag. 1
Defne the followng groupngs of the regons A, A, B, B A = ( B j A = B j) ( Bj A = B j ) A = ( B j A = B j) ( Bj A = B j ) B = ( A j B = A j) ( Aj B = A j ) B = ( A j B = A j) ( Aj B = A j ) (.e. for each A regon A s the unon of all the B regons dsjont from t, smlarly for the B). The space of geodescs on whch T wll be defned s G = ( A A ) ( A A ) = ( B B ) ( B B ) wth (t w, t T (w, z) = (z 1 w j, z j ) = z) = (t w, T B (z)) z B (s w, s z) = (s w, T B (z)) z B j=1 j=2 or equvalently T 1 (s w, s (w, z) = ( w j, w 1 z j ) = z) = (T A (w), s z) w A (t w, t z) = (T A (w), t z) w A j=2 j=1. One can verfy that T s a bjecton T (B B ) = A A, T (B B ) = A A (notng G = (A A A A ) = (B B B B )). The space of geodescs has an sometry nvarant measure z w 4 dudvdxdy (w = u + v, z = x + y) and snce T s a bjecton defned pecewse by sometres, ths measure s T -nvarant on G. The pushforward of ths measure onto the frst and second coordnates wll provde nvarant measures µ A, µ B for T A and T B dudv A dµa = fa(w)dudv = z w 4 dxdy w A dudv A z w 4 dxdy w A, dxdy B dµ B = f B (z)dxdy = z w 4 dudv w B dxdy B z w 4 dudv w B. Computng these ntegrals on the trangular (A ) and quadrangular (B ) regons gves hyperbolc area π f A (w) = 4(v w A 2) 2 1 π w A, 4v 2 2 d = w (1/2 + 5 2/8), ρ = 2/8 w A 3 d = w (1/2 + 3 2/8), ρ = 2/8 w A πρ 2 4 f A (w) = (ρ 2 d 2 ) 2, d, ρ = d = w (1 + 3 2/4), ρ = 2/4 w A 5 d = w 3 2/4, ρ = 2/4 w A, 6 d = w (1 + 2/4), ρ = 2/4 w A 7 d = w 2/4, ρ = 2/4 w A 8 2
f B (z) = f B (z) = πρ 2 (ρ 2 d 2 ) 2, d, ρ = π 4(x 1) 2 z B 3 π 4x 2 z B 4 d = z (1/2 + 2), ρ = 1/2 w B 1 d = z 1/2, ρ = 1/2 w B 2 d = z (1/2 + 2), ρ = 1/4 w B 5 d = z (1/2 + 2), ρ = 1/4 w B 6 wth the mass of each pece beng µ A (A ) = π2 /4, µ B (B ) = π2 /2. On the crcular regons A, B, we have G(u, v) w A f A (w) = 4, G(mw) mw w A,, where f B (z) = H(x, y) z B2 H(nz) nz z B, h(a, b) = arctan(a/b) 4a 2 1 4ab, H(a, b) = h(a, b) + h(1 a, b) + h(a 2 a + b 2, b), G(a, b) = H(b/ 2, a/ 2) + H((b 1)/ 2, a/ 2) and the m and n are chosen to take A A or B B to A 4 A 4 or B 2 B 2. The ntegrals are obtaned by notng that for any moebus transformaton m φ(z) = z w 4 dudv φ(mz) mz = z w 4 dudv. mr The followng act as permutatons of A A or B B and generate the symmetres (octahedral/cubc): n (3574) = m (1326) = 1 w 2, m (165) = 1 z 2, n (167) = n (458) = R 1 w 1, m (12) = w + 2, m (34)(56) = w + 1, 1 z 1, n (12)(34)(57)(68) = z + 2, n (58)(67) = z + 1. The mass of the crcular regons s µ A (A ) = π 2 /2, µ B (B ) = π 2 /4. Hence the total masses are µ A (C) = 5π 2, µ B (C) = 11π 2 /2 Notes Relatng the maps to Schmdt s n [7], we have (τ complex conjugaton) s 1 = G 2 4 ( τ), s 2 = G 2 2 ( τ), s 3 = G 2 1 ( τ), s 4 = τ, s 5 = G 2 3 ( τ), s 6 = Z 2 ( τ), smlar to the gaussan case. t 1 = V 2 1 τ, t 2 = τ, t 3 = C 2 τ, t 4 = W 2 1 τ, t 5 = W 2 2 τ, t 6 = V 2 2 τ, t 7 = V 2 3 τ, t 8 = W 2 3 τ, 3
A 1 2 A1 1 + 2 A 8 A 5 A 3 A 4 A 6 A 5 A 3 A 4 A 6 A 7 A 2 0 1 A 2 4
2 B 1 1 + 2 B 1 B 8 B 5 B 3 B 4 B 6 B 5 B 3 B 4 B 6 B 7 B 2 0 1 B 2 5
6
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References [1] Nakada, Htosh, On Ergodc Theory of A. Schmdt s Complex Contnued Fractons over Gaussan Feld, Mh. Math. 105, 131 150 (1988) [2] Nakada, Htosh The metrcal theory of complex contnued fractons, Acta Arth. 56 (1990), no. 4, 279 289 [3] Nakada, Htosh On metrcal theory of Dophantne approxmaton over magnary quadratc feld, Acta Arth. 51 (1988), no. 4, 399 403 [4] Asmus L. Schmdt, Dophantne Approxmaton of Complex Numbers, Acta Math. 134, 1 85 (1975) [5] Asmus L. Schmdt, Ergodc Theory for Complex Contnued Fractons, Mh. Math. 93, 39 62 (1985) [6] Schmdt, A. L., Farey trangles and Farey quadrangles n the complex plane, Math. Scand., 21 (1967), 241 295 [7] A. L. Schmdt, Dophantne approxmaton n the feld Q( 2), Journal of Number Theory 131 (2011), 1983 2012 8