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A "non-commutative Fibonacci" (or "reverse Fibonacci") sequence. Often written as: a, b, ab, bab, abbab, bababbab, abbabbababbab, bababbababbabbababbab, abbabbababbabbababbababbabbababbab, bababbababbabbababbababbabbababbabbababbababbabbababbab, ...
Converges in the appropriate topology. - Dylan Thurston, Jan 28 2005
Do a web search on bababbababbabbababbab to get further links.
Comments from A. N. W. Hone, Jan 28 2005: [Start] Write the recurrence symbolically as g_{n+1} = g_{n-1}g_n. Then the determinant d_n = det g_n is given by d_n = d_0^{f_{n-2}} d_1^{f_{n-1}} where f_{n+1} = f_n+f_{n-1}, f_0 = f_1 = 1 are the Fibonacci numbers.
To avoid getting involved with the Baker-Campbell-Hausdorff identity, I now restrict to SL(2), or to make life easier make it SU(2) (which is isomorphic over C). Then we can explicitly write g as an exponential of Lie algebra elements:
g_n = exp (i theta_n v_n cdot sigma ), where theta_n is an angle, v_n is a unit vector and sigma = ( sigma_1, sigma_2, sigma_3)^T is a vector of Pauli spin matrices.
Moreover the adjoint action on su(2) (viewing the coordinates in su(2) as giving points in 3D space) means that g_n gives a rotation through - theta_n /2 about the v_n axis.
So from the double cover of SO(3) by SU(2), we can view the g_n as a sequence of "Fibonacci rotations."
Furthermore, in SU(2) we can write explicitly g_n = cos theta_n + i sin theta_n v_n cdot sigma so the recurrence can be decoupled as
cos theta_{n+1} = cos theta_n + cos theta_{n-1} - sin theta_{n-1} sin theta_n (v_{n-1} cdot v_n),
sin theta_{n+1} v_{n+1} = cos theta_{n-1} sin theta_n v_n + cos theta_n sin theta_{n-1} v_{n-1} - sin theta_{n-1} sin theta_n ( v_{n-1} wedge v_n ) [End]
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