On hyperbolic once-punctured-torus bundles IV: Automata for lightning curves

Let (A, B, C):=(A, B, C, D|A2=B2=C2=ABCD=1). Let R and L denote the automorphisms of (A, B, C) determined by (A,B,C)R=(A,BCB,B), (A,B,C)L=(B,BAB,C). Let (a1, b1, a2, b2, ap, bp) be a non-empty, even-length, positive-integer sequence, let F denote Ra1Lb1Ra2Lb2.RapLbp, and let (A, B, C, F) denote the semidirect product (F|).(A, B, C). In an influential unfinished work, Jørgensen constructed a discrete faithful representation ρF:(A,B,C,F)→PSL2(C). The group (A, B, C, F) then acts conformally on the Riemann sphere Ĉ via ρF. Using results of Thurston, Minsky, McMullen, Bowditch, and others, Cannon-Dicks showed that Ĉ has a CW-structure formed from three closed two-cells, denoted [A], [B] and [C], that are Jordan disks satisfying the ping-pong conditions A[A]=[B]∪[C], B[B]=[C]∪[A], and C[C]=[A]∪[B]. Further, Cannon-Dicks expressed the resulting theta-shaped one-skeleton as the union of two arcs, denoted ∂-A and ∂+B, and expressed each of these lightning curves as limit sets of finitely generated subsemigroups of (A, B, C, F). The foregoing results had previously been obtained by Alperin-Dicks-Porti for F=RL by elementary methods. Independently, Mumford, Scorza, Series, Wright, and others studied more general lightning curves that arise as limits of sequences of finite chains of round disks in Ĉ Later, Cannon-Dicks showed that the set of (D, F)-translates of ∂-A∪∂+B gives a tessellation CW(F) of C with tiles that are Jordan disks.In this article, we find that classic Adler-Weiss automata codify ∂-A and ∂+B in terms of ends of trees. The ∂+B-automaton distinguishes a tree of words in a certain finite alphabet S that is a subset of (A, B, C, F). The ∂-A-automaton distinguishes a tree of words in the finite alphabet S-1. The automata allow depth-first searches which give drawings of ∂-A and ∂+B that, while requiring less computer time and memory, are more detailed than those that have hitherto been obtained.We show that the limit set of the semigroup generated by S is ∂+B and the limit set of the semigroup generated by S-1 is ∂-A. We use this to show that the Hausdorff dimensions of ∂-A and ∂+B are equal. We raise the problem of whether or not the common Hausdorff dimension can be calculated by applying a famous technique of McMullen to the ∂-A-automaton.We note that the improved drawings of ∂-A and ∂+B give improved drawings of the planar tessellation CW(F). We review Riley's sufficient condition for the columns of CW(F) to be vertical. We review Helling's description of Jørgensen's ρRLn and Hodgson-Meyerhoff-Weeks' ρRL∞, and we draw CW(RL100) together with something we call CW(RL∞). © 2011 Elsevier B.V.