The Maximal Number of Transverse Self-Intersections of Geodesics ...

4
y
2
–3 –2 –10
1 2 3
x
4 5
–2
–4
Lemma 3.10 Words of the form W = a k b n 2 a n 2 −k , 2 ≤ k ∈ Z ≤ n − 2 have roots that follow
the fixed point iteration of H = z +1
z +2 .
Proof. By symmetry (Lemma 3.6) and by Lemma 3.9.
Lemma 3.11 Words with n ≥ 4, neven of the form W = b n 2 −1 a n 2 b have roots in [−1, 0]
and [1, ∞).
Outline of proof: Take a piece of the word, for instance W 0 = a n 2 −1 b n 2 −1 . Find the
roots of this. The roots start out in (0.565,φ− 1) and (φ, 1.768), by the previous lemma.
By looking at the iteration patterns of the b(x) function, we see that one root converges to
−φ +1, while the other is diverging, and always greater than 1.
Lemma 3.12 Words with n ≥ 4, neven of the form W = a n 2 −1 b n 2 a have roots in [0, 1] and
(−∞, −1].
Proof. By the previous lemma, since W has symmetric roots (Lemma 3.6), the roots of
W = a n 2 −1 b n 2 a are in the negative of the regions of b n 2 −1 a n 2 b.
3.2
14