Confirming Brennan’s Conjecture Numerically on a Counterexample to Thurston’s K = 2 Conjecture

Figures & data

Fig. 1 The gap $G_{\theta }$ between $\Phi ={\overline{f}}_{\theta }(\text{id})\Phi$ and ${\overline{f}}_{\theta }(B)\Phi$ in the image of ${\overline{\psi }}_{\theta }$.

Fig. 2 Domain Ω.

Fig. 3 The crosses represent values of parameters ${\lambda }_i^A,a_i^A$ of a disk automorphism estimated using smaller 100-point polygonal approximations ${\widehat{\Omega }}_i^{\mathrm{s}}$ to Ω. The circles are the estimates of λ^A and a^A obtained from $\widehat{\Omega }$.

Fig. 4 The crosses represent values of parameters ${\lambda }_i^B,a_i^B$ of a disk automorphism estimated using smaller 100-point polygonal approximations ${\widehat{\Omega }}_i^{\mathrm{s}}$ to Ω. The circles are the estimates of λ^B and a^B obtained from $\widehat{\Omega }$.

Fig. 5 The plot of $\hspace{0.17em}\log \hspace{0.17em}{S}_n^{(4)}$ as a function of n for $n\le 17$. The solid line is the least-squares best linear fit to the shown data points.

Fig. 6 The plots of $\hspace{0.17em}\log \hspace{0.17em}{S}_n^{(p)}$ as a function of n for $n\le 16$ and $p=5.52,5.54$. The solid lines are the least-squares best linear fits to the data points $(n,\hspace{0.17em}\log \hspace{0.17em}{S}_n^{(p)})$ for $n\ge 6$, for $p=5.52,5.54$.