Local contact sub-Finslerian geometry for maximum norms in dimension 3

Figure 1.  Evolution of the front at $r\neq0$ fixed. In red dot lines and in black the extremals with initial speed $G_1$, in full line the front at 4 different times, with four colors corresponding to the four possible initial speeds

Figure 2.  The conjugate locus and three points of view of the non singular part of the sphere in the nilpotent case

Figure 3.  $C_1 > 0$ and $C_2 > 0$: closure of the cut locus at $z$ fixed

Figure 4.  $C_1 > 0$ and $C_2 > 0$: closure of the cut locus at $z$ fixed

Figure 5.  The upper part of the cut locus

Figure 6.  The front before $t = 8\rho$ when $C_1 > 0$ and $C_2 < 0$

Figure 7.  $C_1 > 0$ and $C_2 < 0$ : picture of the front at times with $T_2 = 0$ and $T_3 < T_{3c}$, $T_3 = T_{3c}$ and $T_3 = T_{3b}$ when $4 b_{110}+8c_{110}c_{200}-8c_{210}+4C_2 < 0$

Figure 8.  $C_1 > 0$ and $C_2 < 0$: picture of the front at times with $T_2 = 0$ and $T_3 < T_{3c}$, $T_3 = T_{3c}$ and $T_3 = T_{3g}$ when $4 b_{110}+8c_{110}c_{200}-8c_{210} < 0$ and $4 b_{110}+8c_{110}c_{200}-8c_{210}+4C_2 > 0$

Figure 9.  Picture of the cut locus when $C_1 > 0$ and $C_2 < 0$

Figure 10.  The front before $t = 8\rho$ when $C_1 < 0$ and $C_2 < 0$

Figure 11.  $C_1 < 0$ and $C_2 < 0$: evolution of the front when $|T_{3e}-T_{3f}| > \tau_3$

Figure 12.  $C_1 < 0$ and $C_2 < 0$: evolution of the front when $|T_{3e}-T_{3f}| < \tau_3$

Figure 13.  Possible cut loci when $ C_1<0 $ and $ C_2<0 $

Figure 14.  Extremals when $|f_{51}| < f_{41}$

Figure 15.  Extremals when $|f_{51}| < -f_{41}$

Figure 16.  Extremals when $|f_{41}| < f_{51}$

Figure 17.  Extremals when $|f_{41}| < -f_{51}$

Figure 18.  Part of the cut locus generated by the extremal with $\lambda_z(0)\sim0$ when $|f_{41}| < -f_{51}$ and $|f_{52}| < f_{42}$