Earthquakes on the Once-Punctured Torus

Figures & data

Fig. 1 Level set $\kappa (x,y,z)=-2$ in ${\mathbb{R}}^3$.

Fig. 2 Example forward and backward earthquake deformations about α (red), β (blue), and $\alpha \beta$ (green) on ${\mathcal{T}}_{\text{tr}}$. Starting points are given by the respective sets ${\mathcal{S}}_{\alpha }$, ${\mathcal{S}}_{\beta }$, and ${\mathcal{S}}_{\alpha \beta }$ from equation (3.4)(3.4) $$\begin{array}{cc}{\mathcal{S}}_{\alpha }=\hfill & \left\{\left(3,3,3\right),\left(2\sqrt{2},2\sqrt{2},4\right),\left(10,10,-10\left(-5+\sqrt{23}\right)\right)\right\},\hfill \\ {\mathcal{S}}_{\beta }=\hfill & {\Sigma }_{\text{Rot}}^{-1}{\mathcal{S}}_{\alpha },\hfill \\ {\mathcal{S}}_{\alpha \beta }=\hfill & {\Sigma }_{\text{Rot}}{\mathcal{S}}_{\alpha }.\hfill \end{array}$$(3.4) .

Fig. 3 Maximal collar neighborhood for α on $S_{1,1}$ (left) and unfolded into a tube (right).

Fig. 4 The maximal collar neighborhood for α on $S_{1,1}$ unfolded into a tube (left) and half of the neighborhood unfolded in the universal cover ${\mathbb{H}}^2$ (right).

Fig. 5 Example forward and backward earthquake deformations about α (red), β (blue), and $\alpha \beta$ (green) on the image of ${\mathcal{T}}_{\text{tr}}$ in ${\mathbb{R}}_{+}^3$ using triangle lengths. Starting points are given by applying ν from equation (3.5)(3.5) $$\begin{array}{cc}\nu :{\mathcal{T}}_{\text{tr}}\hfill & \to {\mathcal{T}}_{\mathcal{l}}\hfill \\ \left(\begin{array}{c}x\\ y\\ z\end{array}\right)\hfill & \mapsto \left(\begin{array}{c}{\cosh }^{-1}\left(\frac{x}{2}\right)\\ {\cosh }^{-1}\left(\frac{y}{2}\right)\\ {\cosh }^{-1}\left(\frac{z}{2}\right)\end{array}\right)\hfill \end{array}$$(3.5) to the respective sets ${\mathcal{S}}_{\alpha },{\mathcal{S}}_{\beta }$, and ${\mathcal{S}}_{\alpha \beta }$ from equation (3.4)(3.4) $$\begin{array}{cc}{\mathcal{S}}_{\alpha }=\hfill & \left\{\left(3,3,3\right),\left(2\sqrt{2},2\sqrt{2},4\right),\left(10,10,-10\left(-5+\sqrt{23}\right)\right)\right\},\hfill \\ {\mathcal{S}}_{\beta }=\hfill & {\Sigma }_{\text{Rot}}^{-1}{\mathcal{S}}_{\alpha },\hfill \\ {\mathcal{S}}_{\alpha \beta }=\hfill & {\Sigma }_{\text{Rot}}{\mathcal{S}}_{\alpha }.\hfill \end{array}$$(3.4) .

Fig. 6 Example forward and backward earthquake deformations about α (red), β (blue), and $\alpha \beta$ (green) using Fenchel-Nielsen length-twist coordinates. Starting points are given by applying ζ from equation (3.9)(3.9) $$\begin{array}{cc}\zeta :{\mathcal{T}}_{\text{tr}}\hfill & \to {\mathcal{T}}_{\text{FN}}\hfill \\ \left(\begin{array}{c}x\\ y\\ z\end{array}\right)\hfill & \mapsto \left(\begin{array}{c}2{\cosh }^{-1}\left(\frac{x}{2}\right)\\ 2{\cosh }^{-1}\left(\frac{y}{2\coth \left({\cosh }^{-1}\left(\frac{x}{2}\right)\right)}\right)\end{array}\right).\hfill \end{array}$$(3.9) to the respective sets ${\mathcal{S}}_{\alpha },{\mathcal{S}}_{\beta }$, and ${\mathcal{S}}_{\alpha \beta }$ from equation (3.4)(3.4) $$\begin{array}{cc}{\mathcal{S}}_{\alpha }=\hfill & \left\{\left(3,3,3\right),\left(2\sqrt{2},2\sqrt{2},4\right),\left(10,10,-10\left(-5+\sqrt{23}\right)\right)\right\},\hfill \\ {\mathcal{S}}_{\beta }=\hfill & {\Sigma }_{\text{Rot}}^{-1}{\mathcal{S}}_{\alpha },\hfill \\ {\mathcal{S}}_{\alpha \beta }=\hfill & {\Sigma }_{\text{Rot}}{\mathcal{S}}_{\alpha }.\hfill \end{array}$$(3.4) .

Fig. 7 Illustration of map $E_{\mathbf{p}}$ from equation (3.11)(3.11) $$\begin{array}{cc}{E}_{\mathbf{u}}:\mathcal{M}\mathcal{L}\subseteq H_1\left(S_{1,1};\mathbb{R}\right)\cong {\mathbb{R}}^2\hfill & \to \mathcal{E}\subseteq {\mathcal{T}}_{\text{FN}}\hfill \\ \mathcal{L}\hfill & \mapsto E_{\mathcal{L}}^{\text{FN}}(\mathbf{u}),\hfill \end{array}$$(3.11) taking laminations with basepoint (0, 0) in ${\mathbb{R}}^2$ (top left) to forward earthquakes with starting point p in ${\mathcal{T}}_{\text{FN}}$ (top right) going via the space of projective laminations (bottom). Clockwise, example laminations are given by simple closed curves β (blue), $\alpha {\beta }^5,{\alpha }^2{\beta }^3,\alpha \beta$, ${\alpha }^3{\beta }^2,{\alpha }^5\beta$, α (red), ${\alpha }^5{\beta }^{-1},{\alpha }^3{\beta }^{-2},\alpha {\beta }^{-1}$, ${\alpha }^2{\beta }^{-3},\alpha {\beta }^{-5}$. This illustrates the behaviour from Remark 3.4.1.

Fig. 8 Some example earthquake deformations about ${\gamma }_n=\alpha \beta {\alpha }^{n-1}$ for in trace coordinates (top) and triangle length coordinates (bottom). Examples shown are for $n=1,2,3,4$ in green, yellow, orange, pink, respectively. Starting points are given by ${\mathcal{S}}_{\alpha }$ (top) and $\nu {\mathcal{S}}_{\alpha }$ (bottom) from equations (3.4)(3.4) $$\begin{array}{cc}{\mathcal{S}}_{\alpha }=\hfill & \left\{\left(3,3,3\right),\left(2\sqrt{2},2\sqrt{2},4\right),\left(10,10,-10\left(-5+\sqrt{23}\right)\right)\right\},\hfill \\ {\mathcal{S}}_{\beta }=\hfill & {\Sigma }_{\text{Rot}}^{-1}{\mathcal{S}}_{\alpha },\hfill \\ {\mathcal{S}}_{\alpha \beta }=\hfill & {\Sigma }_{\text{Rot}}{\mathcal{S}}_{\alpha }.\hfill \end{array}$$(3.4) and (3.5)(3.5) $$\begin{array}{cc}\nu :{\mathcal{T}}_{\text{tr}}\hfill & \to {\mathcal{T}}_{\mathcal{l}}\hfill \\ \left(\begin{array}{c}x\\ y\\ z\end{array}\right)\hfill & \mapsto \left(\begin{array}{c}{\cosh }^{-1}\left(\frac{x}{2}\right)\\ {\cosh }^{-1}\left(\frac{y}{2}\right)\\ {\cosh }^{-1}\left(\frac{z}{2}\right)\end{array}\right)\hfill \end{array}$$(3.5) .

Fig. 9 Some example earthquake deformations about ${\gamma }_n=T^n(\alpha )$ for $n=1,2,3,4$ in trace coordinates (top) and triangle length coordinates (bottom). Examples shown are for $n=1,2,3,4$ in green, turquoise, light blue, and cyan, respectively. Starting points are given by ${\mathcal{S}}_{\alpha }$ (top) and $\nu {\mathcal{S}}_{\alpha }$ (bottom) from equations (3.4)(3.4) $$\begin{array}{cc}{\mathcal{S}}_{\alpha }=\hfill & \left\{\left(3,3,3\right),\left(2\sqrt{2},2\sqrt{2},4\right),\left(10,10,-10\left(-5+\sqrt{23}\right)\right)\right\},\hfill \\ {\mathcal{S}}_{\beta }=\hfill & {\Sigma }_{\text{Rot}}^{-1}{\mathcal{S}}_{\alpha },\hfill \\ {\mathcal{S}}_{\alpha \beta }=\hfill & {\Sigma }_{\text{Rot}}{\mathcal{S}}_{\alpha }.\hfill \end{array}$$(3.4) and (3.5)(3.5) $$\begin{array}{cc}\nu :{\mathcal{T}}_{\text{tr}}\hfill & \to {\mathcal{T}}_{\mathcal{l}}\hfill \\ \left(\begin{array}{c}x\\ y\\ z\end{array}\right)\hfill & \mapsto \left(\begin{array}{c}{\cosh }^{-1}\left(\frac{x}{2}\right)\\ {\cosh }^{-1}\left(\frac{y}{2}\right)\\ {\cosh }^{-1}\left(\frac{z}{2}\right)\end{array}\right)\hfill \end{array}$$(3.5) .

Fig. 10 Example forwards (left) and backwards (right) earthquake deformations about ${\gamma }_n=T^n(\alpha )$ for $n=0,1,2,3$ and ${\delta }_n=T^n(\beta )$ for n = 0, 1, 2 with starting point $(2{\cosh }^{-1}\left(\frac{3}{2}\right),2{\cosh }^{-1}\left(\frac{\sqrt{5}}{2}\right))$. The dashed black line denotes an estimation of the limiting earthquake between γ₃ and δ₂.

Fig. 11 Forward and backward earthquake deformations about curves in $\mathcal{C}$ on ${\mathcal{T}}_{\text{tr}}$ in trace coordinates given starting points (3, 3, 3) (left), $\left(2\sqrt{2},2\sqrt{2},4\right)$ (centre), and $(10,10,-10(-5+\sqrt{23}))$ (right).

Fig. 12 Forward and backward earthquake deformations about curves in $\mathcal{C}$ on ${\mathcal{T}}_{\text{tr}}$ in spherical trace coordinates given starting points corresponding to trace coordinates (3, 3, 3) (left), $\left(2\sqrt{2},2\sqrt{2},4\right)$ (centre), and $(10,10,-10(-5+\sqrt{23}))$ (right).

Fig. 13 Forward and backward earthquake deformations about curves in $\mathcal{C}$ on ${\mathcal{T}}_{\text{tr}}$ in inverted trace coordinates given starting points corresponding to trace coordinates (3, 3, 3) (left), $\left(2\sqrt{2},2\sqrt{2},4\right)$ (centre), and $(10,10,-10(-5+\sqrt{23}))$ (right).

Fig. 14 Forward and backward earthquake deformations about curves in $\mathcal{C}$ on ${\mathbb{R}}_{+}^3$ in triangle length coordinates given starting points corresponding to trace coordinates (3, 3, 3) (left), $\left(2\sqrt{2},2\sqrt{2},4\right)$ (centre), and $(10,10,-10(-5+\sqrt{23}))$ (right).

Fig. 15 Forward and backward earthquake deformations about curves in $\mathcal{C}$ in Fenchel-Nielsen coordinates given starting points corresponding to trace coordinates (3, 3, 3) (left), $\left(2\sqrt{2},2\sqrt{2},4\right)$ (centre), and $(10,10,-10(-5+\sqrt{23}))$ (right).

Fig. 16 Forward and backward earthquake deformations about curves in $\mathcal{C}$ on the simplex given starting points corresponding to trace coordinates (3, 3, 3) (left), $\left(2\sqrt{2},2\sqrt{2},4\right)$ (centre), and $(10,10,-10(-5+\sqrt{23}))$ (right).

Fig. 17 Forward earthquake deformations about curves in $\mathcal{C}$ in Fenchel-Nielsen coordinates given starting points $({\cosh }^{-1}(3),0)$ (left), $({\cosh }^{-1}(3),1)$ (right).