# Thursday, February 17 :::{.remark} Let $\Sigma \subseteq (Y^3, \xi)$. - Characteristic foliation: $F = \xi \intersect \T\Sigma$, complicated but necessary - Dividing set: a multicurve, simpler ::: :::{.theorem title="?"} If $\Sigma$ is convex with a dividing set $\Gamma$ and $F$ is any foliation divided by $\Gamma$, there is a $C^0\dash$small isotopy $\phi_t$ wt - $\phi_0(\Sigma) = \Sigma, \phi_t(\Gamma) = \Gamma$ - $\phi_t(\Sigma)$ is convex for all $t \in [0, 1]$ - The characteristic foliation of $\phi_1(\Sigma)$ is $F$. ::: :::{.remark} Idea: dividing sets give ways to detect overtwisted contact structures. ::: :::{.remark} If $\Sigma = S^2$ and $\size \Gamma \geq 2$, then $(Y, \xi)$ is overtwisted. Recall that an overtwisted disc is an embedded $D^2$ with Legendrian boundary such that $\tb(\bd D) = 0$ and $\mathrm{tw}(\xi, \bd D)$. ![](figures/2022-02-17_11-24-46.png) Spheres can have exactly one dividing component. ::: :::{.exercise title="?"} Generalize to an arbitrary number of components $\size \Gamma = n$. ::: :::{.remark} Same if $\Sigma \neq S^2$ and $\Gamma$ contains a contractible curve. Contrapositively, if $(Y, \xi)$ is tight, then either - $\Sigma = S^2$ and $\Gamma$ is connected, or - $\Sigma \neq S^2$ and $\Gamma$ has no contractible components. ::: :::{.exercise title="?"} Consider tight contact structures on $S^3$. Choose Darboux $B^3$ neighborhoods at the ends, and note the interior is $S^2\times [0, 1]$: ![](figures/2022-02-17_11-37-35.png) The $S^3\times \ts{t_0}$ slices can be perturbed to be complex. So there is only one tight contact structure on $S^3$. ::: :::{.remark} What can $F$ look like on an $S^2$ in a tight $(Y,\xi)$? $F$ can be perturbed to be Morse-Smale. - There are a finite number of elliptic/hyperbolic singularities - There are nondegenerate periodic orbits, either attracting or repelling - There are no saddle-saddle arcs - The limit sets are singularities or periodic orbits Dimension 3: strange attractors! Two types of limit sets: - $\omega$ limit sets: $x\in Y$ where there exists a sequence $\ts{t_1 < \cdots }$ with $\phi(t_k)\to x$. - $\alpha$ limit sets: $x\in Y$ where there exists a sequence $\ts{t_1 > \cdots }$ with $\phi(t_k)\to x$. ::: :::{.remark} For $S^2$, take $S^+$ with an outward pointing vector field. ![](figures/2022-02-17_11-50-05.png) There are no periodic orbits since $(Y, \xi)$ is tight. The only limit sets are singular points. $\chi(D) = 1 = \size e - \size h$. Stable manifold of $h$: $\Stab_h$ are $x\in D^2$ such that there exists a flow like with $\phi(0) = x$ and $\phi(t) \to h$ Form a 1-complex $\Union_h \cl_X(\Stab_h)$ -- this contains no cycles, thus this is a tree, and the dividing set is a neighborhood of the tree. ![](figures/2022-02-17_11-58-12.png) ::: :::{.proposition title="?"} If $F$ on $\Sigma$ is Morse-Smale, then it admits dividing curves. ::: :::{.proof title="?"} Let $G = \Union_h \cl(\Stab_h) \union \Union_e e_t$ along with all of the repelling periodic orbits. Then $\Gamma = \bd \nu(G)$ divides $F$. ::: :::{.theorem title="?"} If $\Sigma$ is orientable, then there is a $C^\infty$ small perturbation of $F$ such that it is Morse-Smale. ::: :::{.proposition title="?"} Every oriented $\Sigma \subseteq (Y, \xi)$ can be perturbed to be convex. ::: :::{.proof title="?"} Near $\Sigma$, $\alpha = \beta_t + \alpha_t \dt$ and $\beta_0$ define $F$. By Peixoto there exists $\tilde \beta_t$ such that$\tilde \beta_t$ defines a Morse-Smale $F$. For $\norm{\beta - \tilde\beta}_{C^\infty} \ll \eps$, $\tilde \alpha = \tilde \beta_t + \alpha_t \dt$ is contact. Then $\alpha_s = s\tilde\alpha + (1-s)\alpha$ is a path of contact forms, so by Gray stability there is an isotopy $\phi_s$ such that $\phi_s^*(\alpha_s) = \lambda_s \alpha$ and we can take $\phi_1(\Sigma)$ to be our surface. ::: :::{.proposition title="?"} If $(\Sigma, \tilde F)$ admits dividing curves, then it is convex. :::