Read-back: hryniewicz_criterion

Binders and hypotheses. Let $H:\mathbb R^4\to\mathbb R$ be any function, and assume:

• $(hS)$ the level $\Sigma=H^{-1}(1)$ is a strictly star-shaped level of $H$ ($H$ is $C^\infty$, each ray from the origin meets $\Sigma$ exactly once, and $DH(x)\,x>0$ on $\Sigma$);
• $(hdc)$ $H$ is dynamically convex (every periodic orbit, of every period, with every linearized flow, has index $\mu\ge3$; see below);
• $P$ is a periodic orbit of $X_H$ on $\Sigma$ (a trajectory $x_P$ with a period $T_P>0$);
• $(hP)$ $P$ is prime: $x_P(t)\neq x_P(0)$ for $0<t<T_P$.

Conclusion (an equivalence).

$$\Bigl(P \text{ bounds a disk-like global section}\Bigr)\iff\Bigl(P\text{ is unknotted}\ \text{ and }\ P \text{ has self-linking number } -1\Bigr),$$

where:

• left side: there exists $e:\mathbb R^2\to\mathbb R^4$, $C^\infty$, injective with injective derivative on $\overline{\mathbb D}$, with $e(\overline{\mathbb D})\subseteq\Sigma$, $e(S^1)=\operatorname{im}P$, $X_H(e(u))\notin\operatorname{range}De(u)$ for all $u$ in the open disk $\mathbb D$, and such that every trajectory $y$ with $y(0)\notin\operatorname{im}P$ meets $e(\overline{\mathbb D})$ at some time $>a$ and at some time $<a$, for every $a\in\mathbb R$;
• right side, first part: there exists a $C^\infty$ $e:\mathbb R^2\to\mathbb R^4$, injective with injective derivative on $\overline{\mathbb D}$, with $e(\overline{\mathbb D})\subseteq\Sigma$ and $e(S^1)=\operatorname{im}P$;
• right side, second part: there is $\varepsilon_0>0$ such that for every $\varepsilon\in(0,\varepsilon_0)$ the loops

$$\ell_P(s)=\frac{x_P(T_Ps)}{|x_P(T_Ps)|},\qquad \ell_P^\varepsilon(s)=\frac{x_P(T_Ps)+\varepsilon\,\xi_1(x_P(T_Ps))}{\bigl|x_P(T_Ps)+\varepsilon\,\xi_1(x_P(T_Ps))\bigr|}$$

have linking number $-1$ (some unit $N$ avoided by both loops on $\mathbb R$ exists, and for every such $N$ the Gauss integral $G_N(\ell_P,\ell_P^\varepsilon)=-1$).

Edge cases. The left side contains no orientation condition and no condition on the direction of the boundary orbit; the self-linking number on the right is computed with the specific frame vector $\xi_1$ and with the given parametrization $s\mapsto x_P(T_Ps)$, $s\in[0,1]$. Both directions of the equivalence are asserted under $(hS)$, $(hdc)$, $(hP)$.

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Unfolded definitions used above

Throughout, $\mathbb R^4$ is the space of functions $\{0,1,2,3\}\to\mathbb R$ (with the sup norm as its normed-space structure), points are written $x=(x_0,x_1,x_2,x_3)$, and $\mathbb R^2$ (points $u=(u_0,u_1)$) is likewise $\{0,1\}\to\mathbb R$. $DH(x)$ denotes the Fréchet derivative of $H$ at $x$, which is by convention the zero map wherever $H$ is not differentiable; similarly for every other derivative below. Divisions by $0$ and inverses of $0$ evaluate to $0$.

Basic objects.

• Liouville form and standard symplectic form:

$$\lambda_x(v)=\tfrac12\,(x_0v_1-x_1v_0+x_2v_3-x_3v_2),\qquad \omega_0(u,v)=u_0v_1-u_1v_0+u_2v_3-u_3v_2 .$$

• Euclidean inner product and norm: $\langle x,y\rangle=\sum_{i=0}^3 x_iy_i$, $|x|=\sqrt{\langle x,x\rangle}$.
• Partial derivatives $\partial_iH(x)=DH(x)\,e_i$ ($e_i$ the $i$-th standard basis vector), and the Hamiltonian vector field

$$X_H(x)=\bigl(-\partial_1H(x),\ \partial_0H(x),\ -\partial_3H(x),\ \partial_2H(x)\bigr).$$

• Energy surface $\Sigma=\{x\in\mathbb R^4: H(x)=1\}$.
• Strictly star-shaped level (hypothesis $hS$): (i) $H:\mathbb R^4\to\mathbb R$ is $C^\infty$ on all of $\mathbb R^4$; (ii) for every $u\neq 0$ there is exactly one $r>0$ with $H(ru)=1$; (iii) for every $x$ with $H(x)=1$, $DH(x)\,x>0$. (Consequence of (iii): $H(0)\neq 1$, so $0\notin\Sigma$, and $X_H\neq 0$ on $\Sigma$.)
• Trajectory: a map $y:\mathbb R\to\mathbb R^4$ that is differentiable at every $t\in\mathbb R$ with $y'(t)=X_H(y(t))$, and with $H(y(t))=1$ for all $t\in\mathbb R$.
• Periodic orbit $P$: a pair $(x_P,T_P)$ consisting of a trajectory $x_P:\mathbb R\to\mathbb R^4$ and a number $T_P>0$ with $x_P(t+T_P)=x_P(t)$ for all $t$. $T_P$ is a period, not necessarily the minimal one; the same geometric orbit gives many different $P$ (any time shift, any positive multiple of the minimal period). Its image is $\operatorname{im}P=\{x_P(t):t\in\mathbb R\}$.
• Prime: $x_P(t)\neq x_P(0)$ for all $0<t<T_P$ (i.e. $T_P$ is the minimal period).
• Two fixed linear maps $Q_1(x)=(-x_2,\,x_3,\,x_0,\,-x_1)$ and $Q_2(x)=(x_3,\,x_2,\,-x_1,\,-x_0)$, and the frame vectors

$$\xi_1(x)=\frac{1}{|x|}\Bigl(Q_2x-\frac{DH(x)\,Q_2x}{DH(x)\,x}\,x\Bigr),\qquad \xi_2(x)=\frac{1}{|x|}\Bigl(Q_1x-\frac{DH(x)\,Q_1x}{DH(x)\,x}\,x\Bigr)$$

(with the junk convention $1/0=0$).

Dynamical convexity (hypothesis $hdc$).

• $\rho(s)=(\cos s,\sin s)\in\mathbb R^2$.
• Given a path of real $2\times2$ matrices $\varphi:\mathbb R\to M_2(\mathbb R)$ and $s\in\mathbb R$, an angle lift from $s$ is a function $\theta:\mathbb R\to\mathbb R$, continuous on $[0,1]$, with $\theta(0)=s$, such that for every $t\in[0,1]$ there is $r>0$ with $\varphi(t)\,\rho(s)=r\,\rho(\theta(t))$. (No continuity or invertibility of $\varphi$ is assumed; if $\varphi(t)\rho(s)=0$ for some $t\in[0,1]$, no lift from $s$ exists.)
• Winding set $W(\varphi)=\bigl\{\tfrac{\theta(1)-s}{2\pi}:\ s\in[0,2\pi],\ \theta \text{ an angle lift of }\varphi\text{ from }s\bigr\}\subseteq\mathbb R$.
• Index: let $a=\inf W(\varphi)$, $b=\sup W(\varphi)$ (each set to $0$ if $W(\varphi)$ is empty or unbounded on that side), and $k=\lceil b\rceil$ (least integer $\ge b$). Then

$$\mu(\varphi)=\begin{cases}2(k-1)&\text{if } a\le k-1,\\ 2k-1&\text{otherwise.}\end{cases}$$

In particular, if $W(\varphi)=\emptyset$ then $a=b=0$, $k=0$ and $\mu(\varphi)=-1$.

• A linearized flow along a curve $x:\mathbb R\to\mathbb R^4$ is a map $Y:\mathbb R\to L(\mathbb R^4,\mathbb R^4)$ (continuous linear maps) with $Y(0)=\mathrm{Id}$ and, for every $t\in\mathbb R$, $Y$ differentiable at $t$ with $Y'(t)=DX_H(x(t))\circ Y(t)$, where $DX_H$ is the Fréchet derivative of the vector field $X_H$.
• Projection along $X_H$: $\pi_x(v)=v-\dfrac{\lambda_x(v)}{\lambda_x(X_H(x))}\,X_H(x)$.
• Coordinates: $c_x(v)=\bigl(\omega_0(v,\xi_2(x)),\ \omega_0(\xi_1(x),v)\bigr)\in\mathbb R^2$.
• For a periodic orbit $P$ and a map $Y$, the matrix path $\Phi_{P,Y}:\mathbb R\to M_2(\mathbb R)$ has $j$-th column ($j=1,2$)

$$\Phi_{P,Y}(\tau)\,e_j=c_{x_P(T_P\tau)}\Bigl(\pi_{x_P(T_P\tau)}\bigl(Y(T_P\tau)\,\xi_j(x_P(0))\bigr)\Bigr).$$

• $H$ is dynamically convex means: for every periodic orbit $P$ (any period $T_P$, so including all multiple covers and time shifts) and every linearized flow $Y$ along $x_P$, $\mu(\Phi_{P,Y})\ge 3$. (If for some $P$ no linearized flow exists, the condition is vacuous for that $P$.)

Disks and global sections.

• $\overline{\mathbb D}=\{u: u_0^2+u_1^2\le1\}$, $\mathbb D=\{u:u_0^2+u_1^2<1\}$, $S^1=\{u:u_0^2+u_1^2=1\}$, and $c(t)=(\cos 2\pi t,\ \sin2\pi t)$.

• Smooth disk embedding $e:\mathbb R^2\to\mathbb R^4$: $e$ is $C^\infty$ on all of $\mathbb R^2$, injective on $\overline{\mathbb D}$, and $De(u)$ is injective for every $u\in\overline{\mathbb D}$.

• Disk-like global section map for $(H,P)$: a map $e:\mathbb R^2\to\mathbb R^4$ such that

  1. $e$ is a smooth disk embedding;
  2. $e(\overline{\mathbb D})\subseteq\Sigma$;
  3. $e(S^1)=\operatorname{im}P$ (set equality);
  4. for every $u\in\mathbb D$ (open disk), $X_H(e(u))\notin \operatorname{range} De(u)$;
  5. for every trajectory $y$ with $y(0)\notin\operatorname{im}P$ and every $a\in\mathbb R$, there exist $t>a$ with $y(t)\in e(\overline{\mathbb D})$ and $t'<a$ with $y(t')\in e(\overline{\mathbb D})$ (the hitting set is the image of the closed disk).

  No orientation condition is imposed.

• $P$ bounds a disk-like global section: some disk-like global section map for $(H,P)$ exists.

• $P$ is unknotted: there is a smooth disk embedding $e$ with $e(\overline{\mathbb D})\subseteq\Sigma$ and $e(S^1)=\operatorname{im}P$ (conditions 1–3 above only).

• Adapted disk open book for $(H,P)$: a map $F:\mathbb R\times\mathbb R^2\to\mathbb R^4$, $F(\vartheta,u)=F_\vartheta(u)$, such that

  1. $F$ is jointly $C^\infty$;
  2. $F_{\vartheta+1}=F_\vartheta$ for all $\vartheta$;
  3. every page $F_\vartheta$ is a disk-like global section map for $(H,P)$;
  4. (binding) $F_\vartheta(c(t))=x_P(T_P\,t)$ for all $\vartheta,t\in\mathbb R$;
  5. (positivity) for all $\vartheta$ and $u\in\mathbb D$, $\omega_0\bigl(\partial_{u_0}F_\vartheta(u),\ \partial_{u_1}F_\vartheta(u)\bigr)>0$;
  6. (fibration) every $y\in\Sigma\setminus\operatorname{im}P$ equals $F(\vartheta,u)$ for exactly one pair $(\vartheta,u)$ with $\vartheta\in[0,1)$, $u\in\mathbb D$;
  7. (regularity) for all $\vartheta\in\mathbb R$ and $u\in\mathbb D$, the derivative $DF(\vartheta,u):\mathbb R\times\mathbb R^2\to\mathbb R^4$ is injective.

  "$P$ has an adapted disk open book" means such an $F$ exists. (Since each page is injective on $\overline{\mathbb D}$, the binding condition forces $t\mapsto x_P(T_Pt)$ to be injective on $[0,1)$, i.e. $P$ prime.)

Linking and self-linking.

• $\nu(x)=x/|x|$ (with $\nu(0)=0$).
• For $N\in\mathbb R^4$, the map $\sigma_N(y)=\dfrac{y-\langle y,N\rangle N}{1-\langle y,N\rangle}$ (equal to $0$ when $\langle y,N\rangle=1$), with values in $\mathbb R^4$.
• $\operatorname{vol}_N(a,b,c)=\det$ of the $4\times4$ matrix with rows $-N,a,b,c$.
• Gauss integral of two curves $\gamma_1,\gamma_2:\mathbb R\to\mathbb R^4$ with respect to $N$: writing $\alpha(s)=\sigma_N(\gamma_1(s))$, $\beta(t)=\sigma_N(\gamma_2(t))$,

$$G_N(\gamma_1,\gamma_2)=\frac1{4\pi}\int_0^1\!\!\int_0^1 \frac{\operatorname{vol}_N\bigl(\alpha'(s),\ \beta'(t),\ \alpha(s)-\beta(t)\bigr)}{|\alpha(s)-\beta(t)|^3}\,dt\,ds,$$

where derivatives are $0$ at non-differentiability points, the integrand is $0$ where $\alpha(s)=\beta(t)$, and an interval integral of a non-integrable function is $0$.

• $\gamma_1,\gamma_2$ have linking number $n\in\mathbb Z$: (a) there exists $N$ with $|N|=1$ such that $\gamma_1(s)\neq N$ and $\gamma_2(s)\neq N$ for all $s\in\mathbb R$; and (b) for every $N$ with $|N|=1$ avoiding both curves on all of $\mathbb R$ in that sense, $G_N(\gamma_1,\gamma_2)=n$. (Avoidance is required over all $s\in\mathbb R$; the integral uses only $[0,1]$. The curves are not required to be closed or smooth.)
• Orbit loop $\ell_P(s)=\nu\bigl(x_P(T_Ps)\bigr)$; push-off loop $\ell^{\varepsilon}_P(s)=\nu\bigl(x_P(T_Ps)+\varepsilon\,\xi_1(x_P(T_Ps))\bigr)$. Over $s\in[0,1]$ these traverse the period $T_P$, which is the minimal period only if $P$ is prime.
• $P$ has self-linking number $n$: there is $\varepsilon_0>0$ such that for every $\varepsilon\in(0,\varepsilon_0)$, the curves $\ell_P,\ell_P^\varepsilon$ have linking number $n$ (in the sense above).
• $P$ links $Q$ nontrivially: there is an integer $n\neq0$ such that $\ell_P,\ell_Q$ have linking number $n$.