Read-back: Def_HryniewiczCriterion_ConleyZehnder

Namespace HryniewiczCriterion; uses the notions of Def_HryniewiczCriterion_Basic: $\mathbb R^4$ with coordinates $x_0,\dots,x_3$; $\lambda_x(v)=\tfrac12(x_0v_1-x_1v_0+x_2v_3-x_3v_2)$; $\omega_0(u,v)=u_0v_1-u_1v_0+u_2v_3-u_3v_2$; $X_H(x)=(-\partial_1H,\partial_0H,-\partial_3H,\partial_2H)(x)$ (all partials $0$ where $H$ is not differentiable); periodic orbits $P=(x_P,T_P)$ with $T_P>0$ any (not necessarily minimal) period; and the vectors $\xi_1(x)=|x|^{-1}\bigl(Q_2x-\tfrac{DH_x(Q_2x)}{DH_x(x)}x\bigr)$, $\xi_2(x)=|x|^{-1}\bigl(Q_1x-\tfrac{DH_x(Q_1x)}{DH_x(x)}x\bigr)$ with $Q_1x=(-x_2,x_3,x_0,-x_1)$, $Q_2x=(x_3,x_2,-x_1,-x_0)$ (quotient $:=0$ if $DH_x(x)=0$; $\xi_i(0)=0$). For $x\neq0$, $\omega_0(\xi_1(x),\xi_2(x))=1$.

Throughout, $\varphi:\mathbb R\to\mathbb R^{2\times2}$ is an arbitrary function (no continuity, invertibility or symplecticity is assumed); only its values on $[0,1]$ are ever used.

rotationVector

$$e(s)=(\cos s,\ \sin s)\in\mathbb R^2,\qquad s\in\mathbb R .$$

IsAngleLift

For $\varphi$, a starting angle $s\in\mathbb R$ and a function $\theta:\mathbb R\to\mathbb R$, "$\theta$ is an angle lift of $\varphi$ from $s$" means:

• $\theta$ is continuous on $[0,1]$;
• $\theta(0)=s$;
• for every $t\in[0,1]$ there is $r>0$ with

$$\varphi(t)\,e(s)=r\,e(\theta(t)) ,$$

i.e. $\theta(t)$ is a polar angle of the vector $\varphi(t)e(s)$ (matrix times column vector).

Consequences: a lift can exist only if $\varphi(t)e(s)\neq0$ for all $t\in[0,1]$ and $\varphi(0)e(s)$ is a positive multiple of $e(s)$. Because $\theta(t)$ is determined modulo $2\pi$ at each $t$, the restriction of a lift to $[0,1]$ is unique when it exists. Values of $\theta$ outside $[0,1]$ are unconstrained and irrelevant.

windingInterval

$$W(\varphi)=\Bigl\{\tfrac{\theta(1)-s}{2\pi}\ :\ s\in[0,2\pi],\ \theta \text{ an angle lift of }\varphi\text{ from }s\Bigr\}\subseteq\mathbb R .$$

Starting angles range over the closed interval $[0,2\pi]$ (both endpoints, which give the same direction). The set is empty if no starting direction admits a lift (e.g. if $\varphi(t)$ kills some direction, or $\varphi$ jumps). Despite its name, nothing in the definition makes it an interval.

czIndexOfPath

Let $a=\inf W(\varphi)$, $b=\sup W(\varphi)$ (real infimum/supremum, with the junk convention: $\inf$ is $0$ if $W$ is empty or unbounded below; $\sup$ is $0$ if $W$ is empty or unbounded above). Let $k=\lceil b\rceil\in\mathbb Z$. Then

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

Case-by-case:

• $b\notin\mathbb Z$: $k-1=\lfloor b\rfloor$; $\mu=2\lfloor b\rfloor$ if $a\le\lfloor b\rfloor$ (i.e. $[a,b]$ contains the integer $\lfloor b\rfloor$), else $\mu=2\lfloor b\rfloor+1$.
• $b\in\mathbb Z$: $\mu=2(b-1)$ if $a\le b-1$, else $\mu=2b-1$. In particular if $a=b\in\mathbb Z$ then $\mu=2b-1$.

Junk values: if $W(\varphi)=\varnothing$ then $a=b=0$, $k=0$, and $\mu=-1$. If $W$ is nonempty but unbounded above, $b$ is taken as $0$ (so $k=0$) and $\mu\in\{-2,-1\}$ according to whether $a\le-1$.

IsLinearizedFlow

For $H$, a curve $x:\mathbb R\to\mathbb R^4$ and a family $Y:\mathbb R\to L(\mathbb R^4,\mathbb R^4)$ of linear maps: $Y(0)=\mathrm{id}$ and, for every $t\in\mathbb R$, $Y$ is differentiable at $t$ with

$$Y'(t)=DX_H\bigl(x(t)\bigr)\circ Y(t),$$

where $DX_H(p)$ is the Fréchet derivative of the map $X_H:\mathbb R^4\to\mathbb R^4$ at $p$ (junk value $0$ where $X_H$ is not differentiable; then the equation reads $Y'(t)=0$). The curve $x$ is not required to be a trajectory here.

reebProjection

For $x,v\in\mathbb R^4$:

$$\pi_x(v)=v-\frac{\lambda_x(v)}{\lambda_x(X_H(x))}\,X_H(x).$$

This is the projection along $X_H(x)$ onto $\ker\lambda_x$, provided $\lambda_x(X_H(x))=\tfrac12DH_x(x)\neq0$. Junk value: if $\lambda_x(X_H(x))=0$, the quotient is $0$ and $\pi_x(v)=v$.

xiCoords

For $x,v\in\mathbb R^4$, the pair

$$c_x(v)=\Bigl(\omega_0\bigl(v,\xi_2(x)\bigr),\ \omega_0\bigl(\xi_1(x),v\bigr)\Bigr)\in\mathbb R^2 .$$

Since $\omega_0(\xi_1,\xi_2)=1$ (for $x\neq0$), a vector $v=\alpha\,\xi_1(x)+\beta\,\xi_2(x)$ has $c_x(v)=(\alpha,\beta)$. For a general $v$ the formula is still applied as written (it is a linear map $\mathbb R^4\to\mathbb R^2$). At $x=0$ it gives $(0,0)$.

linearizedXiPath

For $H$, a periodic orbit $P=(x_P,T_P)$ and a family $Y$, the path $\Phi:\mathbb R\to\mathbb R^{2\times2}$ given by: for $\tau\in\mathbb R$ and $i,j\in\{1,2\}$ (written $0,1$ in the code),

$$\Phi(\tau)_{ij}=\Bigl[c_{x_P(T_P\tau)}\Bigl(\pi_{x_P(T_P\tau)}\bigl(Y(T_P\tau)\,\xi_j(x_P(0))\bigr)\Bigr)\Bigr]_i .$$

So column $j$ is: take the frame vector $\xi_j$ at the starting point $x_P(0)$, push it by $Y$ at time $T_P\tau$, project along $X_H$ onto $\ker\lambda$ at the current point $x_P(T_P\tau)$, and express it in the coordinates $c$ at that point. Time is rescaled so that $\tau\in[0,1]$ covers $t\in[0,T_P]$, i.e. one full traversal of the period $T_P$ carried by $P$ (which may be a multiple of the minimal period).

IsDynamicallyConvex

$H$ is dynamically convex iff for every periodic orbit $P$ of $H$ (every curve and every positive period, including multiple covers and any stationary orbits) and every $Y$ satisfying the linearized-flow condition along $x_P$,

$$\mu\bigl(\Phi_{H,P,Y}\bigr)\ \ge\ 3 ,$$

with $\mu$ = czIndexOfPath and $\Phi$ = linearizedXiPath above.

Vacuity/edge cases: the condition holds vacuously if $H$ has no periodic orbits; for a given $P$ it holds vacuously if no $Y$ satisfies the linearized-flow condition; if several such $Y$ exist, all must satisfy the bound. No smoothness, star-shapedness or other hypothesis on $H$ is part of this definition. If the winding set of $\Phi$ is empty, $\mu=-1$ and the bound fails.