## Admissible perturbations of the tangent map

Franks’s Lemma is a major tool in the study of differentiable dynamical systems. It says that along a simple orbit segment $E=\{x,fx,\cdots,f^nx\}$, the perturbation of $A\sim D_xf^n$ can be realized via a perturbation of the map $g\sim f$ (which preserves the orbit segment). Moreover, such a perturbation is localized in a neighborhood of $E$, and it can be made arbitrarily $C^1$-close to $f$.

There have been various generalizations of Franks’ Lemma. Some constraints have been noticed when generalizing to geodesic flows and billiard dynamics, since one can’t perturb the dynamics directly, but have to make geometric deformations. See D. Visscher’s thesis for more details.

Let $Q$ be a strictly convex domain, $x$ be the orbit along the/a diameter of $Q$. Clearly $x$ is 2-period. Let $r\le R$ be the radius of curvatures at $x, fx$, respectively. Then
$D_xf^2=\frac{1}{rR}\begin{pmatrix}2d(d-r-R)+rR & 2d(d-R)\\ 2(d-r)(d-r-R) & 2d(d-r-R)+rR\end{pmatrix}$, where $d$ stands for the diameter of $Q$.
Note that the two entries on the diagonal are always the same. Therefore any linearization with different entries on the diagonal can’t be realized as the tangent map along a periodic billiard orbit of period 2. In other words, even through there are three parameters that one can change: the distance $d$, the radii of curvature at both ends $r,R$, the effects lie in a 2D-subspace $\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:ad-bc=1, a=d\}$ of the 3D $\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:ad-bc=1\}$.

Visscher was able to prove that generically, for each periodic orbit of period at least 3, every small perturbation of $D_xF^3$ is actually realizable by deforming the boundary of billiard table. For more details, see Visscher’s paper:

A Franks’ lemma for convex planar billiards.

In the following we discuss something else.

1. Notes from Donnay’s paper (see also here). Let $Q$ be a connected bounded domain on the plane, and $F:M\to M$ be the billiard map induced on $Q$. Let $x(s)\in M$ be a smooth curve, and $x_{i}(s)=F^{i}x(s)$ for $i=\pm 1$. Assume $Q'$ is a 2-nd order deformation of $Q$ around the base point $p_0=\pi(x(0))$, that is, $p_0\in \partial Q'$,
$T_{p_0}Q'=T_{p_0}Q$, but $\kappa'(p_0)\neq \kappa(p_0)$. Assume the perturbation is so localized such that $x_{\pm}(0)$ are not effected. Let $F'$ be the new billiard map, which preserves the orbit segment $x_{-1}(0)\to x(0)\to x_{1}(0)$. But we always have that $y(s)=F'(x_{-1}(s))$ and $z(s)=F'^{-1}(x_1(s))$ are transverse at $x(0)$ in $M$.

Denote  $X=\dot x(0), Y=\dot y(0), Z=\dot z(0)$  the three tangent vectors at $y(0)=z(0)=x(0)$. It suffices to show $Y\neq Z$. Note that

–the backward focusing function of $Y$ satisfies $f_-(Y,F')=f_-(X,F)$;

–the forward focusing function of $Z$ satisfies $f_+(Z,F')=f_+(X,F)$.

–the mirror formula says that $\displaystyle \frac{1}{f_-(X,F)} + \frac{1}{f_+(X,F)} = \frac{2}{d'(x_0)}$.

Then we have $\displaystyle \frac{1}{f_-(Y,F')} + \frac{1}{f_+(Z,F')} \neq \frac{2\kappa'(p_0)}{\sin\theta_0}$ (since we assumed $\kappa'(p_0) \neq \kappa(p_0)$). Therefore  $Y\neq Z$.

For certainty let’s assume $\kappa'(p_0)>\kappa(p_0)$. Then we have $\displaystyle \frac{1}{f_-(Y,F')} + \frac{1}{f_+(Z,F')} < \frac{2}{d'(x_0)}$, or equally, $f_+(Y,F'): the $y$-beam focuses before the $z$-beam.

Another way to see this is via computing the slopes. In a proper coordinate system, the slope of a vector $V\in T_xM$ satisfies the following $\displaystyle f_{\pm}(V)=\frac{d(x)}{1\pm m(X)}$. One can show that $m(Z) < m(X) < m(Y)$.

2. Notes from Donnay’s paper. Let $M$ be a smooth Riemannian surface, $x_0\in M$, and $U$ be a small neighborhood of $x_0$, $\gamma:(-3,3)\to M$ be a geodesic passing through $x_0$ with $\gamma(0)=x_0$, $\gamma(t)\notin U$ for all $|t|\ge 1$. Consider a normal variation $\gamma_s:(-1,1)\to M$ of geodesics with $\gamma_0=\gamma$ such that the Jacobi field $J(t)\neq 0$ for all $t$. Let $\phi^g_t$ be the geodesic flow on $TM$, $\gamma_{s,\pm1}=\phi^g_{\pm 1}(\dot \gamma_s)$ be two families of geodesics through $x_{\pm 1}=\gamma(\pm 1)$.
Let $g'$ be 2nd order perturbation of the metric $g$ inside $U$ along the geodesic $\gamma$ such that $\kappa(g',x)> \kappa(g,x)$ for all $x \in U$ (the existence of such a perturbation needs be verified). In particular $\gamma(-3,3)\to M$ is still a geodesic with respect to $g'$. However, $Y_s=\phi_{1}^{g'}(\dot \gamma_{s,-1})$ and $Z_s=\phi_{-1}^{g'}(\dot \gamma_{s,1})$ are now transverse at $\dot \gamma(0)$. To show this, we need to compare the solutions of Riccati equations (for $u=\frac{J'}{J}$)

• $\dot u_Y=-\kappa'-u_Y^2$ and $\dot u_X=-\kappa-u^2$ for $t\in(-1,0)$, with the initial condition $u_Y(-1)=u_X(-1)$;
• $\dot u_Z=-\kappa-uZ^2$ and $\dot u_X=-\kappa'-u_X^2$ for $t\in(0,1)$, with the final condition $u_Z(1)=u_X(1)$.

Take the first one for example. Let $\delta(t)=u_X(t)-u_Y(t)$ and $\sigma(t)=u_X(t)+u_Y(t)$. Then $\dot \delta(t)=-\Delta_\kappa(t)-\sigma(t)\cdot \delta(t)$; $\delta(-1)=0$. A solution is given by
$\displaystyle \delta(t)=e^{-\int_{-1}^t \sigma(s) ds}\cdot\int_{-1}^t(-\Delta_\kappa(s)\cdot e^{\int_{-1}^s \sigma(r) dr}ds)$. Since $\Delta_\kappa(t)=\kappa(t)-\kappa'(t) < 0$, one gets $\delta(t) > 0$ for all $t\in(-1,0)$ and hence $u_X(0) > u_Y(0)$. Similarly (via time-reversal) we have $u_Z(0) > u_X(0)$ and
$u_Y(0) > u_X(0) > u_Z(0)$.

3. Let $M$ be a closed manifold, and $\phi_t$ be a nonsingular flow on $M$. A cross-section $\Sigma$ is a co-dimension 1 closed submanifold transverse to the flow, such that every flow-line returns to $\Sigma$ after leaving it. More precisely, there exists a smooth function $t:\Sigma\to[c,C]$, and a diffeomorphism $f$ on $\Sigma$ such that $f(x)=\phi_{t(x)}(x)$. Such $f$ is called the first-return map of $\phi$, and $\phi$ is called the suspension of $f$.

The existence of such sections is rare. For example, if $\pi_{d-1}(M)=0$, then no such section could exist, since every closed $(d-1)$-dimensional closed submanifold in $M$ is a boundary and separates $M$.

A Birkhoff section is a cross section $\Sigma$ with boundary $\partial \Sigma\neq\emptyset$ such that

• $\partial \Sigma$ is invariant under the flow;
• $\text{Int}(\Sigma)$ is transverse to the flow;
• the angle between $\dot \phi(x)$ and $T\partial \Sigma$ tends to 0 at the same rate that $x\to \partial \Sigma$.

4. Consider the complex projective plane $\mathbb{C}P^2=(\mathbb{C}^3\backslash \{o\})/\mathbb{C}^\ast$. There is an injective complex plane $(x,y)\in\mathbb{C}^2\mapsto[x,y,1]$, whose complement $\{[x,y,0]\in \mathbb{C}P^2\}$ is a copy of $\mathbb{C}P^1$, which a complex projective line of points at infinity. Consider a regular circle in $\mathbb{R}^2$: $(x-a)^2+(y-b)^2=r^2$. We can complexify this equation and convert it into a homogeneous equation on $\mathbb{C}P^2$: $x^2+y^2-2axz-2byz-r^2 z^2=0$. The solution of this equation is called the complexification of a real circle.
Note that the two points $(1,i,0)$ and $(1,-i,0)$ are the two common solutions (for any choice of $a,b,r$). These two are called the circular points at infinity, or isotropic points in $\mathbb{C}P^2$.

5. Let $f:X\to X$ be a homeomorphism, $x_0$ be a non-periodic point. Kodama and Matsumoto constructed a cocycle $\phi$ with $\displaystyle \sum_{\mathbb{Z}}e^{\phi(x_0,n)}<+\infty$. Such $\phi$ is given by $\sum_{_k\ge 1}\phi_k$ which converges uniformly. More precisely, for each $k\ge 1$,
let $U_k$ be a sufficiently small neighborhood of $x_0$ such that $f^iU_k$, $i=-2^k,\cdots,0,\cdots, 2^k$ are all disjoint. Let $\phi_k:U_k\to[0,(3/4)^k]$ be a bump function with $\phi_k(x_0)=(3/4)^k$. Then we extend the domain of $\phi_k$ such that

• for $x\in f^iU_k, 1\le i\le 2^k$: $\phi_k(x):=-\phi(f^{-i}x)$;
• for $x\in f^{-i}U_k, 1\le i\le 2^k$: $\phi_k(x):=\phi(f^{i}x)$;
• $\phi=0$ elsewhere.

A special feature of such a cocycle is that $\phi_k(x_0,n)=-|n|\cdot (3/4)^k$ for all $|n|\le 2^k$; and $\phi_k(x_0,n)\le 0$ for all $n$.

Now let $\phi=\sum_k \phi_k$. Note that $\displaystyle \phi(x_0,n)=\sum_k \phi_k(x_0,n)\le \phi_l(x_0,n)= -|n|\cdot(3/4)^l$ for any $2^l \ge |n|$. In particular one can pick $l$ with $2^{l-1}\le |n| < 2^{l}$, and get $\phi(x_0,n)\le - 2^{l-1}\cdot(3/4)^l=-(3/2)^l/2$. Therefore
$\displaystyle \sum_{\mathbb{Z}}e^{\phi(x_0,n)}=1+\sum_{l\ge 1}\sum_{2^{l-1}\le |n| < 2^{l}}e^{\phi(x_0,n)} \le 1+\sum_{l\ge 1}2^{l}\cdot e^{-(3/2)^l/2}<+\infty$.