Appendix A. Technical Proofs

Here, we provide some rather technical proofs to avoid interruptions of the text flow.

1.1 A.1. Proof of Lemma 2.4

First, we introduce a smooth involution \(}: \mathbb \times \partial M \partial _\textrm M \rightarrow \mathbb \times \partial M \partial _\textrm M\) by putting \(}(t,x,v) \mathrel (t,x,v')\). Due to the transversality between the vectors in \(\partial M \partial _\textrmM\) and \(T (\partial \Omega )\) the inverse function theorem implies that the geodesic flow

$$\begin \varphi ^g:\mathbb \times (\partial M \partial _\textrmM)\rightarrow S\Sigma \end$$

restricts to a diffeomorphism from an open neighborhood N of \(\\times (\partial M \partial _\textrmM)\) in \(\mathbb \times (\partial M \partial _\textrmM)\) onto an open neighborhood of \(\partial M \partial _\textrmM\) in \(S\Sigma \). By replacing N with its intersection with the open set \((\varphi ^)^(S\Sigma \partial _\textrmM)\) we achieve that \(\varphi ^g(N)\) is disjoint from \(\partial _\textrmM\). By shrinking N further we achieve that N is \(}\)-invariant and satisfies convexity w.r.t. the flow, i.e., a point \((t,x,v)\in \mathbb \times (\partial M \partial _\textrmM)\) with \(t\ge 0\) lies in N iff all points (s, x, v) and \((s,x,v')\) with \(s\in [0,t]\) lie in N, and similarly for \(t < 0\).

Now for every \((x,v)\in \partial _\textrmM\) and small \(t > 0\) we have that \(\varphi ^g_t(x,v)\not \in M\), while for every \((x,v)\in \partial _\textrm M\) and small \(t<0\) we have that \(\varphi ^g_t(x,v)\not \in M\). This shows that

$$\begin \begin \&=N^+_\textrm\cup N^-_\textrm,\\ \&=N^-_\textrm\cup N^+_\textrm. \end \end$$

(46)

Recalling (4) and taking into account that \(N\subset \mathbb \times (\partial M \partial _\textrmM)\) is open and the projection \(\mathbb \times (\partial M \partial _\textrmM) \rightarrow \mathbb \) is an open map, (46) shows that every point \((t,x,v)\in N\) satisfies \(t_-(x,v)<t<t_+(x,v)\). This proves the inclusion \(N\subset D\) and recalling (7) we get (13).

Let \((t_1,x_1,v_1),(t_2,x_2,v_2)\in N_\textrm\) satisfy \(\varphi _(x_1,v_1)=\varphi _(x_2,v_2)\). Then by (46) the signs of \(t_1\) and \(t_2\) must coincide and by (13) we have

$$\begin \varphi ^g_(x_1,v_1)=\varphi ^g_(x_2,v_2),\qquad &t_1\le 0,t_2\le 0,\\ \varphi ^g_(x_1,v_1')=\varphi ^g_(x_2,v_2'), &t_1> 0,t_2> 0.\end\right. } \end$$

We conclude that \((x_1,v_1) = (x_2,v_2)\) because \(\varphi ^g\) and \(\varphi ^g\circ }\) are injective on \(N_\textrm\). This proves that \(\varphi \) is injective on \(N_\textrm\). We argue analogously for \(N_\textrm\).

That \(\varphi :N\rightarrow M\) is open follows once we know that \(\varphi (N_\mathrm )\) are open in M and \(\varphi |_}^:\varphi (N_\mathrm )\rightarrow N_\mathrm \) are smooth, because \(N=N_\textrm\sqcup N_\textrm\). By (13), combined with the fact that \(}\) interchanges \(N_\textrm\) with \(N_\textrm\) as well as \(N^\pm _\textrm\) with \(N^\pm _\textrm\), one has

$$\begin \begin \varphi (N^-_\textrm)&= M\cap \varphi ^g(N_\textrm),&\varphi (N^+_\textrm)&= M\cap \varphi ^g(N_\textrm),\\ \varphi (N_\textrm N^-_\textrm)&= M\cap \varphi ^g(N_\textrm N^-_\textrm),\qquad&\varphi (N_\textrm N^+_\textrm)&= M\cap \varphi ^g(N_\textrm N^+_\textrm). \end\nonumber \\ \end$$

(47)

This shows that the sets on the left-hand sides of the equalities are open in M because they are intersections with M of open subsets of \(S\Sigma \), \(\varphi ^g:N\rightarrow S\Sigma \) being an open map. As we already know that \(\varphi \) is injective on \(N_\mathrm \), we now see that each of the sets \(\varphi (N_\mathrm )\subset M\) decomposes into a disjoint union of two open subsets of M as follows:

$$\begin \varphi (N_\textrm)= \varphi (N^-_\textrm)\sqcup \varphi (N_\textrm N^-_\textrm),\qquad \varphi (N_\textrm)= \varphi (N^+_\textrm)\sqcup \varphi (N_\textrm N^+_\textrm). \end$$

To prove that \(\varphi |_}^\) is smooth we use that \((\varphi ^g)^:\varphi ^g(N)\rightarrow N\) and \((\varphi ^g\circ })^:\varphi ^g(N)\rightarrow N\) are smooth: By (13) the map \(\varphi |_}^\) coincides with \((\varphi ^g)^\) on \(\varphi (N^-_\textrm)\) and with \((\varphi ^g\circ })^\) on \(\varphi (N_\textrm N^-_\textrm)\). The latter two disjoint open sets cover \( \varphi (N_\textrm)\) proving that \(\varphi |_}^\) is smooth. We argue analogously for \(\varphi |_}^\).

Finally, we note that \(\varphi (N)\) is disjoint from \(\partial _\textrmM\) since \(\varphi ^g(N)\) is disjoint from \(\partial _\textrmM\). \(\square \)

1.2 A.2. Proof of Lemma 2.7

Given \((t,x, v)\in \varphi ^(O)\) with \(t\ge 0\), suppose first that the compact trajectory segment \(\varphi ([0,t]\times \)\) lies in \(\mathring\), so that it agrees with a trajectory segment of the geodesic flow: \(\varphi ([0,t]\times \)=\varphi ^g([0,t]\times \)\subset \mathring\). Then it follows from the continuity of \(\varphi ^g\) that there is an \(\varepsilon >0\) and an open set \(U\subset \mathring\) containing (x, v) such that \(\varphi ^g([t-2\varepsilon ,t+2\varepsilon ]\times U)\subset \mathring\cap O\). Now \(t_-(},})< t-\varepsilon \), \(t_+(},})>t+\varepsilon \) for all \((},})\in U\). It follows that \((t-\varepsilon ,t+\varepsilon )\times U\subset \varphi ^(O)\), so that (t, x, v) is an interior point of \(\varphi ^(O)\).

As a second case, suppose that \(t>0\), \(\varphi ([0,t)\times \)\subset \mathring\), and \((x_0,v_0) \mathrel \varphi _t(x,v)\in (\partial M \partial _\textrm M)\cap O\). Then \(\varphi _t(x,v)\in \partial _\textrm M\cap O\) because the trajectory is incoming. Recalling the definition of \(\varphi \) departing from (5), the half-open trajectory segment \(\varphi ([0,t)\times \)\) agrees with a trajectory segment of the geodesic flow: \(\varphi ([0,t)\times \)=\varphi ^g([0,t)\times \)\subset \mathring\). Let \(N_\textrm\subset \mathbb \times \partial _\textrm M\) be an open set as in Lemma 2.4. Then, since \(N_\textrm\) is a neighborhood of \(\\times \partial _\textrm M\) in \(\mathbb \times \partial _\textrm M\) and by the continuity of \(\varphi ^g\) we can choose a small open subset \(S_\textrm\subset \partial _\textrm M\) and a small \(\delta >0\) such that \(S_\textrm\) contains \((x_0,v_0)=\varphi _t(x,v)\), \(}_\textrm\subset \partial _\textrm M\cap O\) is compact, \((-\delta ,\delta )\times }_\textrm\subset N_\textrm\), and \(\varphi ^g((-\delta , 0] \times S_\textrm)\subset O\). Then, (13) gives us \(\varphi ((-\delta ,0]\times S_\textrm)=\varphi ^g((-\delta ,0]\times S_\textrm)\subset O\). In addition, we introduce the open set \(S_\textrm \mathrel \\}\subset \partial _\textrm M\) which contains \((x_0,v_0')\). Then, by the reflection-symmetry of \((\partial M \partial _\textrm M)\cap O\) we have \(\overline_\textrm\subset \partial _\textrm M\cap O\). Using again the continuity of \(\varphi ^g\) and (13) we can achieve, shrinking \(\delta \) if necessary, that \(\varphi ^g([0,\delta )\times S_\textrm)=\varphi ([0,\delta )\times S_\textrm)\subset O\).

Now, the continuity of \(\varphi ^g\) and the fact that \(\varphi ((-\delta ,\delta )\times S_\textrm)\cap \mathring\) is open in \(S\Sigma \) by Lemma 2.4iii) imply that there is a small \(\varepsilon \in (0,t)\) and a small open set \(U\subset \mathring\) containing (x, v) such that \(\varphi ^g([0,t-\varepsilon ]\times U)\subset \mathring\) and \(\varphi _^g(U)\subset \varphi ((-\delta ,0)\times S_\textrm)\cap O\). Then, by (5) we have \([0,t-\varepsilon ]\times U\subset D\) and \(\varphi ([0,t-\varepsilon ]\times U)=\varphi ^g([0,t-\varepsilon ]\times U)\). See Fig. 8 for an illustration.

Fig. 8

Illustration of the argument in the proof of Lemma 2.7. The transversality properties of \(\varphi \) established in Lemma 2.4 allow us to extend the trajectories of all points in U to the time interval \([0,t+\varepsilon )\)

Lemma 2.4iv) and the flow property \(\varphi _s(\varphi _(},}))=\varphi _(},})\) now imply the inclusion \((t-\varepsilon ,t+\varepsilon )\times U\subset \varphi ^(O)\), so that (t, x, v) is an interior point of \(\varphi ^(O)\).

If the trajectory segment \(\varphi ([0,t]\times \)\) does not lie entirely in \(\mathring\), we can cut it into finitely many segments each of which lies either entirely in \(\mathring\) or intersects \(\partial M \partial _\textrm M\) precisely in one of its endpoints. It then suffices to repeat the arguments above inductively finitely many times to show that (t, x, v) is an interior point of \(\varphi ^(O)\). The case \(t<0\) is treated analogously, finishing the proof that \(\varphi ^(O)\) is open.

Finally, (14) follows immediately from a combination of (11) with the presupposed reflection-symmetry of \(O\cap (\partial M \partial _\textrm M)\). This completes the proof.

\(\square \)

1.3 A.3. Proof of Proposition 3.3

The proof of Proposition 3.3 relies on the following auxiliary result which describes some general technical properties of smooth models for the non-grazing billiard flow:

Lemma A.1

Let \((\mathcal ,\pi ,\phi )\) be a smooth model for \(\varphi \). Then \(\pi \) is a proper map, one has

$$\begin \pi (\partial _\textrmM)=\pi (\partial _\textrmM) \mathrel \mathcal ,\qquad \qquad \mathcal =\pi (\mathring)\sqcup \mathcal , \end$$

the set \(\mathcal \) is a codimension 1 submanifold of \(\mathcal \), and the maps \(\pi |_M}:\partial _\mathrm M\rightarrow \mathcal \) are diffeomorphisms. In particular, for every set \(}\subset \mathcal \) the set \(\pi ^(})\cap (\partial M \partial _\textrm M)\) is reflection-symmetric. Furthermore, the flow \(\phi \) is transversal to \(\mathcal \).

Proof

From (21), (7), and the continuity of \(\pi \) and \(\phi \) it follows that

$$\begin \pi (x,v)=\pi (x,v')\qquad \forall \; (x,v)\in \partial M \partial _\textrmM. \end$$

(48)

Since the tangential reflection \((x,v)\mapsto (x,v')\) interchanges \(\partial _\textrmM\) and \(\partial _\textrmM\), this shows that \(\pi (\partial _\textrmM)=\pi (\partial _\textrmM)\). From the fact that \(M \partial _\textrmM=\mathring\sqcup \partial _\textrmM\sqcup \partial _\textrmM\) and the surjectivity of \(\pi \) it follows that \(\mathcal =\pi (\mathring)\cup \mathcal \). Suppose that there is a point \(p\in \mathcal \cap \pi (\mathring)\). Then there are points \(}\in \mathring\) and \(}'\in \partial M \partial _\textrm M\) such that \(\pi (})=\pi (}')=p\). Since each trajectory of the non-grazing billiard flow \(\varphi \) intersects \(\partial M \partial _\textrm M\) only at a discrete set of time parameters, we can find an \(\varepsilon >0\) such that \(\varphi _\varepsilon (}),\varphi _\varepsilon (}')\in \mathring\). Then (21) implies

$$\begin \pi |_}(\varphi _\varepsilon (}))=\phi _\varepsilon (p)=\pi |_}(\varphi _\varepsilon (}')), \end$$

and the injectivity of \(\pi |_}\) and \(\varphi _\varepsilon \) yields \(}=}'\), contradicting the fact that \(\mathring\) and \(\partial M \partial _\textrm M\) intersect trivially. This proves that \(\mathcal \cap \pi (\mathring) = \emptyset \) hence \(\mathcal =\pi (\mathring)\sqcup \mathcal \).

To prove that \(\mathcal \) is a smooth submanifold of \(\mathcal \), let \(N_\textrm\subset \mathbb \times \partial _\textrm M\) be an open set as in Lemma 2.4 and put \(} \mathrel (\textrm_\mathbb \times \pi )(N_\textrm)\subset \mathbb \times \mathcal \). Then \(\\times \mathcal \subset }\subset }\) by Lemma 2.4i). Let \((x,v)\in \partial _\textrmM\) and choose a small open set \(S\subset \partial _\textrmM\) containing (x, v) and a small \(\varepsilon >0\) such that \((-\varepsilon ,\varepsilon )\times S\subset N_\textrm\). This is possible because \(N_\textrm\) is a neighborhood of \(\\times \partial _\textrm M\) in \(\mathbb \times \partial _\textrm M\) by Lemma 2.4i). Consider the set \(N^-_\textrm\) from (12) and recall from (13) that \(\varphi |_}=\varphi ^g|_}\). Now, since \(\pi |_}\) is a diffeomorphism and \(\varphi ^g\) is an open map, we get that \(\pi |_}(\varphi ^g((-3\varepsilon /4,-\varepsilon /4)\times S))\) is an open subset of \(\mathcal \). Consequently, the set

$$\begin } \mathrel \phi _(\pi |_}(\varphi ^g((-3\varepsilon /4,-\varepsilon /4)\times S)))\subset \mathcal \end$$

is open in \(\mathcal \) because \(\phi _\) is an open map. Moreover, using (13) and Lemma 2.4v), we see that \(\phi _\circ \pi |_}\circ \varphi ^g|_\) is a diffeomorphism from \((-3\varepsilon /4,-\varepsilon /4)\times S\) onto \(}\).

However, thanks to (21), (13), and the flow property of \(\varphi \) we have

$$\begin & }=(\pi \circ \varphi )((-\varepsilon /4,\varepsilon /4)\times S),\\ & \phi _\circ \pi |_}\circ \varphi ^g|_=\pi \circ \varphi |_, \end$$

which shows that \(\mathcal \subset }\) and that \(\pi \circ \varphi |_:(-\varepsilon /4,\varepsilon /4)\times S\rightarrow }\) is a diffeomorphism mapping \(\\times S\) onto \(}\cap \mathcal \mathrel }\). In particular, the identity \(\pi |_S=\phi _\circ \pi |_}\circ \varphi ^g_|_S\) shows that \(\pi |_S:S\rightarrow }\) is a diffeomorphism. Applying (21) again we find that \(\phi |_}}:(-\varepsilon /4,\varepsilon /4)\times }\rightarrow }\) is a diffeomorphism. This shows that \(\phi \) is transversal to \(\mathcal \) at each point in \(}\).

Since \((x,v)\in \partial _\textrmM\) was arbitrary and we know that \(\pi (\partial _\textrmM)=\mathcal \), we have proved that \(\mathcal \) is a smooth submanifold of \(\mathcal \) of the same dimension as \(\partial _\textrmM\), i.e., of codimension 1, that \(\pi |_M}:\partial _\textrmM \rightarrow \mathcal \) is a local diffeomorphism, and that \(\phi \) is transversal to \(\mathcal \).

To prove that \(\pi |_M}\) is injective, it suffices to observe that given \((x_1,v_1), \)\( (x_2,v_2)\in \partial _\textrmM\) we can repeat the above argument with an \(S\subset \partial _\textrmM\) containing both \((x_1,v_1)\) and \((x_2,v_2)\) and some small enough \(\varepsilon >0\) depending on \((x_1,v_1)\) and \((x_2,v_2)\); then the same trick of writing \(\pi |_S=\phi _\circ \pi |_}\circ \varphi ^g_|_S\) shows that the assumption \(\pi (x_1,v_1)=\pi (x_2,v_2)\) implies \((x_1,v_1)=(x_2,v_2)\) by injectivity of \(\phi _\), \(\pi |_}\), and \(\varphi ^g_\).

The restriction \(\pi |_M}\) is treated analogously, finishing the proof that the maps \(\pi |_M}:\partial _\mathrm M\rightarrow \mathcal \) are diffeomorphisms.

Finally, that \(\pi \) is proper follows from the continuity of \(\pi \) and the observation that by the above the inverse image \(\pi ^(p)\) of any point \(p\in }\) contains at most 2 points. \(\square \)

We are now in a position to prove Proposition 3.3:

Proof of Proposition 3.3

Using Lemma A.1, we define \(F:\mathcal \rightarrow \mathcal '\) by

$$\begin F(p) \mathrel (\pi '\circ \pi |_}^)(p),& p\in \pi (\mathring),\\ (\pi '\circ \pi |_M}^)(p), & p\in \pi (\partial _\textrmM)=\mathcal . \end\right. } \end$$

Then, F satisfies by construction the relation \(F\circ \pi =\pi '\) and F is bijective with inverse

$$\begin F^(p')= (\pi \circ \pi '|_}^)(p'),& p'\in \pi '(\mathring),\\ (\pi \circ \pi '|_M}^)(p'), & p'\in \pi '(\partial _\textrmM)=\mathcal '. \end\right. } \end$$

From (21) and the relations \(\mathcal = (\textrm_\mathbb \times \pi )(D)\), \(\mathcal ' = (\textrm_\mathbb \times \pi ')(D)\) we get \((\textrm_\mathbb \times F)(})=}'\) and \(F\circ \phi =\phi '\circ (\textrm_\mathbb \times F)|_}}\).

It remains to prove that F and \(F^\) are smooth. By Definition 3.1 and Lemma A.1, the maps \(F:\pi (\mathring)\rightarrow \pi '(\mathring)\) and \(F:\mathcal \rightarrow \mathcal '\) are diffeomorphisms; in particular, F and \(F^\) are smooth on the open sets \(\pi (\mathring)\subset \mathcal \) and \(\pi '(\mathring)\subset \mathcal '\), respectively. To prove smoothness of F and \(F^\) near \(\mathcal \) and \(\mathcal '\), we note that since \(}\) and \(}'\) are open and \(\phi \), \(\phi '\) are flows transversal to \(\mathcal \) and \(\mathcal '\), respectively, by Lemma A.1, the inverse function theorem implies that there is an open set \(U\subset \mathcal \) containing \(\mathcal \), an open set \(V\subset (\mathbb \times \mathcal )\cap \mathcal \) containing \(\\times \mathcal \), an open set \(U'\subset \mathcal '\) containing \(\mathcal '\), and an open set \(V'\subset (\mathbb \times \mathcal ')\cap \mathcal '\) containing \(\\times \mathcal '\) such that

$$\begin \phi : V\rightarrow U,\qquad \phi ':V'\rightarrow U' \end$$

are smooth local coordinate charts on their respective domains. In fact, since \(F\circ \phi =\phi '\circ (\textrm_\mathbb \times F)|_}}\), we can achieve

$$\begin V'=(\textrm_\mathbb \times F|_\mathcal )(V) \end$$

by shrinking V or \(V'\). To show that F is smooth on U and \(F^\) is smooth on \(U'\), it now suffices to prove that

$$\begin (\textrm_\mathbb \times F|_\mathcal ^)\circ \phi '|_^\circ F\circ \phi : V\rightarrow V, \quad (\textrm_\mathbb \times F|_\mathcal )\circ \phi |_^\circ F^\circ \phi ': V'\rightarrow V' \end$$

are smooth maps. However, the latter are nothing but the identity maps as one sees by employing again the relation \(F\circ \phi =\phi '\circ (\textrm_\mathbb \times F)|_}}\).

Finally, the uniqueness of F follows from the fact that by Lemma A.1 the complement \(\mathcal \mathcal \) is dense in \(\mathcal \) and F is uniquely determined by \(\pi \) and \(\pi '\) on \(\mathcal \mathcal \). \(\square \)

1.4 A.4. Proof of Proposition 4.5

Consider the gluing regions \(}_g=\pi _g(\partial M \partial _\textrmM)\subset \mathcal _g\) and \(}_=\pi _(\partial M \partial _\textrmM)\subset \mathcal _\), respectively. Then, by the fact that \(\pi _g|_}\) and \(\pi _|_}\) are diffeomorphisms onto \(\mathcal _g }_g\) and \(\mathcal _ }_\), respectively, we immediately get the built-in diffeomorphism

$$\begin \pi _|_}\circ \pi _g|_}^: \mathcal _g }_g\limits ^} \mathcal _ }_ \end$$

(49)

that makes the analogue of the diagram (34), with \(M \partial _\textrmM\) replaced by \(\mathring\), commute. Moreover, we see that any diffeomorphism \(\mathcal _g\cong \mathcal _\) making (34) commute coincides with (49) on the dense set \(\mathcal _g }_g\), so that it is uniquely determined by this property.

In order to extend (49) to a (necessarily unique) diffeomorphism \(\mathcal _g\cong \mathcal _\), let \(N_g,N_\subset \mathbb \times (\partial M \partial _\textrmM)\) be two sets as in Lemma 2.4, applied separately for g and \(g'\), respectively, and consider the intersection \(N_\textrm \mathrel (N_g)_\textrm\cap (N_)_\textrm=N\cap (\mathbb \times \partial _\textrmM)\), where \(N \mathrel N_g\cap N_\). Then the set \(O \mathrel \varphi _g(N_\textrm)\cap \varphi _(N_\textrm)\) is an open neighborhood of \(\partial M \partial _\textrmM\) in \(M \partial _\textrmM\) by Lemma 2.4. In particular, \(O\cap (\partial M \partial _\textrmM)=\partial M \partial _\textrmM\) is invariant under tangential reflection with respect to g and \(g'\). Thus, by Lemma 2.7 (also applied separately for g and \(g'\)), the sets

$$\begin N_ \mathrel \varphi _g^(O)\cap N_\textrm,\qquad N_ \mathrel \varphi _^(O)\cap N_\textrm \end$$

are open in \(\mathbb \times \partial _\textrmM\), and by definition of the quotient topologies on \(\mathcal _g\) and \(\mathcal _\) the identity map \(O\rightarrow O\) descends to a homeomorphism

$$\begin }_g\cong }_ \end$$

(50)

between the open neighborhood \(}_g \mathrel \pi _g(O)\) of \(}_g\) in \(\mathcal _\) and the open neighborhood \(}_ \mathrel \pi _(O)\) of \(}_\) in \(\mathcal _\). Moreover, by definition of the quotient maps \(\pi _g\) and \(\pi _\), the diffeomorphism (49) and the homeomorphism (50) agree on \(}_g\cap (\mathcal _g }_g)\), so that they glue to a global homeomorphism \(\mathcal _g\cong \mathcal _\). It remains to prove that the latter is a diffeomorphism, which reduces to proving that the map (50) and its inverse are smooth. By definition of the smooth structures on \(\mathcal \) and \(\mathcal _\), this reduces to checking that

$$\begin \begin&\varphi _g|_}^\circ \varphi _|_}:N_\rightarrow N_, \\&\quad \varphi _|_}^\circ \varphi _g|_}: N_\rightarrow N_ \end \end$$

(51)

are smooth maps. By (13), the maps (51) can be expressed in terms of the geodesic flows \(\varphi ^g\), \(\varphi ^\) and the reflection maps \(R_,R_: \mathbb \times (\partial M \partial _\textrmM)\rightarrow \mathbb \times (\partial M \partial _\textrmM)\), \((t,x,v)\mapsto (t,x,R_v)\), \((t,x,v)\mapsto (t,x,R_v)\), by

$$\begin (\varphi _g|_}^\circ \varphi _|_})(t,x,v)= (\varphi ^g|_}^\circ \varphi ^|_})(t,x,v), & t\le 0,\\ (R_\circ \varphi ^g|_)}^\circ \varphi ^|_(N_)}\circ R_)(t,x,v), & t>0,\end\right. }\\ (\varphi _|_}^\circ \varphi _g|_})(t,x,v) =(\varphi ^|_}^\circ \varphi ^g|_})(t,x,v), & t\le 0,\\ (R_\circ \varphi ^|_(N_)}^\circ \varphi ^g|_(N_)}\circ R_)(t,x,v), & t>0.\end\right. } \end$$

Since \(\varphi ^g|_}\) and \(\varphi ^|_}\) are diffeomorphisms onto their images, the above maps are smooth iff their post-compositions with \(\varphi ^g|_}\) and \(\varphi ^|_}\) are smooth, respectively. In view of the definitions of \(R_\) and \(}_\), the proof is finished. \(\square \)

Appendix B. Construction of Smooth Model Bundles

Here, we provide a concrete construction of smooth models in the vector-valued setting, i.e., of smooth model bundles. The reader who is not interested in the vector-valued case may safely skip this appendix. The construction follows ideas very similar to those employed in the above construction of smooth models for non-grazing billiard flows.

We begin by reminding the reader of some notation used in the main text: There we introduced the non-grazing billiard flow \(\varphi \) acting on the phase-space \(M \partial _\textrm M\), defined on the domain \(D\subset \mathbb \times (M \partial _\textrm M)\) from (10) which is open by Lemma 2.7. For the analytic treatment of this dynamical system, we constructed a model manifold \(\mathcal \) together with a smooth surjection \(\pi : M \partial _\textrm M \rightarrow \mathcal \) and a smooth model flow \(\phi \) on \(\mathcal \), defined on the domain \(}\subset \mathbb \times }\), such that \(\pi \circ \varphi _t = \phi _t\circ \pi \). This was necessary because \(\varphi \) is non-smooth (in fact non-continuous and not even a flow) due to the presence of the instantaneous boundary reflections \(R: \partial M \partial _\textrm M \rightarrow \partial M \partial _\textrm M,\, (x, v) \mapsto (x, v')\).

For the remainder of this appendix, we now assume a smooth \(\mathbb \)-vector bundle

$$\begin \pi _}}}: \widetilde} \rightarrow M \partial _\textrm M \end$$

of rank r to be given, the fibers of which we denote by \(\widetilde}}_ \mathrel \pi _}}}^(\)\).

Furthermore we require a first-order differential operator \(\widetilde}\) acting on smooth sections of \(\widetilde}\) and satisfying the following Leibniz rule:

$$\begin \widetilde}\big ( }\cdot } \big ) = (}})\cdot } + }\cdot \widetilde}},\quad \forall }\in \textrm^\infty _\textrm(M \partial _\textrmM),\, }\in \textrm^\infty (M\partial _\textrm M, \widetilde}),\nonumber \\ \end$$

(52)

where \(}\) denotes the billiard generator defined in Sect. 2.4. An additional piece of data necessary for the construction of a smooth model for \(\widetilde}\) is a bundle isomorphism

$$\begin \kappa : \widetilde}\big |_ M} \longrightarrow \widetilde}\big |_ M} \end$$

such that \(\pi _}}}\circ \kappa = R\circ \pi _}}}\big |_}}}^(\partial _\textrm M)}\) holds. For example, such an isomorphism exists if both \(\widetilde}\big |_ M}\) and \(\widetilde}\big |_ M}\) can be trivialized: Then, we can simply define \(\kappa \) as the composition of the first trivialization, the map \(R\times \textrm_^r}\), and the inverse of the second trivialization.

Before proving our main theorem, we first have to describe a dynamical quantity associated with the above data, namely the billiard parallel transport. Morally, it is derived from the operator \(\widetilde}\) in the same intuitive manner as the billiard flow is derived from the geodesic flow:

Lemma B.1

There exists a unique map \(}: \widetilde} \rightarrow \widetilde}\) on the flow domain

$$\begin \widetilde} \mathrel \big \\partial _\textrm M:\, (t, x, v)\in D ~\text ~ e\in \widetilde}_ \big \} ~, \end$$

called the billiard parallel transport, with the following properties:

(1)

For each \((t, x, v)\in D\) the map

$$\begin }_: \widetilde}_ \longrightarrow \widetilde}_,\qquad e\mapsto }_(e) \mathrel }(t, e), \end$$

(53)

is a well-defined linear isomorphism.

(2)

\(}\) is a flow up to composition with \(\kappa \) on boundary fibers. More precisely, one has

$$\begin }_=\textrm_}_}\qquad \forall \;(x,v)\in M\partial _\textrmM \end$$

(54)

and if \((t,x, v)\in D\) and \(t'\in \mathbb \) are such that \((t',\varphi _t(x, v))\in D\), then the following generalization of (8) holds:

$$\begin }_\circ }_=}_,& t + t' \ne 0 \text \varphi _(x,v)\in \mathring\text \\ &t<0,(x, v) \in \partial _\textrm M \text t>0,(x, v) \in \partial _\textrm M,\\ \kappa |_}_}, & t+t'=0 \text t>0,(x, v) \in \partial _\textrm M,\\ \kappa ^|_}_},& t+t'=0 \text t<0,(x, v) \in \partial _\textrm M. \end\right. }\nonumber \\ \end$$

(55)

(3)

For each \(}\in \textrm^\infty (M\partial _\textrm M, \widetilde})\) the map

$$\begin \varphi ^(\mathring)\rightarrow \widetilde},\qquad (t,x,v)\mapsto }_\big (}(x,v)\big ), \end$$

is smooth.

(4)

For each \((x,v)\in \mathring\) and \(}\in \textrm^\infty (M\partial _\textrm M, \widetilde})\) one has

$$\begin (\widetilde}})(x,v)=\frac\Big |_}_}(\varphi _t(x, v)). \end$$

Proof

As a preliminary step we embed a neighborhood of \(M\partial _\textrmM\) in \(S\Sigma \) into a closed manifold N and extend \(\widetilde}\) and \(\widetilde}\) arbitrarily to N such that near \(M\partial _\textrmM\) they satisfy the Leibniz rule (52) with \(}\) replaced by the geodesic vector field \(X^g\) (recall from (20) that \(}\) agrees with \(X^g\) on billiard functions). We continue to denote these extensions by \(\widetilde}}, \widetilde}\) and obtain a well-defined transfer operator \(\exp (-t \widetilde})\) acting on smooth sections of \(\widetilde}}\).

Given \((t,x, v)\in D\) we begin by assuming that \(t \ge 0\) is small enough such that \(\varphi _s(x, v)\), \(s\in [0, t]\), intersects \(\partial M \partial _\textrm M\) only at its endpoint \(\varphi _t(x, v)\), if at all. Then, two cases must be distinguished:

(1)

\((x, v)\in \mathring\): Given \(e\in \widetilde}_\) choose \(}\in \textrm^\infty (N,\widetilde}})\) with \(}(x, v) = e\) and supported in \(\mathring\). Then, we define

$$\begin }_(e) \mathrel \big (\hspace\exp (-t \widetilde}) } \big )(\varphi _t(x, v)), \end$$

which is independent of the choice of \(}\) by the Leibniz rule (52), which in turn applies by the support property of \(}\) and where we use that any smooth function supported in \(\mathring\) is a billiard function.

(2)

\((x, v)\in \partial M \partial _\textrmM\): If \((x, v)\in \partial _\textrm M\), given \(e\in \widetilde}_\), choose \(}\in \textrm^\infty (N,\widetilde}})\) with \(}(x, v) = e\) by multiplying some local frame with cutoffs that restrict to billiard functions on \(M \partial _\textrmM\). Again we define

$$\begin }_(e) \mathrel \big (\hspace\exp (-t \widetilde}) } \big )(\varphi _t(x, v)), \end$$

independently of the choice of \(}\). If instead \((x, v)\in \partial _\textrm M\) we proceed in the same way but with \(\kappa (e)\) instead of e.

This construction can analogously be transferred to sufficiently small \(t < 0\).

Now, without these smallness assumptions on t, we use that by definition of D and \(\varphi \) there exists a unique finite sequence \(t_0, t_1,..., t_N\) with \(t=t_0+\cdots +t_N\) such that \(\varphi _t(x, v)\) can be written in terms of the geodesic flow \(\varphi ^g\) as \(\varphi _t(x, v) = \varphi _^g\circ R\circ \cdots \circ R\circ \varphi _^g(x, v)\). We then define the billiard parallel transport of \(e\in \widetilde}_\) as

$$\begin }_(e) \mathrel }_\circ \cdots \circ }_(e). \end$$

The claimed Properties (1), (2) of \(}\) are now satisfied by construction and Properties (3), (4) follow from the properties of \(\exp (-t \widetilde})\) since \(\varphi \) is smooth on \(\varphi ^(\mathring)\). Finally, Property (1), Eq. (54), and Property (4) determine \(}\) uniquely on \(\varphi ^(\mathring)\times \widetilde}\) and (55) then implies that the full map \(}\) is unique because its values at points outside \(\varphi ^(\mathring)\times \widetilde}\) are determined by \(\kappa \) and values at points inside \(\varphi ^(\mathring)\times \widetilde}\). \(\square \)

Definition B.2

We call the operator

$$\begin \mathcal :D\times \textrm^\infty (M\partial _\textrm M, \widetilde})&\longrightarrow \Gamma (M\partial _\textrm M, \widetilde})\\ ((t, x, v),})&\longmapsto }_(x, v), t}(}(\varphi _(x, v))) \mathrel (\mathcal _t})(x,v) \end$$

the billiard transfer operator associated with \(\widetilde}\). Here, \(\Gamma (M\partial _\textrm M, \widetilde})\) denotes the set of arbitrary sections of \(\widetilde}\) without any continuity or smoothness assumption.

This terminology is of course motivated by Lemma B.1, which implies that for each \(}\in \textrm^\infty (M\partial _\textrm M, \widetilde})\) the map

$$\begin \varphi ^(\mathring)\cap (\mathbb \times \mathring) \rightarrow \widetilde}|_},\qquad (t,x,v)\mapsto (\mathcal _t})(x,v), \end$$

is smooth and one has

$$\begin (\widetilde}})(x,v)=-\frac\Big |_(\mathcal _t})(x,v)\quad \forall \; (x,v)\in \mathring. \end$$

(56)

We can now introduce the vector-valued equivalent of the billiard functions:

Definition B.3

The set of smooth billiard sections of \(\widetilde}}\) is

$$\begin \textrm^\infty _\textrm(M\partial _\textrm M, \widetilde}}) \mathrel \left\}\in \textrm^\infty (M \partial _\textrm M) \,\big |\, \big ( (t,x,v)\mapsto (\mathcal _t})(x,v)\big )\in \textrm^\infty (D, \widetilde}}) \right\} . \end$$

An elementary property of the smooth billiard sections is that they are stable with respect to multiplication by smooth billiard functions—in other words, \(\textrm^\infty _\textrm(M\partial _\textrm M, \widetilde}})\) is a \(\textrm^\infty _\textrm(M\partial _\textrm M)\)-module. The main significance of the smooth billiard sections is that they are preserved by the operator \(\widetilde}\) and the latter acts on them by differentiation of the billiard transfer operator:

$$\begin \widetilde}: \textrm^\infty _\textrm(M\partial _\textrm M, \widetilde}})\longrightarrow \textrm^\infty _\textrm(M\partial _\textrm M, \widetilde}}),\qquad (\widetilde}})(x,v)=-\frac\Big |_(\mathcal _t})(x,v). \end$$

This follows from (56). It provides the vector-valued generalization of the formula (19).

With this data as our point of departure, we can now prove our main theorem in the vector-valued situation:

Theorem B.4

(Existence of smooth model bundles). There exists a smooth vector bundle \(\pi _\mathcal : \mathcal \rightarrow \mathcal \) and a smooth surjection \(\Pi :\widetilde}} \rightarrow \mathcal \) such that the diagram