We now apply Proposition 3.8 for the chain-complexes
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ and
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ for all
$\kappa\in\mathrm{Spec}(\mathcal{Y}_{\oplus})$. Notice that the construction
of homotopies $G$ and $\widehat{G}$ does not depend on $\kappa$, but only on
$\epsilon$ and $H_{\ominus}$. We can thus choose generically the same pairs
$(G,J^{s}_{t})$ and $(\widehat{G},\widehat{J}^{s}_{t})$ to induce maps from
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ to
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$, and from
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ to
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$, for all
$\kappa\in\mathrm{Spec}(\mathcal{Y}_{\oplus})$.

We then obtain chain maps
$\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ and
$\Psi_{\widehat{G}}:CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$. Since
$\Psi_{\widehat{G}}\circ\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ is chain isotopic to the
identity and the differential $d^{J_{t}}$ vanishes, we conclude that the chain
map
$\Psi_{\widehat{G}}\circ\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ is the identity. It
follows that $\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ is injective and that
$\Psi_{\widehat{G}}:CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ is surjective.

Since $\Psi_{g}$ is injective we know that the dimension of
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ is $\geq n_{\kappa}$.
We let the $m_{\kappa}\geq n_{\kappa}$ be the dimension of
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$, and since
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ is the
$\mathbbm{Z}_{2}$-vector space over
$\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$, we
conclude that
$\\#\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})=m_{\kappa}$.
We denote by $\\{\sigma^{\kappa}_{1},...,\sigma^{\kappa}_{m_{\kappa}}\\}$ the
elements of
$\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$.

We now introduce some terminology. For each $j\in\\{1,...,n_{\kappa}\\}$, the
image $\Psi_{G}(\gamma^{\kappa}_{j})$ can be written in a unique way as a sum
of orbits in
$\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$: we use
here that we are working with $\mathbbm{Z}_{2}$-coefficients. We say that an
orbit $\sigma^{\kappa}_{l}$ appears in $\Psi_{G}(\gamma^{\kappa}_{j})$, if the
orbit $\sigma^{\kappa}_{l}$ is one of the orbits which appear in the
expression of $\Psi_{G}(\gamma^{\kappa}_{j})$ written in the base
$\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ of
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$.

We define in a similar way orbits of
$\mathcal{P}^{1}_{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ which appear
in $\Psi_{\widehat{G}}(\sigma^{\kappa}_{l})$ for each
$\sigma^{\kappa}_{l}\in\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$.
We make the following claim:

Claim 1: It is possible to choose an injective map
$\mathfrak{f}_{\kappa}:\\{1,...,n_{\kappa}\\}\to\\{1,...,m_{\kappa}\\}$ and a
bijective map
$\mathfrak{g}_{\kappa}:\\{1,...,n_{\kappa}\\}\to\\{1,...,n_{\kappa}\\}$ such
that for each $i\in\\{1,...,n_{\kappa}\\}$:

  * •

the orbit $\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}$ appears in
$\Psi_{G}(\gamma^{\kappa}_{i})$, and the orbit
$\gamma^{\kappa}_{\mathfrak{g}_{\kappa}(i)}$ appears in
$\Psi_{\widehat{G}}(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)})$.

It is clear that the claim will follow from the following combinatorial lemma,
whose proof is presented in Appendix A.

######  Lemma 5.1.

Let $V$ and $R$ be finite dimensional $\mathbbm{Z}_{2}$-vector spaces whose
dimensions we denote by $n$ and $m$, respectively. Let $\\{v_{1},...,v_{n}\\}$
and $\\{r_{1},...,r_{m}\\}$ be basis of $V$ and $R$, respectively, and
$\mathfrak{F}:V\to R$ and $\mathfrak{G}:R\to V$ be linear maps such that
$\mathfrak{G}\circ\mathfrak{F}$ is an isomorphism. Then, it is possible to
find an injective map $\mathfrak{f}:\\{1,...,n\\}\to\\{1,...,m\\}$ and a
bijective map $\mathfrak{g}:\\{1,...,n\\}\to\\{1,...,n\\}$ such that for each
$i\in\\{1,...,n\\}$:

  * •

the element $r_{\mathfrak{f}(i)}$ appears in $\mathfrak{F}(v_{i})$, and the
element $v_{\mathfrak{g}(i)}$ appears in $\mathfrak{G}(r_{\mathfrak{f}(i)})$.

Step 2.

For each $\gamma^{\kappa}_{i}\in\mathcal{Y}^{\kappa}_{\oplus}$ we consider the
$1$-periodic orbit $\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}$. Since
$\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}$ appears in
$\Psi_{G}(\gamma^{\kappa}_{i})$, there exists a Floer cylinder
$u^{1}_{\kappa,i}$ of $(G,J^{s}_{t})$ which is negatively asymptotic to
$\gamma^{\kappa}_{i}$ and positively asymptotic to
$\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}$.

Similarly, since $\gamma^{\kappa}_{\mathfrak{g}_{\kappa}(i)}$ appears in
$\Psi_{\widehat{G}}(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)})$, there exists
a Floer cylinder $u^{2}_{\kappa,i}$ of $(\widehat{G},\widehat{J}^{s}_{t})$
which is negatively asymptotic to $\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}$
and positively asymptotic to $\gamma^{\kappa}_{\mathfrak{g}_{\kappa}(i)}$.

Since the maps $\mathfrak{g}_{\kappa}$ are permutations, there exists a
natural number $M\geq 1$ such that for every
$\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ the $M$-th iterate
$\mathfrak{g}^{M}_{\kappa}$ equals the identity. For each pair
$\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ and
$i\in{1,...,n_{\kappa}}$, we let
$\widetilde{u}^{1}_{\kappa,i}:\mathbbm{R}\times S^{1}\to\mathbbm{R}\times
S^{1}\times\mathbb{D}$ be the lift of the Floer cylinder ${u}^{1}_{\kappa,i}$.
The cylinder $\widetilde{u}^{1}_{\kappa,i}$ is
$\widetilde{J}_{G,J^{s}_{t}}$-holomorphic, where $\widetilde{J}_{G,J^{s}_{t}}$
is the almost complex structure on $\mathbbm{R}\times S^{1}\times\mathbb{D}$
constructed in Section 4.

Claim 2: We claim that the cylinders $\widetilde{u}^{1}_{\kappa,i}$ have no
intersections. More precisely, we have that

  * •

if $\kappa\neq\kappa^{\prime}$ are elements of
$\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$, then for every
$i\in\\{1,...,n_{\kappa}\\}$ and $j\in\\{1,...,n_{\kappa^{\prime}}\\}$, the
cylinders $\widetilde{u}^{1}_{\kappa,i}$ and
$\widetilde{u}^{1}_{\kappa^{\prime},j}$ have no intersections,

  * •

if $\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ and $i\neq j$ are
elements of $\\{1,...,n_{\kappa^{\prime}}\\}$, then
$\widetilde{u}^{1}_{\kappa,i}$ and $\widetilde{u}^{1}_{\kappa,j}$ have no
intersections.

Before proving the claim we explain why it implies Theorem 3. Denote by
$\mathcal{Y}_{\ominus}$ the collection of orbits
$\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}$,
$\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus}),i\in\\{1,...,n_{\kappa}\\}$.
For each $\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$,
$i\in\\{1,...,n_{\kappa}\\}$ and $s\in\mathbbm{R}$ we let
$\xi^{s}_{\kappa,i}:S^{1}\to S^{1}\times\mathbb{D}$ be defined by

$\xi^{s}_{\kappa,i}(\cdot):=\widetilde{u}^{1}_{\kappa,i}(s,\cdot).$

It is easy to see that $\xi^{s}_{\kappa,i}$ is a knot embedded in
$S^{1}\times\mathbb{D}$ for each $s\in\mathbbm{R}$ which intersect each of
disks $\\{t\\}\times\mathbb{D}$ transversely and only once. We also define
$\xi^{\oplus}_{\kappa,i}:S^{1}\to S^{1}\times\Sigma$ by

$\xi^{\oplus}_{\kappa,i}(t):=(t,\gamma^{\kappa}_{i}(t)),$

and $\xi^{\ominus}_{\kappa,i}:S^{1}\to S^{1}\times\Sigma$ by

$\xi^{\ominus}_{\kappa,i}(t):=(t,\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}(t)).$

The braid $\mathcal{B}(\mathcal{Y}_{\oplus})$ equals the disjoint union
$\cup_{\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})}\cup_{i\in\\{1,...,n_{\kappa}\\}}\xi^{\oplus}_{\kappa,i}$.

Since $\mathfrak{g}_{\kappa}$ is always a bijection for each $\kappa$, the
knots $\xi^{\ominus}_{\kappa,i}$ are also disjoint. It follows that
$\mathcal{B}(\mathcal{Y}_{\ominus})=\cup_{\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})}\cup_{i\in\\{1,...,n_{\kappa}\\}}\xi^{\ominus}_{\kappa,i}$
is a braid with the same number of strands as
$\mathcal{B}(\mathcal{Y}_{\oplus})$.

The asymptotic behaviour of the Floer cylinders $\widetilde{u}^{1}_{\kappa,i}$
tells us that

  * •

$\xi^{s}_{\kappa,i}$ converges in $C^{\infty}$ to $\xi^{\oplus}_{\kappa,i}$,

  * •

$\xi^{s}_{\kappa,i}$ converges in $C^{\infty}$ to $\xi^{\ominus}_{\kappa,i}$.

It follows from this that letting
$\xi^{-\infty}_{\kappa,i}:=\xi^{\oplus}_{\kappa,i}$ and
$\xi^{+\infty}_{\kappa,i}:=\xi^{\ominus}_{\kappa,i}$, the families
$(\xi^{s}_{\kappa,i})_{s\in\overline{\mathbbm{R}}}$ define isotopies between
$\xi^{\oplus}_{\kappa,i}$ and $\xi^{\ominus}_{\kappa,i}$.

For each $s\in\overline{\mathbbm{R}}$, let
$\mathcal{B}^{s}=\cup_{\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})}\cup_{i\in\\{1,...,n_{\kappa}\\}}\xi^{s}_{\kappa,i}$.
To prove that $(\mathcal{B}^{s})_{s\in\overline{R}}$ gives a braid isotopy
between $\mathcal{B}(\mathcal{Y}_{\oplus})$ and
$\mathcal{B}(\mathcal{Y}_{\ominus})$, it suffices then to show that

  * •

if $\kappa\neq\kappa^{\prime}$ are elements of
$\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$, then for every
$i\in\\{1,...,n_{\kappa}\\}$ and $j\in\\{1,...,n_{\kappa^{\prime}}\\}$, the
knots $\xi^{s}_{\kappa,i}$ and $\xi^{s}_{\kappa^{\prime},j}$ are disjoint for
each $s\in\mathbbm{R}$,

  * •

if $\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ and $i\neq j$ are
elements of $\\{1,...,n_{\kappa^{\prime}}\\}$, then the knots
$\xi^{s}_{\kappa,i}$ and $\xi^{s}_{\kappa,j}$ are disjoint for each
$s\in\mathbbm{R}$.

But from the definition of the knots $\xi^{s}_{\kappa_{i}}$, it is clear that
these two conditions are equivalent to Claim 5.1.

Step 3.

We thus proceed to prove Claim 5.1, which will imply the theorem.

We consider now for each choice of
$\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ and
$i\in\\{1,...,n_{\kappa}\\}$ the ordered $(2M-2)$-tuple

$(u^{1}_{\kappa,i},u^{2}_{\kappa,i},u^{1}_{\kappa,\mathfrak{g}(i)},u^{2}_{\kappa,\mathfrak{g}(i)},u^{1}_{\kappa,\mathfrak{g}^{2}(i)},u^{2}_{\kappa,\mathfrak{g}^{2}(i)},...,u^{2}_{\kappa,\mathfrak{g}(i)},u^{1}_{\kappa,\mathfrak{g}^{M-1}(i)},u^{2}_{\kappa,\mathfrak{g}^{M-1}(i)}).$

Because of the asymptotic behaviour of the maps
$\widetilde{u}^{\nu}_{\kappa,i}$ they can be compactified to cylinders
$\overline{u}^{\nu}_{\kappa,i}:\overline{\mathbbm{R}}\times
S^{1}\to\overline{\mathbbm{R}}\times S^{1}\times\mathbb{D}$.

We now fix, once and for all, a homeomorphism
$\mathcal{L}:[0,{1}]\to\overline{\mathbbm{R}}$ which is smooth in the interior
of $[0,1]$ and satisfies $\mathcal{L}(0)=-\infty$ and
$\mathcal{L}(1)=+\infty$. For $l\in\\{0,...,M-1\\}$ and $\nu\in\\{1,2\\}$ we
let $\mathcal{L}^{\nu}_{l}:[2l+(\nu-1),2l+\nu]\to\overline{\mathbbm{R}}$ the
homeomorphisms given by
$\mathcal{L}^{\nu}_{l}(u)=\mathcal{L}(u-(2l+(\nu-1)))$.

Using the map $\mathcal{L}^{\nu}_{l}$ we obtain homeomorphisms

${\mathfrak{L}^{\nu}_{l}}:[2l+(\nu-1),2l+\nu]\times
S^{1}\to\overline{\mathbbm{R}}\times S^{1}$

given by ${\mathfrak{L}^{\nu}_{l}}(s,t)=({\mathcal{L}^{\nu}_{l}}(s),t)$ and

${\mathfrak{N}^{\nu}_{l}}:\overline{\mathbbm{R}}\times
S^{1}\times\mathbb{D}\to[2l+(\nu-1),2l+\nu]\times S^{1}\times\mathbb{D}$

given by
$\mathfrak{N}^{\nu}_{l}(s,t,p)=((\mathcal{L}^{\nu}_{l})^{-1}(s),t,p)$. We then
define

$\overline{v}^{\nu,l}_{\kappa,i}:[2l+(\nu-1),2l+\nu]\times
S^{1}\to[2l+(\nu-1),2l+\nu]\times S^{1}\times\mathbb{D}$

by

$\overline{v}^{\nu,l}_{\kappa,i}=\mathfrak{N}^{\nu}_{l}\circ\widetilde{u}^{\nu}_{\kappa,\mathfrak{g}^{l}_{\kappa}(i)}\circ{\mathcal{L}^{\nu}_{l}}.$

We notice that if $i\neq i^{\prime}$, intersections between two cylinders
$\overline{v}^{\nu,l}_{\kappa,i}$ and
$\overline{v}^{\nu^{\prime},l^{\prime}}_{\kappa^{\prime},i^{\prime}}$ can only
occur if $l=l^{\prime}$. In this case, any such intersection is positive,
because of positivity of intersection for holomorphic curves and the method we
used to construct these cylinders from the holomorphic cylinders
$\widetilde{u}^{\nu}_{\kappa,\mathfrak{g}^{l}_{\kappa}(i)}$.

We are ready to define

$\overline{U}^{\kappa}_{i}:[0,2M-2]\times S^{1}\to[0,2M-2]\times
S^{1}\times\mathbb{D}$

by the formula
$\overline{U}^{\kappa}_{i}(s,t)=\overline{v}^{\nu,l}_{\kappa,i}$ if
$s\in[2l+(\nu-1),2l+\nu]$. The cylinders $\overline{U}^{\kappa}_{i}$ should be
thought of as the concatenation of the cylinders forming the ordered tuple
$2M-2$-tuple

$(\overline{u}^{1}_{\kappa,i},\overline{u}^{2}_{\kappa,i},\overline{u}^{1}_{\kappa,\mathfrak{g}(i)},\overline{u}^{2}_{\kappa,\mathfrak{g}(i)},\overline{u}^{1}_{\kappa,\mathfrak{g}^{2}(i)},\overline{u}^{2}_{\kappa,\mathfrak{g}^{2}(i)},...,\overline{u}^{2}_{\kappa,\mathfrak{g}(i)},\overline{u}^{1}_{\kappa,\mathfrak{g}^{M-1}(i)},\overline{u}^{2}_{\kappa,\mathfrak{g}^{M-1}(i)}).$

Because the positive asymptotic limit of an element of the tuple coincides
with the negative asymptotic limit of the next element, the map
$\overline{U}^{\kappa}_{i}$ is indeed continuous, and smooth when $s$ is in
the interior of the intervals $[2l+(\nu-1),2l+\nu]$.

Step 4.

We finish the proof of Claim 5.1.

We argue by contradiction. We assume that there exist $\kappa,\kappa^{\prime}$
and $i,j$ such that either $\kappa\neq\kappa^{\prime}$ or $i\neq j$ and that
$\widetilde{u}^{1}_{\kappa,i}$ and $\widetilde{u}^{1}_{\kappa^{\prime},j}$
intersect. In this case the long cylinders $\overline{U}^{\kappa}_{i}$ and
$\overline{U}^{\kappa^{\prime}}_{j}$ must also intersect.

We claim that all intersections of these cylinders count positively. Indeed,
these intersections must occur in the open sets $(2l+(\nu-1),2l+\nu)\times
S^{1}\times\mathbb{D}$ where the cylinders are smooth and holomorphic,
positivity of intersections for holomorphic curves imply that these
intersections count positively. The reason why the intersections can only
occur in these open sets is that the $l$-th element of the tuples

$(\overline{u}^{1}_{\kappa,i},\overline{u}^{2}_{\kappa,i},\overline{u}^{1}_{\kappa,\mathfrak{g}(i)},\overline{u}^{2}_{\kappa,\mathfrak{g}(i)},\overline{u}^{1}_{\kappa,\mathfrak{g}^{2}(i)},\overline{u}^{2}_{\kappa,\mathfrak{g}^{2}(i)},...,\overline{u}^{2}_{\kappa,\mathfrak{g}(i)},\overline{u}^{1}_{\kappa,\mathfrak{g}^{M-1}(i)},\overline{u}^{2}_{\kappa,\mathfrak{g}^{M-1}(i)})$

and

$(\overline{u}^{1}_{\kappa^{\prime},j},\overline{u}^{2}_{\kappa^{\prime},j},\overline{u}^{1}_{\kappa^{\prime},\mathfrak{g}(j)},\overline{u}^{2}_{\kappa,\mathfrak{g}(j)},\overline{u}^{1}_{\kappa^{\prime},\mathfrak{g}^{2}(j)},\overline{u}^{2}_{\kappa^{\prime},\mathfrak{g}^{2}(j)},...,\overline{u}^{2}_{\kappa^{\prime},\mathfrak{g}^{2}(j)},\overline{u}^{1}_{\kappa^{\prime},\mathfrak{g}^{M-1}(j)},\overline{u}^{2}_{\kappa^{\prime},\mathfrak{g}^{M-1}(j)})$

have different positive and negative asymptotic limits.

Thus, the assumption that $\widetilde{u}^{1}_{\kappa,i}$ and
$\widetilde{u}^{1}_{\kappa^{\prime},j}$ intersect implies that
$\overline{U}^{\kappa}_{i}$ and $\overline{U}^{\kappa^{\prime}}_{j}$ have
positive intersection number. Observe that
$\overline{U}^{\kappa}_{i}(2M-2,\cdot)=\gamma^{\kappa}_{i}(\cdot)$ and
$\overline{U}^{\kappa}_{i}(0,\cdot)=\gamma^{\kappa}_{i}(\cdot)$. Similarly,
$\overline{U}^{\kappa^{\prime}}_{j}(2M-2,\cdot)=\gamma^{\kappa^{\prime}}_{j}(\cdot)$
and
$\overline{U}^{\kappa^{\prime}}_{j}(0,\cdot)=\gamma^{\kappa^{\prime}}_{j}(\cdot)$.

Let $\overline{V}^{\kappa}_{i}:[0,2M-2]\times S^{1}\to[0,2M-2]\times
S^{1}\times\mathbb{D}$ be the trivial cylinder over $\xi^{\oplus}_{\kappa,i}$,
given by $\overline{V}^{\kappa}_{i}(s,t)=(s,t,\gamma^{\kappa}_{i}(t))$.
Because $\mathbb{D}$ is contractible, there exist a homotopy
$H^{\kappa}_{i}:[0,1]\times[0,2M-2]\times S^{1}\to[0,2M-2]\times
S^{1}\times\mathbb{D}$ satisfying:

  * •

$H^{\kappa}_{i}(a,2M-2,\cdot)=(2M-2,\xi^{\oplus}_{\kappa,i}(\cdot))$ for every
$a\in[0,1]$,

  * •

$H^{\kappa}_{i}(a,0,\cdot)=(0,\xi^{\oplus}_{\kappa,i}(\cdot))$ for every
$a\in[0,1]$,

  * •

$H^{\kappa}_{i}(0,\cdot,\cdot)=\overline{V}^{\kappa}_{i}(\cdot,\cdot)$ and
$H^{\kappa}_{i}(1,\cdot,\cdot)=\overline{U}^{\kappa}_{i}(\cdot,\cdot)$.

We refer to $H^{\kappa}_{i}$ as a homotopy of cylinders with the same boundary
between $\overline{V}^{\kappa}_{i}$ and $\overline{U}^{\kappa}_{i}$. We
consider similarly a homotopy of cylinders with the same boundary between
$\overline{V}^{\kappa^{\prime}}_{j}$ and $\overline{U}^{\kappa^{\prime}}_{j}$.

Since either $\kappa\neq\kappa^{\prime}$ or $i\neq j$, the knots
$\xi^{\oplus}_{\kappa,i}$ and $\xi^{\oplus}_{\kappa^{\prime},j}$ are disjoint:
it follows that the cylinders $H^{\kappa}_{i}(a,\cdot,\cdot)$ and
$H^{\kappa^{\prime}}_{j}(a,\cdot,\cdot)$ have disjoint boundaries. The
intersection number of the cylinders $H^{\kappa}_{i}(a,\cdot,\cdot)$ and
$H^{\kappa^{\prime}}_{j}(a,\cdot,\cdot)$ does not depend on $a$. But, this
intersection number is clearly $0$ for $a=0$. But this is in contradiction
with the positivity of the intersection number of
$\overline{U}^{\kappa^{\prime}}_{j}$ and $\overline{U}^{\kappa}_{i}$, which
had followed from the assumption that $\widetilde{u}^{1}_{\kappa,i}$ and
$\widetilde{u}^{1}_{\kappa^{\prime},j}$ intersect.

This contradiction shows that $\widetilde{u}^{1}_{\kappa,i}$ and
$\widetilde{u}^{1}_{\kappa^{\prime},j}$ do not intersect, finishing the proof
of Claim 5.1 and completing the proof of Theorem 3. ∎

Proof of Theorem 2.

The proof of Theorem 2 is identical to the proof of Theorem 3, with the only
modification that one applies Proposition 3.6 where in the previous proof one
applied Proposition 3.8. ∎

##  6\. Proof of Theorem 1

To prove Theorem 1 we must consider several different cases. During the proof
we will refer to the proof steps and arguments used in the proof of Theorem 2,
so the reader should first read the proof of that theorem.

We start treating the case in which $\Sigma=S^{2}$.

###  6.1. Proof of Theorem 1 in case $\Sigma=S^{2}$.

Step 1.

Note that one can choose capping discs $\mathcal{D}_{\gamma_{i}},i=1,\ldots k$
of the orbits in $\mathcal{Y}_{\oplus}$ such that
$\mathcal{A}_{H_{\oplus}}(\gamma_{i},\mathcal{D}_{\gamma_{i}})-\mathcal{A}_{H_{\oplus}}(\gamma_{j},\mathcal{D}_{\gamma_{j}})=\Delta(\gamma_{i},\gamma_{j})$,
for all $i,j\in\\{1,\ldots,k\\}$. Denote by
$\widetilde{\mathcal{Y_{\oplus}}}=\\{(\gamma_{1},\mathcal{D}_{\gamma_{1}}),\ldots,(\gamma_{k},\mathcal{D}_{\gamma_{k}})\\}$
the set of the resulting pairs, and by
$\mathrm{Spec}(\widetilde{\mathcal{Y}_{\oplus}}):=\cup_{1\leq i\leq
k}\mathcal{A}_{H_{\oplus}}(\gamma_{i},\mathcal{D}_{\gamma_{i}})$ their action
values. For each number
$\kappa\in\mathrm{Spec}(\widetilde{\mathcal{Y}_{\oplus}})$ we denote by
$\widetilde{\mathcal{Y}^{\kappa}_{\oplus}}\subset\widetilde{\mathcal{Y}_{\oplus}}$
the subset of elements of $\widetilde{\mathcal{Y}_{\oplus}}$ whose action is
$\kappa$. We denote by $n_{\kappa}$ the cardinality of
$\widetilde{\mathcal{Y}^{\kappa}_{\oplus}}$, and let
$\\{(\gamma^{\kappa}_{1},\mathcal{D}_{\gamma^{\kappa}_{1}}),...,(\gamma^{\kappa}_{n_{\kappa}},\mathcal{D}_{\gamma^{\kappa}_{n_{\kappa}}})\\}$
be the elements of $\widetilde{\mathcal{Y}^{\kappa}_{\oplus}}$.

Fix $\epsilon>0$ as in the statement of the theorem and then consider for
every $\kappa\in\mathrm{Spec}(\widetilde{\mathcal{Y}_{\oplus}})$ the Floer
homology $HF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$. We choose the
smooth $S^{1}$-family $J_{t}$ to be regular for all chain-complexes
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ and
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$, so that we can use
always the same pairs $(H_{\oplus},J_{t})$ to define the Floer differential
$d^{J_{t}}$ on $CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ and
$(H_{\ominus},J_{t})$ to define the Floer differential $d^{J_{t}}$ on
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$. We notice that since
all the elements of $CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ have
the same action the differential vanishes on
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$. Moreover, since
$\mathcal{Y}_{\oplus}$ is $100\epsilon$-isolated, we conclude that the rank of
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ and
$HF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ coincide with
$\\#(\widetilde{\mathcal{Y}^{\kappa}_{\oplus}})$.

We now apply Proposition 3.10 for the chain-complexes
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ and
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ for all
$\kappa\in\mathrm{Spec}(\widetilde{\mathcal{Y}_{\oplus}})$. Notice that the
construction of homotopies $G$ and $\widehat{G}$ does not depend on $\kappa$,
but only on $\epsilon$ and $H_{\ominus}$. We can thus choose generically the
same pairs $(G,J^{s}_{t})$ and $(\widehat{G},\widehat{J}^{s}_{t})$ to induce
maps from $CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ to
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$, and from
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ to
$CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$, for all
$\kappa\in\mathrm{Spec}(\widetilde{\mathcal{Y}_{\oplus}})$.

We then obtain chain maps
$\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ and
$\Psi_{\widehat{G}}:CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$. Since
$\Psi_{\widehat{G}}\circ\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ is chain isotopic to the
identity and the differential $d^{J_{t}}$ vanishes, we conclude that the chain
map
$\Psi_{\widehat{G}}\circ\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ is the identity. It
follows that $\Psi_{G}:CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})\to
CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ is injective and that
$\Psi_{\widehat{G}}:CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})\to
CF^{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ is surjective.

Since $\Psi_{g}$ is injective we know that the dimension of
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ is $\geq n_{\kappa}$.
We let the $m_{\kappa}\geq n_{\kappa}$ be the dimension of
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$, and since
$CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$ is the
$\mathbbm{Z}_{2}$-vector space over
$\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$, we
conclude that
$\\#\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})=m_{\kappa}$.
We write
$\\{(\sigma^{\kappa}_{1},\mathcal{D}_{\sigma^{\kappa}_{1}}),...,(\sigma^{\kappa}_{m_{\kappa}},\mathcal{D}_{\sigma^{\kappa}_{m_{\kappa}}})\\}$
the elements of
$\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$.

For each $j\in\\{1,...,n_{\kappa}\\}$, the image
$\Psi_{G}((\gamma^{\kappa}_{j},\mathcal{D}_{\gamma^{\kappa}_{j}}))$ can be
written in a unique way as a sum of pairs in
$\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$. We say
that a pair $(\sigma^{\kappa}_{l},\mathcal{D}_{\sigma^{\kappa}_{l}})$ appears
in $\Psi_{G}((\gamma^{\kappa}_{j},\mathcal{D}_{\gamma^{\kappa}_{j}}))$, if the
pair $(\sigma^{\kappa}_{l},\mathcal{D}_{\sigma^{\kappa}_{l}})$ is one of the
pairs which appear in the expression of
$\Psi_{G}((\gamma^{\kappa}_{j},\mathcal{D}_{\gamma^{\kappa}_{j}}))$ written in
the base $\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$
of $CF^{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$.

We define in a similar way pairs of
$\mathcal{P}^{1}_{(\kappa-\epsilon,\kappa+\epsilon)}(H_{\oplus})$ which appear
in
$\Psi_{\widehat{G}}((\sigma^{\kappa}_{l},\mathcal{D}_{\sigma^{\kappa}_{l}})$
for each
$(\sigma^{\kappa}_{l},\mathcal{D}_{\sigma^{\kappa}_{l}})\in\mathcal{P}^{1}_{(\kappa-2\epsilon,\kappa+2\epsilon)}(H_{\ominus})$.
Using Lemma 5.1 we know that:

######  Fact 6.1.

It is possible to choose an injective map
$\mathfrak{f}_{\kappa}:\\{1,...,n_{\kappa}\\}\to\\{1,...,m_{\kappa}\\}$ and a
bijective map
$\mathfrak{g}_{\kappa}:\\{1,...,n_{\kappa}\\}\to\\{1,...,n_{\kappa}\\}$ such
that for each $i\in\\{1,...,n_{\kappa}\\}$:

  * •

the pair
$(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)},\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}})$
appears in
$\Psi_{G}((\gamma^{\kappa}_{i},\mathcal{D}_{\gamma^{\kappa}_{i}}))$, and the
orbit
$(\gamma^{\kappa}_{\mathfrak{g}_{\kappa}(i)},\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{g}_{\kappa}(i)}})$
appears in
$\Psi_{\widehat{G}}((\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)},\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}}))$.

Step 2.

For each
$(\gamma^{\kappa}_{i},\mathcal{D}_{\gamma^{\kappa}_{i}})\in\mathcal{Y}^{\kappa}_{\oplus}$
we consider the pair
$(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)},\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}})$.
Since
$(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)},\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}})$
appears in
$\Psi_{G}((\gamma^{\kappa}_{i},\mathcal{D}_{\gamma^{\kappa}_{i}}))$, there
exists a Floer cylinder $u^{1}_{\kappa,i}$ of $(G,J^{s}_{t})$ which is
negatively asymptotic to
$(\gamma^{\kappa}_{i},\mathcal{D}_{\gamma^{\kappa}_{i}})$ and positively
asymptotic to
$(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)},\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}})$.

Similarly, there exists a Floer cylinder $u^{2}_{\kappa,i}$ of
$(\widehat{G},\widehat{J}^{s}_{t})$ which is negatively asymptotic to
$(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)},\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(i)}})$
and positively asymptotic to
$(\gamma^{\kappa}_{\mathfrak{g}_{\kappa}(i)},\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{g}_{\kappa}(i)}})$.

For each pair $\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ and
$i\in{1,...,n_{\kappa}}$, we let
$\widetilde{u}^{1}_{\kappa,i}:\mathbbm{R}\times S^{1}\to\mathbbm{R}\times
S^{1}\times\mathbb{D}$ be the lift of the Floer cylinder ${u}^{1}_{\kappa,i}$.
The cylinder $\widetilde{u}^{1}_{\kappa,i}$ is
$\widetilde{J}_{G,J^{s}_{t}}$-holomorphic, where $\widetilde{J}_{G,J^{s}_{t}}$
is the almost complex structure on $\mathbbm{R}\times S^{1}\times\mathbb{D}$
constructed in Section 4. Reasoning as in Step 2 of the proof of Theorem 3
(see Section 5), we conclude that Theorem 1 in the case $\Sigma=S^{2}$ will
follow if we can show that the cylinders $u^{1}_{\kappa,i}$ and
$u^{1}_{\kappa^{\prime},j}$ have no intersections if either
$\kappa\neq\kappa^{\prime}$ or $i\neq j$.

Step 3.

Since the maps $\mathfrak{g}_{\kappa}$ are permutations, there exists a
natural number $M\geq 1$ such that for every
$\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ the $M$-th iterate
$\mathfrak{g}^{M}_{\kappa}$ equals the identity.

We consider now for each choice of
$\kappa\in\mathrm{Spec}^{1}(\mathcal{Y}_{\oplus})$ and
$i\in\\{1,...,n_{\kappa}\\}$ the ordered $(2M-2)$-tuple

$(u^{1}_{\kappa,i},u^{2}_{\kappa,i},u^{1}_{\kappa,\mathfrak{g}(i)},u^{2}_{\kappa,\mathfrak{g}(i)},u^{1}_{\kappa,\mathfrak{g}^{2}(i)},u^{2}_{\kappa,\mathfrak{g}^{2}(i)},...,u^{2}_{\kappa,\mathfrak{g}(i)},u^{1}_{\kappa,\mathfrak{g}^{M-1}(i)},u^{2}_{\kappa,\mathfrak{g}^{M-1}(i)}).$

Because of the asymptotic behaviour of the maps
$\widetilde{u}^{\nu}_{\kappa,i}$ they can be compactified to cylinders
$\overline{u}^{\nu}_{\kappa,i}:\overline{\mathbbm{R}}\times
S^{1}\to\overline{\mathbbm{R}}\times S^{1}\times\mathbb{D}$.

We now fix, once and for all, a homeomorphism
$\mathcal{L}:[0,{1}]\to\overline{\mathbbm{R}}$ which is smooth in the interior
of $[0,1]$ and satisfies $\mathcal{L}(0)=-\infty$ and
$\mathcal{L}(1)=+\infty$. For $l\in\\{0,...,M-1\\}$ and $\nu\in\\{1,2\\}$ we
let
$\mathcal{L}^{\nu}_{l}:[2l+(\nu-1),2l+\nu]\to\overline{\mathbbm{R}},u\mapsto\mathcal{L}(u-(2l+(\nu-1)))$.

Using the map $\mathcal{L}^{\nu}_{l}$ we obtain homeomorphisms between

${\mathfrak{L}^{\nu}_{l}}:[2l+(\nu-1),2l+\nu]\times
S^{1}\to\overline{\mathbbm{R}}\times S^{1}$

given by ${\mathfrak{L}^{\nu}_{l}}(s,t)=({\mathcal{L}^{\nu}_{l}}(s),t)$ and

${\mathfrak{N}^{\nu}_{l}}:\overline{\mathbbm{R}}\times S^{1}\times
S^{2}\to[2l+(\nu-1),2l+\nu]\times S^{1}\times S^{2}$

given by
$\mathfrak{N}^{\nu}_{l}(s,t,p)=((\mathcal{L}^{\nu}_{l})^{-1}(s),t,p)$. We then
define

$\overline{v}^{\nu,l}_{\kappa,i}:[2l+(\nu-1),2l+\nu]\times
S^{1}\to[2l+(\nu-1),2l+\nu]\times S^{1}\times S^{2}$

by

$\overline{v}^{\nu,l}_{\kappa,i}=\mathfrak{N}^{\nu}_{l}\circ\widetilde{u}^{\nu}_{\kappa,\mathfrak{g}^{l}_{\kappa}(i)}\circ{\mathcal{L}^{\nu}_{l}}.$

We are ready to define

$\overline{U}^{\kappa}_{i}:[0,2M-2]\times S^{1}\to[0,2M-2]\times
S^{1}\times\mathbb{D}$

by the formula
$\overline{U}^{\kappa}_{i}(s,t)=\overline{v}^{\nu,l}_{\kappa,i}$ if
$s\in[2l+\nu,2l+\nu+1]$. One should think of $\overline{U}^{\kappa}_{i}$ as
the concatenation of the cylinders $\overline{v}^{\nu,l}_{\kappa,i}$.

We consider

${W}^{\kappa}_{i}:[0,2M-2]\times S^{1}\to S^{2}$

be the projection of $\overline{U}^{\kappa}_{i}$ to $S^{2}$, i.e.
${W}^{\kappa}_{i}:=\Pi_{S^{2}}\circ\overline{U}^{\kappa}_{i}$ where
$\Pi_{S^{2}}:[0,2M-2]\times S^{1}\times S^{2}\to S^{2}$ is the projection onto
the third coordinate. Let

${v}^{\nu,l}_{\kappa,i}:={u}^{\nu}_{\kappa,\mathfrak{g}^{l}_{\kappa}(i)}\circ{\mathcal{L}^{\nu}_{l}}.$

The cylinder ${v}^{\nu,l}_{\kappa,i}$ is obtained by compactifying the Floer
cylinder $u^{\kappa}_{i}$ and taking its domain to be $[2l+(\nu-1),2l+\nu]$.
It is then clear that

${W}^{\kappa}_{i}:=(\overline{v}^{2,M}_{\kappa,i}\\#\overline{v}^{1,M}_{\kappa,i})\\#(\overline{v}^{2,M-1}_{\kappa,i}\\#\overline{v}^{1,M-1}_{\kappa,i})\\#...\\#(\overline{v}^{2,2}_{\kappa,i}\\#\overline{v}^{1,2}_{\kappa,i})\\#(\overline{v}^{2,1}_{\kappa,i}\\#\overline{v}^{1,1}_{\kappa,i}).$

We need the following definition.

######  Definition 6.1.

Let $(\gamma,[D_{\gamma}])\in\mathcal{P}^{1}(H)$ and
$(\gamma^{\prime},[D_{\gamma^{\prime}}])\in\mathcal{P}^{1}(H^{\prime})$ for
Hamiltonians $H:S^{1}\times S^{2}\to\mathbbm{R}$ and $H^{\prime}:S^{1}\times
S^{2}\to\mathbbm{R}$, and let $V:[0,K]\times S^{1}\to S^{2}$ be a
cylinder444Here $K>0$ is some positive real number. such that $V:\\{0\\}\times
S^{1}=\gamma$ and $V:\\{K\\}\times S^{1}=\gamma^{\prime}$. We say that $V$ is
a cylinder from $(\gamma,[D_{\gamma}])$ and
$(\gamma^{\prime},[D_{\gamma^{\prime}}])$ if the disk $D_{\gamma}\\#V$
obtained by gluing $D_{\gamma}$ and $V$ is homotopic to $D_{\gamma^{\prime}}$
among disks filling $\gamma^{\prime}$.

Claim 3: The cylinder ${W}^{\kappa}_{i}:[0,2M-2]\times S^{1}\to S^{2}$ is a
cylinder from $(\gamma^{\kappa}_{i},[\mathcal{D}_{\gamma^{\kappa}_{i}}])$ to
$(\gamma^{\kappa}_{i},[\mathcal{D}_{\gamma^{\kappa}_{i}}])$.

To prove this claim we need the following straightforward fact.

######  Fact 6.2.

Let $a<b<c$ be real numbers, $V_{1}:[a,b]\to S^{2}$ be a cylinder from
$(\gamma,[D_{\gamma}])$ to $(\gamma^{\prime},[D_{\gamma^{\prime}}])$, and
$V_{2}:[b,c]\to S^{2}$ be a cylinder from
$(\gamma^{\prime},[D_{\gamma^{\prime}}])$ to
$(\gamma^{\prime\prime},[D_{\gamma^{\prime\prime}}])$. Then, the concatenated
cylinder $V_{2}\\#V_{1}:[a,c]\to S^{2}$ is a cylinder from
$(\gamma,[D_{\gamma}])$ to
$(\gamma^{\prime\prime},[D_{\gamma^{\prime\prime}}])$.

By construction, the Floer cylinders $u^{1}_{\kappa,\mathfrak{g}^{l}(i)}$
belong to the moduli space
$\mathcal{M}((\gamma^{\kappa}_{\mathfrak{g}^{l}(i)},[\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{g}^{l}(i)}}]),(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(\mathfrak{g}^{l}(i))},[\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(\mathfrak{g}^{l}(i))}}]),G,J^{s}_{t})$.
It follows that its compactification ${v}^{1,l}_{\kappa,i}$ is a cylinder from
$(\gamma^{\kappa}_{\mathfrak{g}^{l}(i)},[\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{g}^{l}(i)}}])$
to
$(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(\mathfrak{g}^{l}(i))},[\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(\mathfrak{g}^{l}(i))}}])$.
Similarly, one obtains that ${v}^{2,l}_{\kappa,i}$ is a cylinder from
$(\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(\mathfrak{g}^{l}(i))},[\mathcal{D}_{\sigma^{\kappa}_{\mathfrak{f}_{\kappa}(\mathfrak{g}^{l}(i))}}])$
to
$(\gamma^{\kappa}_{\mathfrak{g}^{l+1}(i)},[\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{g}^{l+1}(i)}}])$.

Applying Fact 6.2 multiple times we obtain that the concatenation

${W}^{\kappa}_{i}:=(\overline{v}^{2,M}_{\kappa,i}\\#\overline{v}^{1,M}_{\kappa,i})\\#(\overline{v}^{2,M-1}_{\kappa,i}\\#\overline{v}^{1,M-1}_{\kappa,i})\\#...\\#(\overline{v}^{2,2}_{\kappa,i}\\#\overline{v}^{1,2}_{\kappa,i})\\#(\overline{v}^{2,1}_{\kappa,i}\\#\overline{v}^{1,1}_{\kappa,i})$

is a cylinder from
$(\gamma^{\kappa}_{\mathfrak{(}i)},[\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{(}i)}}])$
to
$(\gamma^{\kappa}_{\mathfrak{g}^{M}(i)},[\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{g}^{M}(i)}}])=(\gamma^{\kappa}_{\mathfrak{(}i)},[\mathcal{D}_{\gamma^{\kappa}_{\mathfrak{(}i)}}])$.

This proves Claim 6.1.

Step 4:

Because ${W}^{\kappa}_{i}$ is a cylinder from
$(\gamma^{\kappa}_{i},[\mathcal{D}_{\gamma^{\kappa}_{i}}])$ to
$(\gamma^{\kappa}_{i},[\mathcal{D}_{\gamma^{\kappa}_{i}}])$, it is homotopic
to the trivial cylinder ${W}^{\kappa,i}_{\rm triv}:[0,2M-2]\times S^{1}\to
S^{2}$ over $\gamma^{\kappa}_{i}$ given by the formula

${W}^{\kappa,i}_{\rm triv}(s,t)=\gamma^{\kappa}_{i}(t),$

among cylinders which map $\\{0\\}\times S^{1}$ and $\\{2M-2\\}\times S^{1}$
to $\gamma^{\kappa}_{i}$.

We explain one way to prove the existence of such a homotopy. The fact that
${W}^{\kappa}_{i}$ is a cylinder from
$(\gamma^{\kappa}_{i},[\mathcal{D}_{\gamma^{\kappa}_{i}}])$ to
$(\gamma^{\kappa}_{i},[\mathcal{D}_{\gamma^{\kappa}_{i}}])$ implies that the
sphere
$S:=\mathcal{D}_{\gamma^{\kappa}_{i}}\\#{W}^{\kappa}_{i}\\#(-\mathcal{D}_{\gamma^{\kappa}_{i}})$
is contractible. Let $p_{0}:=\gamma_{i}^{\kappa}(0)$. The cylinder
${W}^{\kappa}_{i}$ can be thought of as a loop in the free loop space
$\Lambda(S^{2})$ starting and ending at the $\gamma_{i}^{\kappa}$, and we
denote by $[{W}^{\kappa}_{i}]$ the element of
$\pi_{1}(\Lambda(S^{2}),\gamma_{i}^{\kappa})$ represented by
${W}^{\kappa}_{i}$. Let $\Omega_{p_{0}}(S^{2})$ be the based loop space of
$S^{2}$ with basepoint $p_{0}$. From the fibration exact sequence for homotopy
groups associated to the fibration
$\Omega_{p_{0}}(S^{2})\xhookrightarrow{}\Lambda(S^{2})\to S^{2}$ we know that
the inclusion $\Omega_{p_{0}}(S^{2})\xhookrightarrow{}\Lambda(S^{2})$ induces
an isomorphism of fundamental groups. It follows that we can homotopy
${W}^{\kappa}_{i}$ among loops in $\Lambda(S^{2})$ starting and ending at
$\gamma_{i}^{\kappa}$, to a loop $\breve{W}^{\kappa}_{i}$ completely contained
in $\Omega_{p_{0}}(S^{2})$. It is clear that the sphere
$\breve{S}:=\mathcal{D}_{\gamma^{\kappa}_{i}}\\#\breve{W}^{\kappa}_{i}\\#(-\mathcal{D}_{\gamma^{\kappa}_{i}})$
is contractible since it is homotopic to $S$.

We will now use the isomorphism between $\pi_{2}(S^{2})$ and
$\pi_{1}(\Omega(S^{2}))$, where $\Omega({S^{2}})$ is the based loop space of
${S^{2}}$. This isomorphism is not canonical, as it depends on the choice of
loop in $S^{2}$ which will be the base point of $\pi_{1}(\Omega({S^{2}}))$
together with a disk in $S^{2}$ capping this loop. We choose the base point of
$\pi_{1}(\Omega(S^{2}))$ to be the loop $\gamma^{\kappa}_{i}$ and the capping
disk to be $\mathcal{D}_{\gamma^{\kappa}_{i}}$. The isomorphism between
$\pi_{1}(\Omega({S^{2}}),\gamma^{\kappa}_{i})$ and $\pi_{2}(S^{2},p_{0})$ then
identifies
$[\breve{W}^{\kappa}_{i}]\in\pi_{1}(\Omega(S^{2}),\gamma^{\kappa}_{i})$ with
$[\breve{S}]\in\pi_{2}(S^{2},p_{0})$. Since
$[\breve{S}]=0\in\pi_{2}(S^{2},p_{0})$ we obtain that
$[\breve{W}^{\kappa}_{i}]=0\in\pi_{1}(\Omega(S^{2}))$, which implies that
$\breve{W}^{\kappa}_{i}$ is homotopic to the trivial cylinder over
$\gamma^{\kappa}_{i}$ among cylinders which are positively and negatively
asymptotic to $\gamma^{\kappa}_{i}$. Since, as we saw in the previous
paragraph, ${W}^{\kappa}_{i}$ is homotopic to $\breve{W}^{\kappa}_{i}$ among
cylinders which are positively and negatively asymptotic to
$\gamma^{\kappa}_{i}$, it is also homotopic to the trivial cylinder over
$\gamma^{\kappa}_{i}$ among cylinders positively and negatively asymptotic to
$\gamma^{\kappa}_{i}$. This gives us the promised homotopy.

The graph lift of the homotopy between ${W}^{\kappa}_{i}$ and
${W}^{\kappa,i}_{\rm triv}$ gives a homotopy between
$\overline{U}^{\kappa}_{i}$ and the trivial trivial cylinder
$\overline{U}^{\kappa,i}_{\rm triv}:[0,2M-2]\times S^{1}\to[0,2M-2]\times
S^{1}\times S^{2}$ defined by

$\overline{U}^{\kappa,i}_{\rm triv}(s,t)=(s,t,\gamma^{\kappa}_{i}(t)).$

The cylinders $\overline{U}^{\kappa,i}_{\rm triv}$ and
$\overline{U}^{\kappa^{\prime},j}_{\rm triv}$ are disjoint if either
$\kappa\neq\kappa^{\prime}$ or $i\neq j$, which implies that their algebraic
intersection number is $0$. The homotopy between $\overline{U}^{\kappa}_{i}$
and $\overline{U}^{\kappa,i}_{\rm triv}$ and the homotopy between
$\overline{U}^{\kappa^{\prime}}_{j}$ and
$\overline{U}^{\kappa^{\prime},j}_{\rm triv}$ then imply that the algebraic
intersection number of $\overline{U}^{\kappa}_{i}$ and
$\overline{U}^{\kappa^{\prime}}_{j}$ is $0$, if either
$\kappa\neq\kappa^{\prime}$ or $i\neq j$.

Reasoning as in Step 2 of the proof of Theorem 3 we conclude that any
intersection point of $\overline{U}^{\kappa}_{i}$ and
$\overline{U}^{\kappa^{\prime}}_{j}$ counts positively, from where we obtain
that $\overline{U}^{\kappa}_{i}$ and $\overline{U}^{\kappa^{\prime}}_{j}$ are
disjoint. This implies that $\widetilde{u}^{1}_{\kappa,i}$ and
$\widetilde{u}^{1}_{\kappa^{\prime},j}$ have no intersection, if either
$\kappa\neq\kappa^{\prime}$ or $i\neq j$.

Reasoning as in step 2 of the proof of Theorem 3 (see Section 5 ) one obtains
an isotopy between the braid $\mathcal{B}(\mathcal{Y}_{\oplus})$ associated to
$\mathcal{Y}_{\oplus}$, and the braid $\mathcal{B}(\mathcal{Y}_{\ominus})$
associated to
$\mathcal{Y}_{\ominus}=\cup_{\kappa\in\mathrm{Spec}(\widetilde{\mathcal{Y}_{\oplus})}}\cup^{n_{\kappa}}_{i=1}\\{\sigma^{\kappa}_{\mathfrak{f}(i)}\\}$.
∎

###  6.2. Proof of Theorem 1 in case $\Sigma$ has positive genus and the
elements of $\mathcal{Y}_{\oplus}$ are contractible.

Step 1:

We define $\mathrm{Spec}(\mathcal{Y}_{\oplus}):=\cup_{1\leq i\leq
k}\mathcal{A}_{H_{\oplus}}(\gamma_{i})$. For each number
$\kappa\in\mathrm{Spec}(\mathcal{Y}_{\oplus})$ we denote by
$\mathcal{Y}^{\kappa}_{\oplus}\subset\mathcal{Y}_{\oplus}$ the subset of
elements of $\mathcal{Y}_{\oplus}$ whose action is $\kappa$. We denote by
$n_{\kappa}$ the cardinality of $\mathcal{Y}^{\kappa}_{\oplus}$. We denote by
$\\{\gamma^{\kappa}_{1},...,\gamma^{\kappa}_{n_{\kappa}}\\}$ the elements of
$\mathcal{Y}^{\kappa}_{\oplus}$.

We find the maps
$\mathfrak{f}_{\kappa}:\\{1,...,n_{\kappa}\\}\to\\{1,...,m_{\kappa}\\}$ and
$\mathfrak{g}_{\kappa}:\\{1,...,n_{\kappa}\\}\to\\{1,...,n_{\kappa}\\}$ as in
Step 1 of the proof of Theorem 3. We then find the periodic orbits
$\sigma^{\kappa}_{\mathfrak{f}(i)}$ of $H_{\ominus}$.

Following the same reasoning in Step 3 of Section 6.1 we construct the
cylinders

$\overline{U}^{\kappa}_{i}:[0,2M-2]\times S^{1}\to[0,2M-2]\times
S^{1}\times\Sigma,$

and

$W^{\kappa}_{i}:[0,2M-2]\times S^{1}\to\Sigma.$

We will show that the cylinders $\overline{U}^{\kappa}_{i}$ and
$\overline{U}^{\kappa^{\prime}}_{j}$ are disjoint if either
$\kappa\neq\kappa^{\prime}$ or $i\neq j$. Once we establish this, the Theorem
will follow by a reasoning identical to the one in Step 2 of the proof of
Theorem 3 (presented in Section 5).

Step 2:

We fix natural numbers $\kappa,\kappa^{\prime},i,j$, and let either
$\kappa\neq\kappa^{\prime}$ or $i\neq j$. As explained in Section 6.1 the
intersection points of $\overline{U}^{\kappa}_{i}$ and
$\overline{U}^{\kappa^{\prime}}_{j}$ always count positively. Therefore
$\overline{U}^{\kappa}_{i}$ and $\overline{U}^{\kappa^{\prime}}_{j}$ are
disjoint if, and only if, their algebraic intersection number is $0$.

We reason by contradiction assuming that $\overline{U}^{\kappa}_{i}$ and
$\overline{U}^{\kappa^{\prime}}_{j}$ intersect. Let $\widetilde{\Sigma}$ be
the universal covering of $\Sigma$: this induces an obvious covering
$[0,2M-2]\times S^{1}\times\widetilde{\Sigma}$ of $[0,2M-2]\times
S^{1}\times{\Sigma}$. It also induces an obvious covering
$S^{1}\times\widetilde{\Sigma}$ of $S^{1}\times{\Sigma}$.

We can thus find lifts $\widetilde{U}^{\kappa}_{i}(0):[0,2M-2]\times
S^{1}\to[0,2M-2]\times S^{1}\times\widetilde{\Sigma}$ and
$\widetilde{U}^{\kappa^{\prime}}_{j}(0):[0,2M-2]\times S^{1}\to[0,2M-2]\times
S^{1}\times\widetilde{\Sigma}$ of these cylinders to the covering
$[0,2M-2]\times S^{1}\to[0,2M-2]\times S^{1}\times\widetilde{\Sigma}$ of
$[0,2M-2]\times S^{1}\to[0,2M-2]\times S^{1}\times{\Sigma}$, which intersect
each other. The existence of these lifts uses the fact that
$\overline{U}^{\kappa}_{i}$ and $\overline{U}^{\kappa^{\prime}}_{j}$ are
asymptotic to contractible periodic orbits of $H_{\oplus}$: if the orbits were
non-contractible the lifts would not exist.

Associated to the covering $\widetilde{\Sigma}$ of $\Sigma$ there is the group
$\Gamma$ of deck transformations on $\widetilde{\Sigma}$. The group $\Gamma$
is also the group of deck transformations of the covering $[0,2M-2]\times
S^{1}\times\widetilde{\Sigma}$ of $[0,2M-2]\times S^{1}\times{\Sigma}$, and of
the covering $S^{1}\times\widetilde{\Sigma}$ of $S^{1}\times{\Sigma}$.

We denote by $\xi^{\oplus}_{\kappa,i}$ the knot in $S^{1}\times\Sigma$ given
by

$\xi^{\oplus}_{\kappa,i}:=\\{(t,\gamma^{\kappa}_{i}(t)\ |\ t\in S^{1}\\}.$

In an identical way we define $\xi^{\oplus}_{\kappa^{\prime},j}$.

The cylinder $\widetilde{U}^{\kappa}_{i}(0)$ is negatively asymptotic to a
lift $\xi^{\oplus}_{\kappa,i}(0)$ of $\xi^{\oplus}_{\kappa,i}$ in
$S^{1}\times\widetilde{\Sigma}$. It is positively asymptotic to a lift
$\xi^{\oplus}_{\kappa,i}(1)$ of $\xi^{\oplus}_{\kappa,i}$ in
$S^{1}\times\widetilde{\Sigma}$.

Let $\mathcal{T}^{\kappa}_{i}\in\Gamma$ be the deck transformation such that

$\mathcal{T}^{\kappa}_{i}(\xi^{\oplus}_{\kappa,i}(0))=\xi^{\oplus}_{\kappa,i}(1).$

We now define for each $l\in\mathbbm{Z}$

(60)
$\xi^{\oplus}_{\kappa,i}(l)=(\mathcal{T}^{\kappa}_{i})^{l}(\xi^{\oplus}_{\kappa,i}(0)),$

and

(61)
$\widetilde{U}^{\kappa}_{i}(l)=(\mathcal{T}^{\kappa}_{i})^{l}(\widetilde{U}^{\kappa}_{i}(0)).$

It is clear that $\widetilde{U}^{\kappa}_{i}(l)$ is negatively asymptotic to
$\xi^{\oplus}_{\kappa,i}(l-1)$ and positively asymptotic to
$\xi^{\oplus}_{\kappa,i}(l)$.

Using $\widetilde{U}^{\kappa^{\prime}}_{j}(0)$ we define in a similar way
$\mathcal{T}^{\kappa^{\prime}}_{j}$ and the sequence of lifts
$\xi^{\oplus}_{\kappa^{\prime},j^{\prime}}(l)$ and
$\widetilde{U}^{\kappa^{\prime}}_{j}(l)$ for $l\in\mathbbm{Z}$.

Step 3:

Since $\Sigma$ is a surface of positive genus its universal cover
$\widetilde{\Sigma}$ is diffeomorphic to the plane. We can thus consider the
lifts of $\xi^{\oplus}_{\kappa,i}$ and of $\xi^{\oplus}_{\kappa^{\prime},j}$
to $S^{1}\times\widetilde{\Sigma}$ as knots on the plane, and define their
crossing number of a pair of knots $\mathrm{Cross}$ as in Section 5 of [47].
Using positivity of intersections the authors showed in Lemma 5.4 of [47] that
because $\widetilde{U}^{\kappa}_{i}(0)$ and
$\widetilde{U}^{\kappa^{\prime}}_{j}(0)$ have positive intersection number,
the crossing numbers of
$\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(0),\xi^{\oplus}_{\kappa^{\prime},j}(0))$
and
$\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(1),\xi^{\oplus}_{\kappa^{\prime},j}(1))$
satisfy

(62)
$\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(0),\xi^{\oplus}_{\kappa^{\prime},j}(0))>\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(1),\xi^{\oplus}_{\kappa^{\prime},j}(1)).$

Moreover using the fundamental Lemma 5.4 of [47] we obtain the following fact:

######  Fact 6.3.

If $\widetilde{U}^{\kappa}_{i}(0)$ and
$\widetilde{U}^{\kappa^{\prime}}_{j}(0)$ intersect, then

$\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(l-1),\xi^{\oplus}_{\kappa^{\prime},j}(l-1))>\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(l),\xi^{\oplus}_{\kappa^{\prime},j}(l)).$

If $\widetilde{U}^{\kappa}_{i}(0)$ and
$\widetilde{U}^{\kappa^{\prime}}_{j}(0)$ do not intersect, then

$\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(l-1),\xi^{\oplus}_{\kappa^{\prime},j}(l-1))=\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(l),\xi^{\oplus}_{\kappa^{\prime},j}(l)).$

We conclude from this fact that the sequence
$(\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(l),\xi^{\oplus}_{\kappa^{\prime},j}(l)))_{l\in\mathbbm{Z}}$
is a non-increasing sequence of integers.

Endow $\Sigma$ with an auxiliary Riemannian metric $g$ and let $\widetilde{g}$
be the induced metric on the universal cover $\widetilde{\Sigma}$. Take a
Riemannian metric $g_{0}$ on $S^{1}$ and consider $g^{\prime}$ to be the
product metric of $g_{0}$ and $\widetilde{g}$ on
$S^{1}\times\widetilde{\Sigma}$. This Riemannian metric induces a distance
function $d$ on $S^{1}\times\widetilde{\Sigma}$ which we will fix from now on.
Remark that $\Gamma$ acts on $S^{1}\times\widetilde{\Sigma}$ by isometries.

Fix a lift $\widetilde{\xi}^{\oplus}_{\kappa,i}$. There are only finitely many
lifts of $\widetilde{\xi}^{\oplus}_{\kappa^{\prime},j}$ to
$S^{1}\times\widetilde{\Sigma}$ which have non-zero crossing number with
$\widetilde{\xi}^{\oplus}_{\kappa,i}$. The reason for this is that for a lift
of $\widetilde{\xi}^{\oplus}_{\kappa^{\prime},j}$ to have non-zero crossing
number with $\widetilde{\xi}^{\oplus}_{\kappa,i}$, its projection to the the
plane $\widetilde{\Sigma}$ must intersect the projection of
$\widetilde{\xi}^{\oplus}_{\kappa,i}$ to $\widetilde{\Sigma}$: clearly, there
are only finitely many lifts of $\widetilde{\xi}^{\oplus}_{\kappa^{\prime},j}$
with this property. It follows that there exists a constant $\mathrm{D}>0$
such that any lift $\widetilde{\xi}$ of ${\xi}^{\oplus}_{\kappa^{\prime},j}$
such that $d(\widetilde{\xi}^{\oplus}_{\kappa,i},\widetilde{\xi})>D$ satisfies

$\mathrm{Cross}(\widetilde{\xi}^{\oplus}_{\kappa,i},\widetilde{\xi})=0.$

Because $\Gamma$ acts by isometry we obtain that this constant $D$ does not
depend on the lift of ${\xi}^{\oplus}_{\kappa,i}$. This gives us the following
fact:

######  Fact 6.4.

There exists a constant $D>0$ such that if $\widehat{\xi}_{\kappa,i}$ is a
lift of ${\xi}^{\oplus}_{\kappa,i}$ and $\widehat{\xi}_{\kappa^{\prime},j}$ is
a lift of ${\xi}^{\oplus}_{\kappa^{\prime},j}$ such that
$d({\xi}^{\oplus}_{\kappa^{\prime},j},\widehat{\xi}_{\kappa^{\prime},j})>D$
then

$\mathrm{Cross}(\widehat{\xi}_{\kappa,i},\widehat{\xi}_{\kappa^{\prime},j})=0.$

The next fact follows easily from the observation above that there are only
finitely many lifts of $\widetilde{\xi}^{\oplus}_{\kappa^{\prime},j}$ to
$S^{1}\times\widetilde{\Sigma}$ which have non-zero crossing number with
$\widetilde{\xi}^{\oplus}_{\kappa,i}$.

######  Fact 6.5.

Given any lift $\widetilde{\xi}^{\oplus}_{\kappa,i}$ of
${\xi}^{\oplus}_{\kappa,i}$ we have

$\max_{\widetilde{\xi}\mbox{ is a lift of
}{\xi}^{\oplus}_{\kappa^{\prime},j}}\\{|\mathrm{Cross}(\widetilde{\xi}^{\oplus}_{\kappa,i},\widetilde{\xi})|\\}<+\infty.$

Step 4:

We now have to deal with two cases:

  * (A)

The deck transformations $\mathcal{T}^{\kappa}_{i}$ and
$\mathcal{T}^{\kappa^{\prime}}_{j}$ do not coincide.

  * (B)

The deck transfomations $\mathcal{T}^{\kappa}_{i}$ and
$\mathcal{T}^{\kappa^{\prime}}_{j}$ coincide.

We start treating the case (A).

Case (A):

Since $\mathcal{T}^{\kappa}_{i}$ and $\mathcal{T}^{\kappa^{\prime}}_{j}$ do
not coincide, we can find a positive integer $N>0$ such that

$d((\mathcal{T}^{\kappa}_{i})^{l}(\xi^{\oplus}_{\kappa,i}(0)),(\mathcal{T}^{\kappa^{\prime}}_{j})^{l}(\xi^{\oplus}_{\kappa^{\prime},j}(0)))>D\mbox{
for all }l\geq N,$

and

$d((\mathcal{T}^{\kappa}_{i})^{l}(\xi^{\oplus}_{\kappa,i}(0)),(\mathcal{T}^{\kappa^{\prime}}_{j})^{l}(\xi^{\oplus}_{\kappa^{\prime},j}(0)))>D\mbox{
for all }l\leq-N,$

where $D>0$ is the constant given by Fact 6.4.

In particular, we conclude that

$\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(N),\xi^{\oplus}_{\kappa^{\prime},j}(N))=\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(-N),\xi^{\oplus}_{\kappa^{\prime},j}(-N))=0.$

But, as observed above, Fact 6.3 implies that the sequence
$(\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(l),\xi^{\oplus}_{\kappa^{\prime},j}(l)))_{l\in\mathbbm{Z}}$
is a non-increasing sequence of integers. We conclude that for every integer
$-N\leq p\leq N$ we have

$\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(p),\xi^{\oplus}_{\kappa^{\prime},j}(p))=0.$

But this is in contradiction with (62) which we derived from the fact that
$\widetilde{U}^{\kappa}_{i}(0)$ and $\widetilde{U}^{\kappa^{\prime}}_{j}(0)$
have positive intersection number. This concludes the proof in case (A).

Case (B):

In this case we let
$\mathcal{T}:=\mathcal{T}^{\kappa}_{i}=\mathcal{T}^{\kappa^{\prime}}_{j}$.
Since $\widetilde{U}^{\kappa}_{i}(0)$ and
$\widetilde{U}^{\kappa^{\prime}}_{j}(0)$ have positive intersection the same
is true for
$\widetilde{U}^{\kappa}_{i}(l)=\mathcal{T}^{l}(\widetilde{U}^{\kappa}_{i}(0))$
and $\mathcal{T}^{l}(\widetilde{U}^{\kappa^{\prime}}_{j}(l))$ for all
$l\in\mathbbm{Z}$. Lemma 5.4 of [47] then implies that for all
$l\in\mathbbm{Z}$

(63)
$\mathrm{Cross}(\xi^{\oplus}_{\kappa^{\prime},j}(l-1),\xi^{\oplus}_{\kappa^{\prime},j}(l-1))>\mathrm{Cross}(\xi^{\oplus}_{\kappa^{\prime},j}(l),\xi^{\oplus}_{\kappa^{\prime},j}(l)).$

It is then clear that the sequence
$(\mathrm{Cross}(\xi^{\oplus}_{\kappa,i}(l),\xi^{\oplus}_{\kappa^{\prime},j}(l)))_{l\in\mathbbm{Z}}$
is unbounded. But this is in contradiction with Fact 6.5. This establishes the
theorem in case (B), and concludes the proof of Theorem 1 under the hypothesis
the $\Sigma$ has positive genus and $\mathcal{Y}_{\oplus}$ is formed by
contractible $1$-periodic orbits of $H_{\oplus}$. ∎

###  6.3. Proof of Theorem 1 in case $\Sigma$ has genus $\geq 2$ and the
elements of $\mathcal{Y}_{\oplus}$ are non-contractible.

Outline of the proof:

The proof in this case is identical to the proof of Theorem 3. The reason for
this is that if $\gamma$ is a non-contractible loop in $\Sigma$ then any
cylinder in $\Sigma$ which is both positively and negatively asymptotic to
$\gamma$ is homotopic to the trivial cylinder over $\gamma$: this is the case
because $\Sigma$ is atoroidal since it has genus $\geq 2$.

We use the same notation as in the proof of Theorem 3. We then construct for
each $\kappa\in\mathrm{Spec}(\mathcal{Y}_{\oplus})$ and
$i\in\\{1,...,n_{\kappa}\\}$ the cylinder $\overline{U}^{\kappa}_{i}$ as in
the proof of Theorem 3 and obtain from the previous observation that it is
homotopic to the trivial cylinder $\overline{U}^{\kappa,i}_{\rm
triv}:[0,2M-2]\times S^{1}\to[0,2M-2]\times S^{1}\times S^{2}$ defined by

$\overline{U}^{\kappa,i}_{\rm triv}(s,t)=(s,t,\gamma^{\kappa}_{i}(t)).$

Once this is obtained the proof of the present case of Theorem 1 follows the
arguments presented in Step 2 of the the proof of Theorem 3 (presented in
Section 5). ∎

###  6.4. Proof of Theorem 1 in case $\Sigma=T^{2}$ and the elements of
$\mathcal{Y}_{\oplus}$ are non-contractible.

Outline of the proof:

The proof in this case is similar to the one presented in Section 6.1 for the
case $\Sigma=S^{2}$. Namely, we cannot guarantee by purely topological reasons
that the cylinder $\overline{U}^{\kappa}_{i}$ is homotopic to the trivial
cylinder $\overline{U}^{\kappa,i}_{\rm triv}$ among cylinders which are
positively and negatively asymptotic to $\gamma^{\kappa}_{i}$, since $T^{2}$
is toroidal.

However, using an argument identical to the one presented in Step 3 of Section
6.1, we can show that if $[\mathrm{Cyl}_{\gamma^{\kappa}_{i}}]$ is a choice of
capping for $\gamma^{\kappa}_{i}$, then the cylinder ${W}^{\kappa}_{i}$ (which
is the projection of $\overline{U}^{\kappa}_{i}$ to $T^{2}$) is a cylinder
from $(\gamma_{i}^{\kappa},[\mathrm{Cyl}_{\gamma^{\kappa}_{i}}])$ to itself.
It follows from this that ${W}^{\kappa}_{i}$ represents the trivial element in
$\pi_{1}(\Omega_{[\gamma^{\kappa}_{i}]}(T^{2},\gamma^{\kappa}_{i}))$, where
$\Omega_{[\gamma^{\kappa}_{i}]}(T^{2},\gamma^{\kappa}_{i})$ is the connected
component of the loop space of $T^{2}$ that contains $\gamma^{\kappa}_{i}$. We
conclude that ${W}^{\kappa}_{i}$ is homotopic to the trivial cylinder
${W}^{\kappa,i}_{\rm triv}$ over $\gamma^{\kappa}_{i}$. Lifting this homotopy
we obtain the desired homotopy between $\overline{U}^{\kappa}_{i}$ and the
trivial cylinder $\overline{U}^{\kappa,i}_{\rm triv}$ among cylinders which
are positively and negatively asymptotic to $\gamma^{\kappa}_{i}$. Once this
is obtained the proof is completed by using the argument of Step 2 of the
proof of Theorem 5 (presented in Section 5). ∎

##  7\. A variant of Theorems 1 and 2

We explain in this section why the assumption of the orbits to be
$100\epsilon$-separated in the main theorems can be replaced by a slightly
weaker but more technical assumption of being $100\epsilon$-quasi-isolated.

######  Theorem 1*.

The statement in Theorem 1 holds true if the assumption that
$\mathcal{Y}_{\oplus}$ is $100\epsilon$-isolated is replaced by the assumption
that $\mathcal{Y}_{\oplus}$ is $100\epsilon$-quasi-isolated for some pair
$(H_{\oplus},J_{\oplus})$, where $J_{\oplus}$ is a $S^{1}$ family of
compatible almost complex structures.

######  Theorem 2*.

The statement in Theorem 2 holds true if the assumption that
$\mathcal{Y}_{\oplus}$ is $100\epsilon$-isolated is replaced by the assumption
that $\mathcal{Y}_{\oplus}$ is $100\epsilon$-quasi-isolated for some pair
$(H_{\oplus},J_{\oplus})$, where $J_{\oplus}$ is a $S^{1}$ family of
compatible almost complex structures.

The proofs of Theorems 1 and 2 also apply here with small modifications.

###### Proof of Theorem 2*.

Let $\phi_{\oplus}$, $H_{\oplus}$, $\mathcal{Y}_{\oplus}$ and $J_{\oplus}$ as
in the theorem for some $\epsilon>0$, and let $\phi_{\ominus}$ be non-
degenerate Hamiltonian diffeomorphism with
$d_{\mathrm{Hofer}}(\phi_{\ominus},\phi_{\oplus})<\epsilon$. As in the proof
of Theorem 2 one writes $\mathcal{Y}_{\oplus}$ as a disjoint union of sets
$\mathcal{Y}_{\oplus}^{\kappa}$ of orbits $\gamma\in\mathcal{Y}_{\oplus}$ with
$\mathcal{A}_{H_{\oplus}}(\gamma)=\kappa$. One then considers for each
$\kappa$ with $\mathcal{Y}_{\oplus}^{\kappa}\neq\emptyset$ the
$\mathbbm{Z}_{2}$ vector space $\mathcal{B}_{\mathcal{Y}^{\kappa}_{\oplus}}$
generated by $\mathcal{Y}_{\oplus}^{\kappa}$ as subcomplexes of
$CF^{(\kappa-100\epsilon,\kappa+100\epsilon)}(H_{\oplus})$, as done in Section
3.1.3. Proposition 3.7 gives then $H_{\ominus}$ generating $\phi_{\ominus}$
and homomorphisms
$\Psi^{\mathcal{Y}}_{G}:\mathcal{B}_{\mathcal{Y}^{\kappa}_{\oplus}}\to
CF^{(\kappa-99\epsilon,\kappa+99\epsilon)}(H_{\ominus})$ and
$\Psi^{\mathcal{Y}}_{\widehat{G}}:CF^{(\kappa-99\epsilon,\kappa+99\epsilon)}(H_{\ominus})\to\mathcal{B}_{\mathcal{Y}^{\kappa}_{\oplus}}$,
with $\Psi^{\mathcal{Y}}_{\widehat{G}}\circ\Psi^{\mathcal{Y}}_{G}=id$. One
then can proceed in the same way as in the proof of Theorem 2, using the maps
$\Psi^{\mathcal{Y}}_{G}$ and $\Psi^{\mathcal{Y}}_{\widehat{G}}$ instead of the
maps $\Psi_{G}$ and $\Psi_{\widehat{G}}$. Note that the condition that
$|\mathcal{A}_{H_{\oplus}}(\gamma)-\mathcal{A}_{H_{\oplus}}(\gamma^{\prime})|\notin(0,100\epsilon)$
for $\gamma,\gamma^{\prime}\in\mathcal{Y}_{\oplus}$ guarantees that the first
part of Claim 2 in the proof of Theorem 2 will again hold.

∎

###### Proof of Theorem 1*.

Here again, the proof of Theorem 1 applies with the same small modification.
Instead of the homomorphisms $\Psi_{G}$ and $\Psi_{\widehat{G}}$ that are
guaranteed, for an $100\epsilon$-isolated set of orbits with the same action,
by Proposition 3.1 ($\Sigma$ is a closed surface) resp. Proposition 3.10
($\Sigma=S^{2}$), one uses the homomorphims $\Psi^{\mathcal{Y}}_{G}$ and
$\Psi^{\mathcal{Y}}_{\widehat{G}}$ that are given in Proposition 3.4 resp.
Proposition 3.11, for an $100\epsilon$-quasi-isolated set of orbits of the
same action.

∎

##  8\. Proof of Theorem 2.5

In this section we deduce Theorem 2.5 from Theorems 1* and 2*.

We start the with the following proposition. Consider first $\Sigma$ to be a
closed surface. And let $H:\Sigma\times S^{1}\to\mathbbm{R}$ be a normalized
Hamiltonian, and $\phi=\phi^{1}_{H}$ the diffeomorphism generated by $H$. Let
$\mathcal{Y}=\\{\gamma_{1},\ldots,\gamma_{k}\\}$ be a collection of pairwise
freely homotopic, non-degenerate $1$-periodic orbits for $H$.

######  Proposition 8.1.

There are open disks $U_{i}\subset\Sigma$, $i=1,\ldots,n$ with
$\gamma_{i}(0)\in U_{i}$, $i=1,\ldots,k$, $\varepsilon^{\prime}>0$, a non-
degenerate Hamiltonian $H^{\prime}$ that generates a diffeomorphism
$\phi^{\prime}=\phi^{1}_{H^{\prime}}$, and a $S^{1}$-family of compatible
almost complex structures $J^{\prime}=J^{\prime}_{t}$ with the property that

  * •

$d_{\mathrm{Hofer}}(\phi^{\prime},\phi)<{\varepsilon^{\prime}}$,

  * •

$\phi^{\prime}|_{U_{i}}=\phi|_{U_{i}}$, $i=1,\ldots k$,

  * •

$\mathcal{Y}$ is $200\varepsilon^{\prime}$\- quasi-isolated for
$(H^{\prime},J^{\prime})$

###### Proof.

Note that the orbits in $\mathcal{Y}$ are topologically isolated among all the
$1$-periodic orbits for $H$. It is a variant of the standard argument for the
genericity of non-degenerate Hamiltonian diffeomorphisms that for sufficiently
small neigbourhoods $V_{i}$ of $\\{(t,\gamma_{i}(t))\,|\,t\in S^{1}\\}$ in
$S^{1}\times\Sigma$, there is a sequence of non-degenerate Hamiltonians
$H_{j}:S^{1}\times\Sigma\to\mathbbm{R}$, $j\in\mathbbm{N}$, with
$H_{j}|_{V_{i}}=H|_{V_{i}}$, for all $i=1,\ldots,k$, $j\in\mathbbm{N}$, and
such that $H_{j}$ converge to $H$ in $C^{\infty}$. We may choose the $V_{i}$
such that $U_{i}:={\rm pr}_{2}(V_{i}\cap(\\{0\\}\times\Sigma))\subset\Sigma$
are discs, where ${\rm pr}_{2}$ is the projection to the second coordinate,
and such that $\gamma_{i}(0)$ is the only point $z\in U_{i}$ with $\phi(z)=z$.
We can choose a sequence $J_{j}$, $j\in\mathbbm{N}$, of $S^{1}$ families of
compatible almost complex structures on $\Sigma$ such that the pairs
$(H_{j},J_{j})$ are regular, and such that $J_{j}$ converges with all
derivatives to a $S^{1}$-family of compatible almost complex structures $J$.

We claim that there is $\varepsilon^{\prime}>0$ and $j\in\mathbbm{N}$, such
that $H^{\prime}:=H_{j}$ and $J^{\prime}=J_{j}$ satisfy the asserted
properties from the proposition. The condition $a)$ in the definition of
$\varepsilon^{\prime}$-quasi-isolation is clearly satisfied for any
sufficiently small $\varepsilon^{\prime}$, hence if the above does not hold,
then one can pass to a subsequence of $H_{j}$ for which there are non-constant
$u_{j}:\mathbbm{R}\times S^{1}\to M$ that satisfy
$\mathcal{F}_{H_{j},J_{j}}(u_{j})=0$ such that $u_{j}$ are positive or
negative asymptotic to some $\gamma_{i}$, and for which $E(u_{j})\to 0$. Here
the energies of $u_{j}$ are defined with respect to $J_{j}$. W.l.o.g and by
passing further to a subsequence we may assume that $u_{j}$ are all negatively
asymptotic to $\gamma_{i_{0}}$ for some fixed $i_{0}\in\\{1,\ldots,k\\}$. For
any non-constant Floer trajectory, the periodic orbits that appear as its
negative and positive asymptotics do not coincide, so for any
$j\in\mathbbm{N}$, ${u_{j}}$ is positively asympotic to some $1$-periodic
orbit for $H_{j}$ which is different from $\gamma_{i_{0}}$. It follows that
there is $s_{j}$ such that $u_{j}(s_{j},0)$ lies in the boundary $\partial
U_{i_{0}}$ of $U_{i_{0}}$. We may assume that $s_{j}=0$ for all
$j\in\mathbbm{N}$, and, by passing to a further subsequence, that $u_{j}(0,0)$
converges to some $x_{0}\in\partial U_{i_{0}}$. The first derivatives of
$u_{j}$ are uniformely bounded. Otherwise, by a bubbling-off argument, since
$E(u_{j})\to 0$, one will find a non-constant holomorphic sphere with zero
energy, a contradiction. With this, one proves, as it is standard in Floer
theory, that up to a further subsequence, $(u_{j})_{j\in\mathbbm{N}}$
uniformely converges with all their derivatives on compact subsets. Any limit
curve will satisfy $\mathcal{F}_{H,J}(u)=0,E(u)=0$. In particular for such $u$
we have that $\lim_{s\to\infty}u(s,t)=\lim_{s\to-\infty}u(s,t)$ is a periodic
orbit for $H$. But by the above we in particular find such $u$ with
$u(0,0)=x_{0}\in\partial U_{i_{0}}$, which contradicts the fact that
$\phi(z)\neq z$ for all $z\in\partial U_{i_{0}}$. ∎

We have the following variant of the above proposition. For that consider
$\Sigma=\mathbb{D}$. Let $H\in\mathcal{H}_{0}(\mathbb{D})$ and
$\phi\in\mathrm{Ham}_{0}(\mathbb{D})$ generated by $H$. Let
$\mathcal{Y}=\\{\gamma_{1},\ldots,\gamma_{k}\\}$ be a collection of non-
degenerate $1$-periodic orbits for $H$. For $0<\rho<1$ we denote by
$\mathbb{D}_{1-\rho}$ the set $\\{(r,\theta)\in\mathbb{D}\,|\,1-\rho>r\geq
0\\},$ where $(r,\theta)$ denote the polar coordinates.

######  Proposition 8.2.

There are open disks $U_{i}\subset\mathbb{D}$, $i=1,\ldots,n$ with
$\gamma_{i}(0)\in U_{i}$, $i=1,\ldots,k$, and $\varepsilon^{\prime}>0$, $c>0$,
$\rho_{0}>0$ such that for each $0<\rho<\rho_{0}$ there is a non-degenerate
Hamiltonian $H^{\prime}\in\mathcal{H}_{c}(\mathbb{D})$ that generates a
diffeomorphism
$\phi^{\prime}=\phi^{1}_{H^{\prime}}\in\mathrm{Ham}_{c}(\mathbb{D})$, and
there is a $S^{1}$-family of compatible almost complex structures
$J^{\prime}=J^{\prime}_{t}$ with the property that

  * •

$d_{C^{2}}(H|_{\mathbb{D}_{1-\rho}},H^{\prime}|_{\mathbb{D}_{1-\rho}})<{\varepsilon^{\prime}}$,

  * •

$\phi^{\prime}|_{U_{i}}=\phi|_{U_{i}}$, $i=1,\ldots k$,

  * •

there are no periodic orbits of $\phi^{\prime}$ in
$\mathbb{D}\setminus\mathbb{D}_{1-\rho}$,

  * •

$\mathcal{Y}$ is $200\varepsilon^{\prime}$\- quasi-isolated for
$(H^{\prime},J^{\prime})$

###### Proof.

Since the orbits in $\mathcal{Y}$ are non-degenerate, they are contained in
the interior of the support of $H$. Hence we can find for sufficiently small
$c>0$, sets $U_{i}$, $i=1,\ldots,k$ and a sequence
$H_{j}\in\mathcal{H}_{c}(\mathbb{D})$ with

$d_{C^{2}}(H_{j}|_{\mathbb{D}_{1-\frac{1}{j}}},H|_{\mathbb{D}_{1-\frac{1}{j}}})\to
0$

as $j\to\infty$, and that coincide with $H$ on a sufficiently small
neighbourhood of the suspension of $\gamma_{1},\ldots,\gamma_{k}$, and we can
find a sequence of compatible almost complex structures $J_{j}$ such that the
pairs $(H_{j},J_{j})$ are regular, and such that $J_{j}$ converges to a
compatible almost complex structure $J$. We can additionally assume that there
are no $1$-periodic orbits for $H_{j}$ in the complement of
$\mathbb{D}_{1-\frac{1}{j}}$.

Now, similarly as above, we claim that there is $\varepsilon^{\prime}>0$ and
$j_{0}\in\mathbbm{N}$, such that for all $j>j_{0}$, $\mathcal{Y}$ is
$200\varepsilon^{\prime}$\- quasi-isolated for $(H_{j},J_{j})$. If this is not
the case then there is subsequence of $H_{j}$ and a sequence of
$u_{j}:\mathbbm{R}\times S^{1}\to\mathbbm{R}^{2}$ such that
$\mathcal{F}_{H_{j},J_{j}}(u_{j})=0$, $E(u_{j})\to 0$, and w.l.o.g $u_{j}$ are
all negative asymptotic to $\gamma_{i_{0}}$ for some fixed
$i_{0}\in\\{1,\ldots,k\\}$. As before, this gives rise to a contradiction. ∎

###### Proof of Theorem 2.5.

We start with the case that $\Sigma$ is a closed surface. Let
$H_{\oplus},\phi_{\oplus},\mathcal{Y}_{\oplus}=\\{\gamma_{1},\ldots,\gamma_{k}\\}$
given as in the theorem.

We apply Proposition 8.1 with $H=H_{\oplus}$,
$\mathcal{Y}=\mathcal{Y}_{\oplus}$, and obtain $\varepsilon^{\prime}$ as well
as a Hamiltonian $H^{\prime}$ and a $S^{1}$ family of compatible almost
complex structures $J^{\prime}$, such that the conclusions of the proposition
hold for $\varepsilon^{\prime},J^{\prime},H^{\prime}$ and $\mathcal{Y}$.

Apply now Theorem 1* with now $H_{\oplus}=H^{\prime}$,
$J_{\oplus}=J^{\prime}$, $\mathcal{Y}_{\oplus}=\mathcal{Y}$
$\epsilon=2\varepsilon^{\prime}$. Since any ball
$B\in\mathrm{Ham}(\Sigma,\omega)$ of radius $2\varepsilon^{\prime}$ with a
center that lies in the ball $B^{\prime}$ of radius $\varepsilon^{\prime}$
contains $B^{\prime}$, the conclusion of Theorem 1* imply now Theorem 2.5 in
the case of closed surfaces.

Let now $\Sigma=\mathbb{D}$. Let $H_{\oplus}\in\mathcal{H}_{0}(\mathbb{D})$
non-degenerate and $\phi_{\oplus}$ generated by $H_{\oplus}$. Let
$\mathcal{Y}_{\oplus}$ be a collection of non-degenerate $1$-period orbits for
$H_{\oplus}$. We apply Proposition 8.2 with $H=H_{\oplus}$,
$\mathcal{Y}=\mathcal{Y}_{\oplus}$ and obtain $\varepsilon^{\prime}>0$, $c>0$
$\rho_{0}>0$, such that for each $0<\rho<\rho_{0}$ there is a pair
$(H^{\prime},J^{\prime})$ such that the conclusions of the proposition hold.

Let $\epsilon=2\varepsilon^{\prime}$. Let
$\phi\in\mathrm{Ham}_{0}(\mathbb{D})$ be non-degenerate in its support with
$d_{\mathrm{Hofer}}(\phi,\phi_{\oplus})<\epsilon$. Then there is $\rho_{1}$
such that $\mathrm{supp\,}\phi\in\mathbb{D}_{1-\rho_{1}}$. Choose
$0<\rho_{2}<\min\\{\rho_{0},\rho_{1}\\}$. By a sufficiently $C^{2}$ small
approximation of $H^{\prime}$ we can now find a non-degenerate Hamiltonian
$H_{\ominus}\in\mathcal{H}_{c}(\mathbb{D}^{2})$ that generates a
diffeomorphism $\phi_{\ominus}\in\mathrm{Ham}_{c}(\mathbb{D}^{2})$ such that
$F(t,p):=H_{\ominus}(t,(\phi_{\oplus}^{t})(p))-H_{\oplus}(t,(\phi_{\oplus}^{t})(p))$
that generates $\phi_{\oplus}^{-1}\circ\phi_{\ominus}$ satisfies
$\int_{0}^{1}(\max F_{t}-\min F_{t})\,dt<\epsilon$, and that all the
$1$-periodic orbits for $H_{\ominus}$ are also $1$-periodic orbits of
$H^{\prime}$. In particular we have
$d_{\mathrm{Hofer}}(\phi_{\ominus},\phi_{\oplus})<\epsilon$.

Now apply Theorem 2* with $\epsilon$ from above, and now
$H_{\oplus}=H^{\prime}$, $J_{\oplus}=J^{\prime}$,
$\mathcal{Y}_{\oplus}=\mathcal{Y}$, and note that one can choose in the
conclusions of the theorem $H_{\ominus}$ as above. Hence we obtain a
collection of non-degenerate $1$-periodic orbits $\mathcal{Y}_{\ominus}$ such
that

$\mathcal{B}(\mathcal{Y}_{\ominus})\mbox{ is freely isotopic as a braid to
}\mathcal{B}(\mathcal{Y}_{\oplus}).$

Since all $1$-periodic orbits for $H_{\ominus}$ are already $1$-periodic
orbits for $H^{\prime}$, the conclusion follows.

∎

##  Appendix A Combinatorial lemma

In this appendix we prove Lemma 5.1. For the convenience of the reader we
restate it here.

######  Lemma.

Let $V$ and $R$ be finite dimensional $\mathbbm{Z}_{2}$-vector spaces whose
dimensions we denote by $n$ and $m$, respectively. Let $\\{v_{1},...,v_{n}\\}$
and $\\{r_{1},...,r_{m}\\}$ be basis of respectively $V$ and $W$, and
$\mathfrak{F}:V\to W$ and $\mathfrak{G}:W\to V$ be linear maps such that
$\mathfrak{G}\circ\mathfrak{F}$ is an isomorphism. Then, it is possible to
find an injective map $\mathfrak{f}:\\{1,...,n\\}\to\\{1,...,m\\}$ and a
bijective map $\mathfrak{g}:\\{1,...,n\\}\to\\{1,...,n\\}$ such that for each
$i\in\\{1,...,n\\}$:

  * •

the element $r_{\mathfrak{f}(i)}$ appears in $\mathfrak{F}(v_{i})$, and the
element $v_{\mathfrak{g}(i)}$ appears in $\mathfrak{G}(r_{\mathfrak{f}(i)})$.

We start introducing some terminology. Given a $\mathbbm{Z}_{2}$vector space
$B$ of dimension $n$ and a basis $\\{b_{1},...,b_{n}\\}$ of $B$, we say we say
that an element $b_{i}$ of the base $\\{b_{1},...,b_{n}\\}$ appears in an
element $b\in B$, if this element appears when we write $b$ in an unique way
as a sum of elements of $\\{b_{1},...,b_{n}\\}$. For each
$i\in\\{1,\ldots,n\\}$, we let $H^{B}_{i}=\langle
b_{1},\ldots,b_{i-1},b_{i+1},\ldots,b_{n}\rangle$ the subspace generated by
all but the $i$-th basis vector and $pr_{i}:B\to H^{B}_{i}$ be the projection
onto $H^{B}_{i}$ which satisfies $pr_{i}(b_{i})=0$. The element $b_{i}$
appears in $b$, if and only if, $b\notin H^{B}_{i}$.

To prove Lemma 5.1 we need the following preliminary lemma.

######  Lemma A.1.

Let $n\geq 1$, $k\geq 0$. Let $A$ be a $n+k$-dimensional vector space with
basis $e_{1},\ldots e_{n+k}$.

Let $w_{1},\ldots w_{n}\subset V$ be linearly independent vectors of $A$ and
denote $W=\langle w_{1},\ldots,w_{n}\rangle$. Let $Z\subset V$ be a
$k$-dimensional linear subspace that is transverse to $W$, i.e. $W\cap
Z=\\{0\\}$ and $Z\oplus W=A$.

Then there is an injective function
$\iota:\\{1,\ldots,n\\}\to\\{1,\ldots,n+k\\}$ such that for $L=\langle
w_{\iota(1)},\ldots,w_{\iota(n)}\rangle$

  1. (1)

$e_{\iota(j)}$ appears in $w_{j}$,

  2. (2)

$Z$ is transverse to the $n$-dimensional vector space $L$.

###### Proof.

For each $i\in\\{1,\ldots,k+n\\}$, let $H_{i}=\langle
e_{1},\ldots,e_{i-1},e_{i+1},\ldots,e_{n+k}\rangle$ be the subspace generated
by all but the $i$-th basis vector and $pr_{i}:A\to H_{i}$ the canonical
projection.

We give a proof by induction on $n$. For that start with the case $n=1$,
$k\geq 0$. Take a maximal set of basis vectors
$E=\\{e_{j_{1}},\cdots,e_{j_{l}}\\}$ such that
$w=w_{1}\in\cap_{i=1}^{l}H_{j_{l}}=:\widehat{H}$. Since $w\notin Z$, we have
$\widehat{H}\notin Z$, hence there is a basis vector $e_{i_{1}}\notin E$ with
$e_{i_{1}}\notin Z$. By the construction of $E$, $w\notin H_{i_{1}}$, hence we
can define $\iota$ by $\iota(w)=i_{1}$.

Assume now the statement holds for some $n-1\geq 1$, $k\geq 0$. Let
$w_{1},\ldots w_{n}$ and $Z$ satisfy the assumptions of the lemma. Consider
the $n+k-1$ dimensional subspace $Q=\langle w_{1},\ldots,w_{n-1}\rangle\oplus
Z\subset V$. Then $\langle w_{n}\rangle\cap Q=\\{0\\}$, and as above there is
a basis vector $e_{i_{n}}\notin Q$ with $w_{n}\notin H_{i_{n}}$. Consider
$Z^{\prime}=pr_{i_{n}}(Z)$ and $w^{\prime}_{i}=pr_{i_{n}}(w_{i})$ for
$i=1,\ldots,n-1$. Let $W^{\prime}:=\langle
w^{\prime}_{1},\ldots,w^{\prime}_{n-1}\rangle$. Since $e_{i_{n}}\notin Q$ we
have that $Z^{\prime}$ is $k$-dimensional, $W^{\prime}$ is $n-1$ dimensional,
and $Z^{\prime}\cap W^{\prime}=\\{0\\}$. By the induction hypothesis there is
an injective function
$\iota^{\prime}:\\{1,\ldots{n-1}\\}\to\\{1,\ldots,n+k\\}\setminus\\{i_{n}\\}$
such that for $L^{\prime}:=\langle
w^{\prime}_{\iota^{\prime}(1)},\ldots,w_{\iota^{\prime}(n)}\rangle$,

  * •

$w^{\prime}_{j}\notin\langle\\{e_{1},\ldots
e_{k+n}\\}\setminus\\{e_{i_{n}},e_{\iota^{\prime}(j)}\\}\rangle$, $j=1,\ldots
n-1$,555This is equivalent to the fact that $e_{\iota^{\prime}(j)}$ appears in
$w^{\prime}_{j}$, for the space $H_{i_{n}}$ with the basis $\\{e_{1},\ldots
e_{k+n}\\}\setminus\\{e_{i_{n}}\\}$.

  * •

$L^{\prime}\cap Z^{\prime}=\\{0\\}$.

Note, that by definition also $w^{\prime}_{j}\notin\langle\\{e_{1},\ldots
e_{k+n}\\}\setminus\\{e_{\iota^{\prime}(j)}\\}\rangle=H_{\iota^{\prime}(j)}$.
Define now $\iota:\\{1,\ldots,n\\}\to\\{1,\ldots,n+k\\}$ by

$\iota(i)=\begin{cases}\iota^{\prime}(i),&\text{ if }i=1,\ldots n-1\\\
i_{n},&\text{ if }i=n\end{cases}$

Since for $i=1,\cdots,n-1$, , $w^{\prime}_{i}=pr_{i_{n}}(w_{i})$, we also have
$w_{i}\notin H_{\iota(i)}$. Also $L\cap Z=\\{0\\}$, with
$L=L^{\prime}\oplus\langle e_{i_{n}}\rangle$. Indeed, let $v\in L\cap Z$, then
$pr_{i_{n}}(v)=0$ since it belongs to the intersection $L^{\prime}\cap
Z^{\prime}=\\{0\\}$. It follows that $v\in\langle e_{i_{n}}\rangle$, and since
$e_{i_{n}}\notin Z$, one has $v=0$.

∎

We now apply Lemma A.1 to prove Lemma 5.1.

Proof of Lemma 5.1:
By the hypothesis of the lemma, it is clear that $\mathfrak{F}$ is injective
and thus $m\geq n$. Apply first Lemma A.1 with $A=R$,
$w_{1}=\mathfrak{F}(v_{1}),\ldots,w_{n}=f(v_{n})$ and $Z=\ker(\mathfrak{G})$.
This gives an injective map $\mathfrak{f}:\\{1,\cdots,n\\}\to\\{1,\cdots,m\\}$
with $r_{\mathfrak{f}(i)}$ appearing in $\mathfrak{F}(v_{i})$ and such that in
particular $r_{\mathfrak{f}(1)},\cdots,r_{\mathfrak{f}(n)}$ generate a
subspace $C$ that is transverse to the kernel of $\mathfrak{G}$, and hence
$\mathfrak{G}|_{C}:C\to V$ is an isomorphism.

Apply Lemma A.1 again, now with $A=V$ and
$w_{1}=\mathfrak{G}(r_{\mathfrak{f}(1)}),\ldots,w_{n}=\mathfrak{G}(r_{\mathfrak{f}(n)})$
and $Z=\\{0\\}$. This gives $\mathfrak{g}:\\{1,\cdots,n\\}\to\\{1\cdots,n\\}$
bijective, such that $v_{\mathfrak{g}(i)}$ appears in
$\mathfrak{G}(r_{\mathfrak{f}(i)})$. ∎

##  Appendix B Approximation of entropy by the entropy of braid types

Recall that if $\varphi:\Sigma\to\Sigma$ is a diffeomorphism on a compact
surface $\Sigma$ and $\mathcal{P}$ a periodic orbit and
$\overline{P}\subset\Sigma$ the associated set of periodic points. Then
$\Gamma_{\pi_{1}}([\varphi,\mathcal{P}])$ denotes the growth rate of the
induced action of $\varphi$ on the fundamental group of
$\Sigma\setminus\overline{\mathcal{P}}$, see section 1.4. Also recall that
$\Gamma_{\pi_{1}}([\varphi,\mathcal{P}])\leq h_{\mathrm{top}}(\varphi)$.

The aim of this section is to give a proof of the following result.

######  Theorem B.1.

Let $\Sigma$ be a compact surface, and let $\varphi:\Sigma\to\Sigma$ be a
diffeomorphism such that $h:=h_{\mathrm{top}}(\varphi)>0$. Then, for any
$\epsilon>0$ there is a hyperbolic periodic orbit $\mathcal{P}$ of $\varphi$
such that $\Gamma_{\pi_{1}}([\varphi,\mathcal{P}])>h-\epsilon$.

This is variant of a celebrated result of Katok first announced in [34], see
also [35, Supplement], about the approximation of the topological entropy by
the entropies on locally maximal hyperbolic invariant sets on which the
dynamics is conjugated to a subshift of finite type, and in fact the orbits
$\mathcal{P}$ of Theorem B.1 can be found inside those sets. To our knowledge
there is no proof of Theorem B.1 in the literature, so we include it here. In
[26] it was proved that there is a collection of orbits for which the
exponential growth of $\varphi$ relative to it is positive. We apply the
strategy in [26] to the hyperbolic horseshoes that arise from Katok and
Mendoza’s results in [35, Supplement].

We recall relevant results from [35, Suppl.] which are applications of the
theory of non-uniformly hyperbolic dynamics, or Pesin theory, and we refer
also to [11] and references therein. Consider a $\varphi$-invariant ergodic
Borel probability measure $\mu$ on $\Sigma$ and assume it is hyperbolic, i.e.
its Lyapunov exponents satisfy $\chi_{1}<0<\chi_{2}$. See section 2 in [35,
Suppl.] for the definition and existence of Lyapunov exponents of the measure
$\mu$, and important properties. It is shown that for almost every
$x\in\Sigma$ there are so-called Lyapunov charts
$\psi_{x}:B(0,r(x))\subset\mathbbm{R}^{2}\to M$ for $x$, $r(x)>0$, with
respect to which the dynamics has uniformly hyperbolic behaviour. This can be
expressed with the notion of admissible stable and unstable manifolds, defined
locally in these charts, and which are locally preserved by $\varphi$. In
order to do that one restricts $\psi_{x}$ to $[-\kappa,\kappa]^{2}$ for
sufficiently small $\kappa>0$ and considers the regular rectangles
$R(x,\kappa)=\psi_{x}([-\kappa,\kappa]^{2})$ in $M$. We denote the left
boundary of $R$ by $\partial_{l}R$ given by
$\partial_{l}R=\\{\psi_{x}(-\kappa,s)\,|\,s\in[-\kappa,\kappa]\\}$ and denote
the right, bottom and top boundaries of $R=R(x,\kappa)$ by
$\partial_{r}R,\partial_{b}R$, and $\partial_{t}R$, respectively, which are
defined analogously. Let $\gamma\in(0,1)$. A submanifold $W\subset
R(x,\kappa)$ is called an admissible stable $(\gamma,\kappa)$-manifold near x
if $W=\psi_{x}(\\{(\phi(v),v)\,|\,v\in[-\kappa,\kappa]\\})$, where
$\phi:[-\kappa,\kappa]\to[-\kappa,\kappa]$ is a $C^{1}$-map with
$\phi(0)\leq\kappa/4$ and $|D\phi|\leq\gamma$. Analogously $W\subset
R(x,\kappa)$ is called an admissible unstable $(\gamma,\kappa)$-manifold near
x if $W=\psi_{x}(\\{(u,\phi(u))\,|\,v\in[-\kappa,\kappa]\\}$, where
$\phi:[-\kappa,\kappa]\to[-\kappa,\kappa]$ is a $C^{1}$-map with
$\phi(0)\leq\kappa/4$ and $|D\phi|\leq\gamma$. Admissible stable and unstable
$(\gamma,\kappa)$-manifolds near $x$ intersects in exactly one point in
$R(x,\kappa)$ with angle bounded away from zero and so they endow
$R(x,\kappa)$ with a product structure. Moreover one defines admissible
unstable (stable) rectangles as sets bounded by two admissible unstable
(stable) manifolds. That is, an admissible stable $(\gamma,\kappa)$-rectangle
in $R(x,\kappa)$ is a set of the form $V=\psi_{x}(V^{\prime})$,
$V^{\prime}=\\{(u,v)\in[-\kappa,\kappa]^{2}\,|\,u=(1-\tau)\phi_{1}(v)+\tau\phi_{2}(v),\,\tau\in[0,1]\\}$,
where the left boundary
$\partial_{l}V:=\psi_{x}(\\{(\phi_{1}(v),v)\,|\,v\in[-\kappa,\kappa]\\}$ and
the right boundary $\partial_{r}V:=$
$\psi_{x}(\\{(\phi_{2}(v),v)\,|\,v\in[-\kappa,\kappa]\\}$ are two admissible
unstable $(\gamma,\kappa)$ manifolds near $x$ for which
$\phi_{1}(v)<\phi_{2}(v)$ for all $v\in[-\kappa,\kappa]$. Analogously define
admissible stable $(\gamma,\kappa)$-rectangles $H$ in $R(x,\kappa)$ with
bottom (top) boundaries $\partial_{b}H$ ($\partial_{t}H)$. We define
$\partial_{b}V=\partial_{b}R(x,\kappa)\cap V$,
$\partial_{t}V=\partial_{t}R(x,\kappa)\cap V$, and similarly $\partial_{l}H$
and $\partial_{r}H$. The following statement asserts the existence of
rectangular covers and hyperbolic properties of $\phi$ on those rectangles.

######  Proposition B.2.

[35, Theorem S.14] For every $\delta>0$ and $\rho>0$ there is a compact set
$\Lambda_{\delta}$ with $\mu(\Lambda_{\delta})>1-\delta$, and constants
$\beta>0$, $\kappa>0$, $\gamma\in(0,1)$ and regular rectangles
$R(x_{1})=R(x_{1},\kappa),R(x_{2})=R(x_{2},\delta),\ldots,R(x_{t})=R(x_{t},\kappa)$,
$t\in\mathbbm{N}$, for some $x_{1},\ldots,x_{t}\in\Lambda_{\delta}$, such that

  1. (1)

$\Lambda_{\delta}\subset\bigcup_{i=1}^{t}B(x_{i},\beta)$ with
$B(x_{i},\beta)\subset{\rm int}R(x_{i})$

  2. (2)

${\rm diam}R(x_{i})\leq\rho/3$ for $i=1,\ldots,t$

  3. (3)

If $y\in\Lambda_{\delta},\varphi^{n}(y)\in\Lambda_{\delta}$ for some $n>0,y\in
B(x_{i},\beta)$, and $f^{n}(x)\in B(x_{j},\beta)$, then the connected
component $V$ of $R(x_{i})\cap f^{-n}(R(x_{j}))$ containing $y$ is an
admissible stable $(\gamma,\kappa)$ rectangle near $x_{i}$ and $H=f^{n}(V)$ is
an admissible unstable $(\gamma,\kappa)$ rectangle near $x_{j}$.

  4. (4)

diam $f^{k}(V)<\rho$ for all $0\leq k\leq n$.

In $(3)$ above we have that the union $(\partial_{l}V\cup\partial_{r}V)$ map
to the union of $(\partial_{l}H\cup\partial_{r}H)$, etc.

Assume that $h_{\mu}(\varphi)>0$. One can show now the following, see Theorem
S.5.9 in [35] and its proof.

######  Proposition B.3.

Let $\delta>0,\rho>0$ and choose the constants $\beta,\kappa,\gamma$, the set
$\Lambda_{\delta}$ and rectangles $R(x_{1}),\ldots,R(x_{t})$ as in Proposition
B.2. Then for any $\epsilon_{1}>0$ sufficiently small there is an unbounded
set $\mathcal{N}\subset\mathbbm{N}$ such that for all $n\in\mathcal{N}$ there
is $x\in\\{x_{1},\ldots,x_{t}\\}$, and
$M>e^{n(h_{\mu}(\varphi)-\epsilon_{1})}$ disjoint admissible stable
$(\gamma,\kappa)$-rectangles $V_{1},\ldots,V_{M}$ near $x$ that are mapped as
in $(3)$ above by $\varphi^{n}$ to $M$ admissible unstable
$(\gamma,\kappa)$-rectangles $H_{1},\ldots,H_{M}$ near $x$.

Moreover,

$\Lambda_{M}=\bigcap_{k\in\mathbbm{Z}}\left(\varphi^{kn}(V_{1})\cup\cdots\cup\varphi^{kn}(V_{M})\right)$

is a locally maximal hyperbolic invariant set with respect to $\varphi^{n}$,
and $\varphi^{n}|_{\Lambda_{M}}$ is topologically conjugate to a full two-
sided shift in the symbolic space of $M$ symbols. In particular,
$h_{\mathrm{top}}(\varphi^{n}|_{\Lambda_{M}})>h_{\mu}(\varphi)-\epsilon_{1}$.

Note that it follows from the variational principle, the ergodic decomposition
theorem and Ruelle’s inequality that one can approximate
$h_{\mathrm{top}}(\varphi)$ by $h_{\mu}(\varphi)$ of such measures $\mu$ as
considered above, and conclusions of Proposition B.3 imply approximation of
the topological entropy $h_{\mathrm{top}}(\varphi)$ by the topological entropy
of hyperbolic horseshoes of $\varphi$.

The proof of Theorem B.1 will use the above results and a variant of arguments
of Franks and Handel in [26].

###### Proof of Theorem B.1.

Let $\varepsilon>0$ and let $\mu$ be an invariant ergodic hyperbolic measure
with $h_{\mu}(\varphi)>h-\epsilon/3$. Choose $\varepsilon_{1}=\varepsilon/3$.
Let now $\mathcal{N}\subset\mathbbm{N}$ as in Proposition B.3. Choose some
$n\in\mathcal{N}$ with $\frac{\log(20)}{n}<\varepsilon_{1}$. Let
$x\in\\{x_{1},\ldots,x_{t}\\}$ and $M$, $V_{1},\ldots,V_{M}$, $H_{1},\ldots
H_{M}\subset R(x)$ as in Proposition B.3. We say that $H_{i}$ is positively
oriented if $\varphi^{n}(\partial_{t}V_{i})=\partial_{t}H_{i}$ and negatively
oriented if $\varphi^{n}(\partial_{t}V_{i})=\partial_{b}H_{i}$. We say that
$H_{i}$ lies below $H_{j}$ in $R(x)$, $i\neq j$, if every path in $R(x)$ from
$\partial_{b}R(x)$ to $\partial_{t}R(x)$ intersects first $H_{i}$.

For convenience we restrict to a subset $V^{\prime}_{1},\ldots,V^{\prime}_{m}$
of the admissible stable rectangles $V_{1},\ldots,V_{M}$ as well as a subset
$H^{\prime}_{1},\ldots,H^{\prime}_{m}$ of the admissible unstable rectangles
$H_{1},\ldots H_{M}$, both of size $m\geq M/10$ such that

  * •

$\varphi^{n}(V^{\prime}_{i})=H^{\prime}_{i}$ for $i=1,\ldots,m$.

  * •

All rectangles $H^{\prime}_{1},\ldots,H^{\prime}_{m}$ are either all
positively oriented or all negatively oriented.

  * •

Either $H^{\prime}_{1}$ is below $H^{\prime}_{2},\ldots,H^{\prime}_{m}$ and
$H^{\prime}_{m}$ is above $H^{\prime}_{1},\ldots,H^{\prime}_{m-1}$, or
$H^{\prime}_{1}$ is above $H^{\prime}_{2},\ldots,H^{\prime}_{m}$ and
$H^{\prime}_{m}$ is below $H^{\prime}_{1},\ldots,H^{\prime}_{m-1}$.

That we can ensure the last condition, follows from the following observation.

######  Lemma B.4.

Let $x_{1},\cdots,x_{n}$ be a sequence of $n$ pairwise different natural
numbers. Then it has a subsequence of length $\geq\frac{n}{5}$ such that all
elements of the subsequence lie in the closed interval $I$ whose boundary is
given by the first and the last element of that subsequence.

###### Proof.

Let $i_{0},i_{1}$ such that $x_{i_{0}}=\min\\{x_{i}\,|\,i=1,\ldots,n\\}$ and
$x_{i_{1}}=\max\\{x_{i}\,|\,i=1,\ldots,n\\}$. We assume that $i_{0}<i_{1}$,
the other case is analogous. If $i_{1}-i_{0}+1\geq\frac{n}{5}$ the subsequence
$x_{i_{0}},x_{i_{0}+1},\ldots,x_{i_{1}}$ is a subsequence that satisfies the
properties claimed in the lemma. Otherwise, $i_{0}>\frac{2n}{5}$ or
$n-i_{1}>\frac{2n}{5}$. Note that any element of the sequence lies in the
interval $[x_{1},x_{i_{1}}]$ or in the interval $[x_{i_{0}},x_{1}]$. So if
$i_{0}>\frac{2n}{5}$, $[x_{1},x_{i_{1}}]$ or $[x_{i_{0}},x_{1}]$ contains
$>\frac{n}{5}$ elements from $\\{x_{1},\ldots,x_{i_{0}}\\}$. Similarly, if
$n-i_{1}>\frac{2n}{5}$, the interval $[x_{n},x_{i_{1}}]$ or the interval
$[x_{i_{0}},x_{n}]$ contains $>\frac{n}{5}$ elements from
$\\{x_{i_{1}},\ldots,x_{n}\\}$. In all situations these elements form a
subsequence as claimed. ∎

We now proceed with the argument for the case that all $H^{\prime}_{i}$ are
positively oriented, $H^{\prime}_{1}$ is below
$H^{\prime}_{2},\ldots,H^{\prime}_{m}$ and $H^{\prime}_{m}$ is above
$H^{\prime}_{1},\ldots,H^{\prime}_{m-1}$. The other cases are treated
analogously. By abuse of notation we drop the symbol ′, and denote these
rectangles by $V_{1},\ldots,V_{m}$ and $H_{1},\ldots H_{m}$. Let
$\Lambda_{n}=\bigcap_{k\in\mathbbm{Z}}(\varphi^{kn}(V_{1})\cup\ldots\cup\varphi^{kn}(V_{m}))$,
and $\Sigma_{m}$ the set of bi-infinite sequences of $m$ symbols. The map
$\theta:\Lambda_{n}\to\Sigma_{m}$ defined as
$x\mapsto(\theta(x)_{k})_{k\in\mathbbm{Z}}$, with $\theta_{k}(x)=j$ if
$\varphi^{kn}(x)\in V_{j}$, provides a conjugation of $\varphi^{n}$ with the
two-sided shift $\sigma_{m}$ on $\Sigma_{m}$.

For any $a_{0},\ldots a_{k}\in\\{1,\ldots,m\\}$ we denote by $(a_{0}\cdots
a_{k})^{\infty}$ the $k$-periodic orbit of $\sigma_{n}$ in $\Sigma_{m}$ that
is given by an infinite repetition of $a_{0}\cdots a_{k}$ in positive and
negative direction. If we want to refer to a periodic point of that orbit, we
say that $(a_{i}a_{i+1}\cdots a_{k}a_{1}\cdots a_{i-1})^{\infty}$ is the point
that has $a_{i}$ at the $0$th position. By abuse of notation we denote the
periodic orbits or points of $\varphi^{n}$ in $\Lambda_{n}$ that correspond to
those in $\Sigma_{m}$ via $\theta$ also by the symbols $(a_{0}\cdots
a_{k})^{\infty}$ etc.

Consider the periodic orbit $\mathcal{Q}$ of $\varphi^{n}$ given by

$\displaystyle\mathcal{Q}=\left(\prod_{j=2}^{m-1}(mj1j1jmj)\right)^{\infty}$

We write also $\mathcal{Q}$ as the orbit

(64)
$\left(q_{1}^{2},q_{2}^{2},\ldots,q_{8}^{2},q_{1}^{3},q_{2}^{3}\ldots,q_{8}^{3},\ldots,q_{8}^{m-2},q_{1}^{m-1},q_{2}^{m-1},\ldots,q_{8}^{m-1}\right),$

where the periodic point $q_{1}^{2}$ is given in the symbolic expression by

$q_{1}^{2}=\left(\left(\prod_{j=2}^{m-2}(1j1jmjm(j+1))\right)1(m-1)1(m-1)m(m-1)m2\right)^{\infty},$

and the other periodic points $q_{l}^{j}$, $j\in\\{2,\ldots,m-1\\}$,
$l\in\\{1,\ldots,8\\}$, accordingly via cyclic permutations of the symbols. We
note the following, which will be relevant below: For any
$j\in\\{2,\ldots,m-1\\}$

(65) $\displaystyle\begin{split}&q_{1}^{j}\in V_{1}\cap
H_{j}\cap\varphi^{n}(H_{m}),\quad q_{3}^{j}\in V_{1}\cap
H_{j}\cap\varphi^{n}(H_{1})\\\ &q_{5}^{j}\in V_{m}\cap
H_{j}\cap\varphi^{n}(H_{1}),\quad q_{7}^{j}\in V_{m}\cap
H_{j}\cap\varphi^{n}(H_{m}),\end{split}$

and

(66) $q_{2}^{j},q_{4}^{j},q_{6}^{j},q_{8}^{j}\in H_{1}\cup H_{m}.$

We show below that

######  Proposition B.5.

$\Gamma_{\pi_{1}}([\varphi^{n},\mathcal{Q}])\geq\log(m-2)$.

From the Proposition we obtain

(67)
$\displaystyle\begin{split}\Gamma_{\pi_{1}}([\varphi^{n},\mathcal{Q}])&\geq\log(m-2)\geq\log(M/20)\\\
&\geq\log\left(\frac{e^{n(h-2\epsilon_{1})}}{20}\right)\geq
n(h-2\epsilon_{1})-\log(20)\\\ &>n(h-\varepsilon).\end{split}$

The hyperbolic periodic orbit $\mathcal{Q}$ of $\varphi^{n}$ of period
$8(m-2)$ defines a hyperbolic periodic orbit $\mathcal{P}$ of $\varphi$ of
period $8n(m-2)$, with $\overline{\mathcal{Q}}\subset\overline{\mathcal{P}}$
for the associated sets of $\mathcal{Q}$ and $\mathcal{P}$. Then
$\Gamma_{\pi_{1}}([\varphi,\overline{\mathcal{P}}])=\frac{1}{n}\Gamma_{\pi_{1}}([\varphi^{n},\overline{\mathcal{P}}])\geq\frac{1}{n}\Gamma_{\pi_{1}}([\varphi^{n},\overline{\mathcal{Q}}])>h-\varepsilon$,
which will finish the proof of Theorem B.1. ∎

###### Proof of Proposition B.5.

The proof is a variation of an argument from [26].

Identify the universal covering of $\Sigma\setminus\overline{\mathcal{Q}}$
with the Poincaré disk $\mathbbm{H}$. This gives us a choice of hyperbolic
metric on $\Sigma\setminus\overline{\mathcal{Q}}$. Every proper arc in
$\Sigma\setminus\overline{\mathcal{Q}}$, i.e. an arc $\alpha$ that lies in
$\Sigma\setminus\overline{\mathcal{Q}}$ up to its endpoints and whose
endpoints lie in $\overline{\mathcal{Q}}$, defines uniquely a geodesic that
traces a proper arc homotopic to $\alpha$. Also, any closed curve $\gamma$ in
$\Sigma\setminus\overline{\mathcal{Q}}$ defines a closed geodesic in
$\Sigma\setminus\overline{\mathcal{Q}}$ freely homotopic to $\gamma$.

For any $\gamma_{1}$ and $\gamma_{2}$ that are either proper arcs or closed
curves, denote by $I(\gamma_{1},\gamma_{2})$ their geometric intersection
number, i.e. the minimal number of intersections of curves
$\gamma^{\prime}_{1}$ and $\gamma^{\prime}_{2}$, where $\gamma^{\prime}_{1}$
and $\gamma^{\prime}_{2}$ are homotopic to $\gamma_{1}$ and $\gamma_{2}$,
respectively. Similarly, we can define $I(\cdot,\cdot)$ for families of proper
arcs resp. closed curves. We will below consider a proper arc $\tau$ with
endpoints in $\overline{\mathcal{Q}}$ and a collection of proper geodesic arcs
$\mathcal{E}$ such that for all $k\in\mathbbm{N}$,

(68) $\displaystyle I(\varphi^{kn}(\tau),\mathcal{E})\geq(m-2)^{k}.$

From this the Proposition will follow as in [26].

Let us introduce some terminology. Let $S_{\infty}$ the boundary at infinity
of $\mathbbm{H}$. Every geodesic in $\mathbbm{H}$ has two endpoints in
$S_{\infty}$. We say that a rectangle
$Q=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ in $\mathbbm{H}\cup
S_{\infty}$ with vertices in $S_{\infty}$ and adjacent edges
$\alpha_{1},\ldots,\alpha_{4}$ is a geodesic rectangle in $\mathbbm{H}$ if
$\alpha_{1},\ldots,\alpha_{4}$ are geodesics. We keep the information of the
ordering of the edges and say that the left vertical edge of $Q$ is
$\alpha_{1}$, the right vertical edge of $Q$ is $\alpha_{3}$ and the
horizontal edges are $\alpha_{2}$ and $\alpha_{4}$. We say that a geodesic
$\beta$ in $\mathbbm{H}$ intersects $Q$ horizontally from left to right if
$\beta$ intersects each of its vertical edges, first the left and then the
right vertical edge. We say that geodesic rectangle $Q$ intersects
$Q^{\prime}$ horizontally from left to right if both horizontal edges of $Q$,
parametrized from the left vertical edges to the right vertical edges of $Q$,
intersect $Q^{\prime}$ horizontally from left to right. We say that $Q\prec
Q^{\prime}$, if there is a geodesic in $\mathbbm{H}$ that first intersects $Q$
and then $Q^{\prime}$ horizontally from left to right.

Let $j\in\\{2,\ldots,m-1\\}$. Writing $\mathcal{Q}$ as in (64), we let $e_{j}$
be a simple proper arc from $q_{1}^{j}$ to $q_{3}^{j}$ in $H_{j}\cap V_{1}$,
$f_{j}$ a simple proper arc from $q_{3}^{j}$ to $q_{5}^{j}$ in
$H_{j}\cap\varphi^{n}(H_{1})$, $g_{j}$ a simple proper arc from $q_{5}^{j}$ to
$q^{7}_{j}$ in $H_{j}\cap V_{m}$, and $h_{j}$ a simple proper arc from
$q_{7}^{j}$ to $q_{1}^{j}$ in $H_{j}\cap\varphi^{n}(H_{m})$.
$e_{j},f_{j},g_{j}$, resp. $h_{j}$, uniquely define homotopic simple geodesic
arcs $\hat{e}_{j},\hat{f}_{j}$, $\hat{g}_{j}$, resp. $\hat{h}_{j}$. Let
$\hat{L}_{j}$ be the rectangle formed by $\hat{e}_{j},\hat{f}_{j}$,
$\hat{g}_{j}$, and $\hat{h}_{j}$. By the choice of $\mathcal{Q}$, see (65) and
(66), the full rectangles $\hat{L}_{J}$ are up to their vertices completely
contained in $\Sigma\setminus\overline{\mathcal{Q}}$, and hence can be lifted
to $\mathbbm{H}$. Any lift $\widetilde{L}_{j}$ of $\hat{L}_{j}$ to
$\mathbbm{H}$ is a geodesic rectangle $Q=(\alpha_{1},\ldots,\alpha_{4})$ in
$\mathbbm{H}$, where $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ are
suitable lifts of $\hat{e}_{j},\hat{f}_{j}$, $\hat{g}_{j}$, $\hat{h}_{j}$,
respectively. Note that any two lifts of $\hat{L}_{2},\ldots,\hat{L}_{m-1}$ to
$\mathbbm{H}$ are pairwise disjoint.

It is a theorem of Nielsen that any lift of
$\varphi^{n}|_{\Sigma\setminus\mathcal{P}}$ to $\mathbbm{H}$ extends to a
homeomorphism $F$ on $\mathbbm{H}\cup S_{\infty}$, and let $F$ be such an
extension. Let $Q$ be geodesic rectangle in $\mathbbm{H}$, then the image
$F(Q)$ defines uniquely a geodesic rectangle $[F(Q)]$ in $\mathbbm{H}$. Note
that the endpoints of $\varphi^{n}(f_{i})$ and $\varphi^{n}(h_{i})$ for
$i\in\\{2,\ldots,m-1\\}$ lie in $\overline{\mathcal{Q}}\cap(H_{1}\cup H_{m})$.
It is now easy to verify that for all $j\in\\{2,\ldots,m-1\\}$ there is an arc
$e^{\prime}_{j}$ homotopic to $e_{j}$ and an arc $g^{\prime}_{j}$ homotopic to
$g_{j}$ such that for all $i\in\\{2,\ldots,m-1\\}$, $\varphi^{n}(f_{i})$
intersects both $e^{\prime}_{j}$ and $g^{\prime}_{j}$ in exactly one point,
and the order of those intersections coincides with the orientation of $f_{i}$
and $h_{i}$ if parametrized from $e_{i}$ to $g_{i}$. It follows that for any
lift $Q$ of $\hat{L}_{j}$ there are lifts $Q_{i}$ of $\hat{L}_{i}$,
$i=2,\ldots,m-1$, such that the geodesic rectangle $[F(Q)]$ intersects
$Q_{2},\ldots,Q_{m-1}$ all horizontally from left to right.

Let $q,q^{\prime}$ be distinct points in $\overline{\mathcal{Q}}$ and $\sigma$
a geodesic arc in $\Sigma\setminus\overline{\mathcal{Q}}$ tracing a proper arc
from $q$ to $q^{\prime}$. Take a lift $\widetilde{\sigma}$ of $\sigma$ to
$\mathbbm{H}$. Let $D_{j}(\sigma)$ be the number of geodesic rectangles
$Q=(\alpha_{1},\ldots,\alpha_{4})$ that arise as lifts of rectangles
$\hat{L}_{j}$ and which $\sigma$ intersects horizontally from left to right.
Let $D(\sigma)=\sum_{j=2}^{m-1}D_{j}(\sigma)$. These numbers obviously do not
depend on the choice of lift of $\sigma$. Let $\sigma^{\prime}$ be the
geodesic arc in $\Sigma\setminus\mathcal{P}$ that traces a proper arc
homotopic to $\varphi^{n}(\sigma)$. We claim that

(69) $\displaystyle D(\sigma^{\prime})\geq(m-2)D(\sigma).$

This can be seen as follows. Take a lift $\widetilde{\sigma}$ of $\sigma$ to
$\mathbbm{H}$. We can order the lifts of the $\hat{L}_{j}$,
$j\in\\{2,\ldots,m-1\\}$ that $\widetilde{\sigma}$ intersects horizontally
from left to right by $\prec$, i.e. these are geodesic rectangles
$J_{1},\ldots J_{k}$, with $k=D(\sigma)$, and $J_{i}\prec J_{j}$ if $i<j$.
Since $F|_{S_{\infty}}$ keeps the cyclic order of points at infinity of
geodesics, it follows that the geodesic arc in $\mathbbm{H}$ with the same
endpoints as $F(\widetilde{\sigma})$ has to intersect the geodesic rectangles
$[F(J_{1})]$, $[F(J_{2})],\ldots[F(J_{k})]$ horizontally from left to right,
and $[F(J_{i})]\prec[F(J_{j})]$ if $i<j$. Since each of the rectangles
$[F(J_{i})]$ intersects horizontally from left to right suitable lifts
$Q_{2},\ldots,Q_{m-1}$ of $\hat{L}_{2},\ldots,\hat{L}_{m-1}$, (69) follows.

Now choose $q$ and $q^{\prime}$ two distinct points in
$\overline{\mathcal{Q}}\cap(H_{1}\cup H_{m})$ and $\tau$ an arc from $q$ to
$q^{\prime}$ with $D(\tau)=1$. Furthermore, let
$\mathcal{E}=\bigcup_{j=2}^{m-1}\hat{e}_{j}$. From (69) the estimate (68)
follows, that is

$\displaystyle I(\varphi^{kn}(\tau),\mathcal{E})\geq(m-2)^{k}.$

Choose now as in [26] for each $k\in\mathbbm{N}$ a closed curve $\gamma_{k}$
in $\Sigma\setminus\overline{\mathcal{Q}}$ as the frontier of a small contour
of the union of the geodesic representative of $\varphi^{kn}(\tau)$ and its
two endpoints. For those $k\in\mathbbm{N}$ for which
$\varphi^{kn}(q),\varphi^{kn}(q^{\prime})\in H_{1}\cup H_{m}$, the curve
$\gamma_{k}$ and $\mathcal{E}$ have $2I(\varphi^{kn}(\tau),\mathcal{E})$ many
intersections. Moreover there is no bigon formed by the union of $\mathcal{E}$
with $\gamma_{k}$, and hence the number of those intersections is minimal
among all curves homotopic to $\gamma_{k}$. Since $\varphi^{kn}(\gamma_{0})$
is freely homotopic to $\gamma_{k}$ it follows that
$\Gamma_{\pi_{1}}([\varphi^{n},\mathcal{Q}])\geq\log(m-2)$.

∎

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