Given any symbol $s\in\{1,2\}^\mathbf{N}$ s.t. $s$ is not the constant $\{1,1,1...\}$ nor pre-periodic to it, $P_{In},P_0\not\in D_s$. Additionally, $D_s\subseteq\overline{D_{\alpha,t}}$.

To begin, since $s$ is not strictly pre-periodic to $\{1,1,1...\}$, $\exists k>0$ s.t. $\forall n>0,g^k_{p,r_n}(L^n_s)\subseteq L^n_2$. Because $G_{p,r_n}\to R_{p,t}$ in the $C^\infty$ metric on compact $K\subseteq\mathbf{R}^3$, $P_{In},P_{Out}\not\in K$, similarly to the proof $\rho\subseteq\rho_1$ in the proof of Lemma <ref>, we conclude $f^k_{p,t}(D_s\setminus\{P_{In},P_0,P_{Out}\})\subseteq\overline{D_2}$.

We will now prove $P_{In},P_{0}\not\in D_s$ - to do so, assume otherwise. If either $P_0,P_{In}$ lies in $D_s$, then $\forall j>0$, $\overline{f^j_{p,t}(D_s)}$ contains a curve $\gamma\subseteq\overline{H_{p,t}}$ with at least one endpoint at $P_{In}$. Since Lemma <ref> proves $P_{In}\not\in\rho$, we conclude $\forall j>0, f^j_{p,t}(D_s\setminus\{P_{In},P_0,P_{Out}\})$ intersects $D_1$. This is a contradiction to $f^k_{p,t}(D_s\setminus\{P_{In},P_0,P_{Out}\})\subseteq\overline{D_2}$, hence we conclude $P_{In},P_0\not\in D_s$. Additionally, we also conclude $D_s\setminus\{P_{In},P_0,P_{Out}\}=D_s\setminus\{P_{Out}\}$ - which, by previous paragraph, yields $f^k_{p,t}(D_s\setminus\{P_{Out}\})\subseteq \overline{D_2}$. As both $\overline{D_2},\{P_{Out}\}$ are subsets of $\overline{D_{\alpha,t}}$, because $f^{-1}_{p,t}(\overline{D_{\alpha,t}})\subseteq\overline{D_{\alpha,t}}$ (see Lemma <ref>) we conclude $D_s\subseteq\overline{D_{\alpha,t}}$ and Lemma <ref> follows.

Recall we have previously defined a coding function $\pi:I\to\{1,2\}^\mathbf{N}$ for $I$, the invariant set of $f_{p,t}$ in $\overline{H_{p,t}}\setminus\rho$, and that our present goal is to prove  $\pi(D_s)=s$ (see the discussion preceding Lemma <ref>). Lemma <ref> almost allows us to do so - but we still need to prove several facts: namely, that $\pi$ is defined on $D_s$, or at least some subset of it. To this end, let $s,\omega\in\{1,2\}^\mathbf{N}$ be distinct symbols which are both not pre-periodic to the constant $\{1,1,1...\}$. We will now prove that under these assumptions, $D_s\cap D_\omega\subseteq\{\delta_0,P_{Out}\}$:

Let $s,\omega\in\{1,2\}^\mathbf{N},s\ne\omega$ - then, if both $s,\omega$ are not the constant $\{1,1,1...\}$ nor pre-periodic to it, then $D_s\cap D_\omega\subseteq\{\delta_0,P_{Out}\}$.

Assume this is not the case - then, since $D_s,D_\omega$ are the limit sets of $L^n_s,L^n_\omega,n\to\infty$, they are separated by infinitely many components of $\cup_{d>0}f^{-d}_{p,t}(\rho)$ (see Cor.<ref>). Hence, by Cor.<ref> there exist $d\ne j$ and some $x\in f^{-j}_{p,t}(\rho)\cap f^{-d}_{p,t}(\rho)$, $x\ne P_{Out},\delta_0$ (see the illustration in Fig.<ref>). Because the components of $g_{p,r_n}^{-j}(L^n_3)$ and $g^{-k}_{p,r_n}(L^n_3)$ are separated by infinitely many other pre-images of $L^n_3$, we conclude there exist some $k$ for which $x\in f^{-k}_{p,t}(\rho)\cap f^{-k-1}_{p,t}(\rho)$. The same argument also proves $\forall l_1,l_2>0$, $x\in f^{-k-l_1}_{p,t}(\rho)\cap f^{-k-l_2}_{p,t}(\rho)$.

From $\rho\subseteq\rho_1=f^{-1}_{p,t}(\delta)$, we conclude $f^{k+d}_{p,t}(x)\in\rho\cap\delta$ for all $d>0$ - that is, $\delta\cap\rho$ contains some $f_{p,t}$-invariant set. By Lemma <ref> and the definition of $\delta$ we have $f_{p,t}(\delta)\subset\partial H_{p,t}$, $\delta\cap\partial H_{p,t}=\{\delta_0\}$ (see Lemma <ref>) - hence, if there exists an invariant set for $f_{p,t}$ in $\delta\cap\rho$, it must lie in $\{\delta_0\}$. In particular, we must have $\{f^j_{p,t}(x)\}_j\subseteq\{\delta_0\}$.

(520,80)$W^{u}_{In}\cup W^{s}_{Out}$

[Fig31]an intersection between $D_s,D_\omega$ away from $P_{Out},\delta_0$ - this generates a contradiction.

By Lemma <ref>, either $\delta_0=P_{Out}$, or $\delta_0\ne P_{Out}$ - hence, if such an $x$ exists, by $\{f^{k+j}_{p,t}(x)\}_j\subseteq\{\delta_0\}$, as $\delta_0\in\partial H_{p,t}$ (see Lemma <ref>) $\delta_0$ must be a fixed point. By Lemma <ref>, we already know $\delta_0\ne P_{In}$ - hence, the only possibility is $\delta_0=P_{Out}$. Now, let us first recall $P_{Out}$ is a saddle focus with a two-dimensional stable manifold, transverse to $H_{p,t}$ at $P_{Out}$ (see Cor.<ref>). As we assume $x\ne P_{Out}$ we conclude $\forall j>0,f^{k+j}_{p,t}(x)\ne P_{Out}$. This, however, is a contradiction to $\{f^{k+j}_{p,t}(x)\}_j\subseteq\{\delta_0\}$, and Lemma <ref> follows.

By Lemma <ref> we immediately conclude three immediate corollaries. To begin proving them, let $s\in\{1,2\}^\mathbf{N}$ be a symbol distinct from the constant $\{1,1,1...\}$ and not pre-periodic to it. Recall we denote by $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ the one-sided shift - we now generalize Lemma <ref> as follows:

Under the assumptions above, $\forall j>0$, $f^j_{p,t}(D_s)\subseteq D_{\sigma^j(s)}$, and $D_s\setminus\{P_{Out}\}=D_s$.

To begin, let us first recall several facts. As proven in Lemma <ref> $P_{In},P_0\not\in D_s$. Furthermore, recall $P_{Out}$ is a saddle-focus with a two-dimensional stable manifold transverse to $\overline{H_{p,t}}$ at $P_{Out}$ (see Cor.<ref>) - therefore, $\forall x\in\overline{H_{p,t}}$, $f_{p,t}(x)\ne P_{Out}$. Now, further recall that by our choice of $\{G_{p,r_n}\}_n$, we have $G_{p,r_n}\to R_{p,t}$ on every compact $K\subseteq\mathbf{R}^3$ - provided, of course, $P_{In},P_{Out}\not\in K$. Hence, since $\forall n,j>0$ we have $g^j_{p,r_n}(L^n_s)\subseteq L^n_{\sigma^j(s)}$, by $f_{p,t}(x)\ne P_{Out}$ for all $x\in D_s\setminus\{P_{Out}\}$ we conclude that $\forall j>0$ we must have $f^j_{p,t}(D_s\setminus\{P_{Out}\})\subseteq D_{\sigma^j(s)}$.

We will now prove $P_{Out}\not\in D_s$ - by previous paragraph, this will immediately yield $f^j_{p,t}(D_s)\subseteq D_{\sigma^j(s)}$ for all $j>0$, thus concluding the proof. To this end, let $s\in\{1,2\}^\mathbf{N}$ be as above. We must consider two separate cases - either $s$ is periodic, or it is not. If $s$ is not periodic, by the definition of $D_s$ as a limit set, if $P_{Out}\in D_s$ there must exist some periodic $\omega\in\{1,2\}^\mathbf{N}$, distinct from the constant $\{1,1,1...\}$ s.t. $P_{Out}\in D_\omega$. Therefore, to rule out $P_{Out}\in D_s$ it would suffice to prove the case when $s$ is periodic - that is, it would suffice to prove that if $s$ is periodic and distinct from the constant $\{1,1,1...\}$, then $P_{Out}\ne D_s$.

To do so, assume this is not the case. That is, assume $s\in\{1,2\}^\mathbf{N}$, $s\ne\{1,1,1...\}$ is periodic of minimal period $k>0$. Therefore, for any given $j>0$ we have $f^j_{p,t}(D_s\setminus\{P_{Out}\})\subseteq\cup_{0\leq j\leq k-1}D_{\sigma^j(s)}\setminus\{P_{Out}\}$. As $p$ is a trefoil parameter (see Def.<ref>), the existence of a bounded heteroclinic connection between $P_{In},P_{Out}$ implies the trajectories of points sufficiently close to $P_{Out}$ eventually hit transversely a neighborhood of $P_0$ in $H_{p,t}$. Therefore, setting $O(s)=\cup_{0\leq i\leq k-1}D_{\sigma^i(s)}$ we conclude $P_0\in \overline{O(s)}$ - and since $O(s)$ is a finite union of closed sets, we must have  $P_0\in O(s)$. In particular, $\exists i\in\{0,...,k-1\}$ s.t. $P_0\in D_{\sigma^i(s)}$.

However, $P_0\ne\delta_0,P_{Out}$ - and since $\forall n>0$, $L^n_{\sigma^i(s)}$ is separated from the $CD$ side by infinitely many other $L^n_\omega,\omega\in\{1,2\}^\mathbf{N}$, we conclude $\exists \omega_0\in\{1,2\}^\mathbf{N}$, not pre-periodic to $\{1,1,1...\}$, s.t. $P_0\in D_{\sigma^i(s)}\cap D_{\omega_0}$. This is a contradiction to Lemma <ref> - hence, all in all, for every periodic $s\in\{1,2\}^\mathbf{N},s\ne\{1,1,1,...\}$ we have $P_{Out}\not\in D_s$. In particular, $D_s\setminus\{P_{Out}\}=D_s$, and for every $j>0$, $f^j_{p,t}(D_s)\subseteq D_{\sigma^j(s)}$. Cor.<ref>, $\delta_0\ne D_s$ now follows.

We now conclude:

For every $d\geq0$, and every $s\in\{1,2\}^\mathbf{N}$ which is not strictly pre-periodic to $\{1,1,1...\}$, we have $D_s\cap f^{-d}_{p,t}(\rho)\subseteq\{\delta_0\}$.

Assume this is not the case. Therefore, since Cor.<ref> proves $f^d_{p,t}(D_s)\subseteq D_{\sigma^d(s)}$ we must have $\rho\cap D_{\sigma^d(s)}\ne\emptyset$. By Cor.<ref>, if $\rho\cap D_{\sigma^d(s)}$ lies away from $\delta_0$, we immediately have a contradiction to Lemma <ref> - therefore, $\rho\cap D_{\sigma^d(s)}=\{\delta_0\}$. However, Cor.<ref> also proves $P_{Out}\not\in D_{\sigma^d(s)}$, which, by $\delta_0\in \rho\cap D_{\sigma^d(s)}$ trivially yields $\delta_0\ne P_{Out}$ - hence $f_{p,t}(\delta_0)\ne P_{Out}$. Applying Cor.<ref> yet again, we conclude $f_{p,t}(
\delta_0)\in\delta\cap D_{\sigma^{d+1}(s)}$. As $f_{p,t}(\delta_0)\ne\delta_0$, because $D_{\sigma^{d+1}(s)},\delta$ are separated from one another by infinitely many other $D_\omega,\omega\in\{1,2\}^\mathbf{N}$, we immediately have a contradiction to Lemma <ref> and the assertion follows.

Now, let $s,\omega\in\{1,2\}^\mathbf{N}$ be two different symbols, both distinct from the constant $\{1,1,1...\}$ and not pre-periodic to it. From Lemma <ref> and Cor.<ref> we now conclude the following, stronger version of Lemma <ref>:

$\delta_0\not\in D_s$. As a consequence, for any $s,\omega$ as above $D_s\cap D_\omega=\emptyset$, and for every $d>0$, $f^{-d}_{p,t}(\rho)\cap D_s=\emptyset$.

First, recall $\delta_0=\delta\cap\partial H_{p,t}$ (see Lemma <ref>). Now, assume this is not the case - that is, there exists some $s\in\{1,2\}^\mathbf{N}$, s.t. $s$ is not the constant $\{1,1,1...\}$ and not pre-periodic to it, satisfying $\delta_0\in D_s$. Recall $L^n_s\cap\partial H_{p,t}$ is separated from both $C,D$ (in $\partial R_n$) by infinitely many other $L^n_\omega,\omega\in\{1,2\}^\mathbf{N}$ (see the proof of ,Lemma <ref>) - therefore, there exists some $\omega_0$, distinct from the constant $\{1,1,1...\}$ and not pre-periodic to it s.t. $\delta_0\in D_s\cap D_{\omega_0}$. By Cor.<ref>, if $\delta_0\in D_s$ we must also have $\delta_0\ne P_{Out}$ - which implies $f_{p,t}(\delta_0)\ne\delta_0$. Additionally, Cor.<ref> also implies $f_{p,t}(\delta_0)\in D_{\sigma(s)}\cap D_{\sigma(\omega_0)}$. As $P_{Out}\not\in D_{\sigma(s)}\cup D_{\sigma(\omega_0)}$, we conclude $D_{\sigma(s)}, D_{\sigma(\omega_0)}$ must intersect away from both $\delta_0,P_{Out}$. This is a contradiction to Lemma <ref> - which yields $\delta_0\not\in D_s$. As an immediate consequence, by Cor.<ref>, we obtain  $f^{-d}_{p,t}(\rho)\cap D_s=\emptyset$ for all $d\geq0$.

To conclude the proof, consider any $s,\omega\in\{1,2\}^\mathbf{N}$ s.t. both $s,\omega$ are distinct from $\{1,1,1...\}$, and not pre-periodic to it. By Cor.<ref> and previous paragraph we have $\delta_0,P_{Out}\not\in D_s\cup D_\omega$, hence by Lemma <ref> we conclude $D_s\cap D_\omega=\emptyset$, and Cor.<ref> follows.

To continue, recall we denote by $I$ the invariant set of $f_{p,t}$ in $\overline{H_{p,t}}\setminus\rho$. From now on, let $Q$ denote the collection of $D_s$, for all $s\in\{1,2\}^\mathbf{N}$, which are not strictly pre-periodic to the constant $\{1,1,1...\}$. Using Cor.<ref> and <ref>, we can now finally prove the first-return map $f_{p,t}$ is a distorted horseshoe on $Q$:

Assume $s$ is not the constant $\{1,1,1...\}$ nor pre-periodic to the constant symbol $\{1,1,1...\}$ - then $Q$ is invariant under $f_{p,t}$, $Q\subseteq I$, and  $\pi(D_s)=s$.

First recall we denote by $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ the one-sided shift. The invariance of $Q$ follows immediately from Cor.<ref> - as it proves that $\forall s\in\{1,2\}^\mathbf{N}$ which $s$ is not the constant $\{1,1,1...\}$ nor pre-periodic to it, we have $f_{p,t}(D_s)\subseteq D_{\sigma(s)}$. Additionally, by Cor.<ref> for every $d>0$ we have $f^{-d}_{p,t}(\rho)\cap D_s=\emptyset$. This proves $Q\cap(\cup_{d>0}f^{-d}_{p,t}(\rho))=\emptyset$ - or in other words, $Q\subseteq I$ - it therefore remains to prove $\pi(D_s)=s$.

To do so, write $s=\{i_0,i_1,...\}$, for some $s\in\{1,2\}^\mathbf{N}$ distinct from the constant $\{1,1,1...\}$ and not pre-periodic to it. Now, recall $\overline{H_{p,t}}\setminus\rho$ is composed of two subsets, $D_1,D_2$ (see the discussion preceding Lemma <ref>) - by the definition of $D_s$, $\rho$ as limit sets (see Lemma <ref> and the discussion preceding Lemma <ref>) by $D_s\cap\rho=\emptyset$, we must have $D_s\subseteq D_{i_0}$. Repeating this argument, again we conclude $\overline{D_{i_0}}\setminus f^{-1}_{p,t}(\rho)$ contains two pre-compact subsets, $D_{i_0,1},D_{i_0,2}$ s.t. $f_{p,t}(D_{i_0,i})\subseteq D_i,i=1,2$ - see the illustration in Fig.<ref>. Again, by the definition of $D_s$, $f^{-1}_{p,t}(\rho)$ as limit sets (see Cor.<ref>), by $D_s\cap f^{-1}_{p,t}(\rho)=\emptyset$ we conclude $D_s\subseteq D_{i_0,i_1}$. Repeating this argument infinitely many times, for every $n>0$ we construct a domain $D_{i_0,...,i_n},D_s\subseteq D_{i_0,...,i_n}$ s.t. $f^n_{p,t}(D_{i_0,...,i_n})\subseteq D_{i_n}$. All in all,  $D_s\subseteq R_s=\overline{\cap_{n>0} D_{i_0,...,i_n}}\ne\emptyset$. Hence, by construction we immediately have $\pi(D_s)=s$, and Lemma <ref> follows.

(520,110)$W^{u}_{In}\cup W^{s}_{Out}$

[Fig31]$f^{-1}_{p,t}(\rho)$ and the partitioning of $H_{p,t}$.

As the components of $Q$ are $D_s$, for $s\in\{1,2\}^\mathbf{N}$ distinct from the constant $\{1,1,1...\}$ and not pre-periodic to it, combining Lemma <ref> with Lemma <ref> we immediately conclude:

Having proven the invariance of $Q$ and that $\pi$ is defined on it, we can now almost conclude the proof of Th.<ref> - all we need to do is to prove the dynamics of $f_{p,t}$ on $Q$ are chaotic, and that there are infinitely many periodic orbits for $f_{p,t}$ of all minimal periods. We will first prove the first return map $f_{p,t}:\overline{H_{p,t}}\to\overline{H_{p,t}}$ is chaotic. To do so, recall we denote by $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ the one-sided shift. By definition, the coding function $\pi$ already satisfies $\pi\circ f_{p,t}=\sigma\circ\pi$, and $\pi(Q)$ is $\sigma$-invariant is. Furthermore, by Lemma <ref> $\pi(Q)$ contains all the symbols $s\in\{1,2\}^\mathbf{N}$ which are not $\{1,1,1...\}$ nor pre-periodic to it - hence, $\sigma$ is chaotic on $\pi(Q)$. Therefore, by Def.<ref> in order to prove the chaoticity of $f_{p,t}$ on $Q$, it remains to show:

$\pi$ is continuous on $Q$. As a consequence, $f_{p,t}$ is chaotic on $Q$.

Choose $x\in Q$, and consider $\{x_n\}_n\subseteq Q$ s.t. $x_n\to x\in Q$, $x_n\in D_{s_n}$, $x\in D_s$. By definition, $s_n=\pi(x_n)$, $s=\pi(x)$ - therefore, to prove the continuity of $\pi$ we must prove $s_n\rightarrow s$. By Cor.<ref> the components of $Q$ are all compact and disjoint, hence $D_{s_n},D_s$ are also compact and disjoint. Additionally, since $s$ is not the constant $\{1,1,1...\}$ and not pre-periodic to it, given any $\epsilon>0$ there exist $\omega_1,\omega_2\in\{1,2\}^\mathbf{N},\omega_1,\omega_2\ne s$ s.t. $d_C(s,\omega_i)<\epsilon,i=1,2$ (where $d_C$ denotes the word metric on $\{1,2\}^\mathbf{N}$). Now, let $C_s$ denote the component of $\overline{H_{p,t}}\setminus(D_{\omega_1}\cup D_{\omega_2})$ s.t. $D_s\subseteq C_s$ - because $D_{\omega_1},D_{\omega_2}$ are closed, $C_s$ is open (in the relative topology of $\overline{H_{p,t}}$). And since $D_s,D_{\omega_1},D_{\omega_2}$ are all compact, denoting by $d$ the Euclidean distance in $\overline{H_{p,t}}$ we conclude $d(D_s,D_{\omega_i})>0,i=1,2$. Now, since the components of $Q$ are limit sets of components $Q_{r_n}$, for every $\omega\in\{1,2\}^\mathbf{N}$ s.t. $D_\omega\subseteq C_s$, we must have $d_C(\omega,s)<\epsilon$.

(520,110)$W^{u}_{In}\cup W^{s}_{Out}$

[Fig31]$f^{-1}_{p,t}(\rho)$ and the partitioning of $H_{p,t}$.

Therefore, since $x_n\to x,\{x_n\}_n\subseteq Q$, for every sufficiently large $n>0$ we have $D_{s_n}\cap C_s\ne\emptyset$. However, because $d(D_s,D_{\omega_i})>0,i=1,2$, by $x_n\to x,x_n\in D_{s_n}$, for any sufficiently large $n$ we must also have $D_{s_n}\cap D_{\omega_i}=\emptyset,i=1,2$ - which implies that for every sufficiently large $n>0$, $D_{s_n}\subseteq C_s$, hence $d_C(s_n,s)<\epsilon$. As $\epsilon>0$ and $\{x_n\}_n$ were both chosen arbitrarily, $\pi$ is continuous and the chaoticity of $f_{p,t}$ follows.

Having proven the chaoticity of $f_{p,t}$, we turn to the second assertion in Th.<ref> - the existence of infinitely many periodic points for $f_{p,t}$ in $H_{p,t}$, of all periods. To do so, consider a periodic symbol $s\in\{1,2\}^\mathbf{N}$ of minimal period $k>0$, s.t. $s$ is not the constant $\{1,1,1...\}$. We now prove:

Let $s\in\{1,2\}^\mathbf{N}$ be a periodic symbol of minimal period $k>0$, $s\ne\{1,1,1...\}$. Then, $D_s$ contains a periodic point for $f_{p,t}$ of minimal period $k>0$.

Let $s$ be a periodic symbol distinct from $\{1,1,1...\}$. By Cor.<ref>, $\forall d>0$, $f^d_{p,r}(D_s)\subseteq D_{\sigma^d(s)}$. Now, recall Prop.<ref> and Lemma <ref>. As we proved in Lemma <ref>, for every $g_{p,r},r>0$ there exists a periodic point $x_r\in L^r_s$, of minimal period $k$. By Prop.<ref>, Cor.<ref> and Remark <ref>, $D_s$ also contains a periodic point $x$, of some minimal period. Additionally, because $s\ne\{1,1,1,...\}$, Prop.<ref> further implies $x$ cannot be a fixed point for the flow, that is $x\ne P_{Out}, P_{In}$.

As such, it remains to prove the minimal period for $x$ under $f_{p,t}$ is $k$. To do so, recall Cor.<ref> proves for any $s,\omega\in\{1,2\}^\mathbf{N}$ distinct from the constant $\{1,1,1...\}$,  provided $s,\omega$ not strictly pre-periodic to the constant $\{1,1,1...\}$ we have $D_s\cap D_\omega=\emptyset$. Therefore, since $\forall j>0,f^j_{p,r}(D_s)\subseteq D_{\sigma^j(s)}$ (see Cor.<ref>) we conclude that if $s\in\{1,2\}^\mathbf{N}$ is periodic of minimal period $k$ then $\forall 0<j<k$, $D_{s}\cap f^j(D_s)=\emptyset$. As such, we conclude the minimal period for $x$ is $k$ and Lemma <ref> follows.

We can now conclude the proof of Th.<ref>. Denote by $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ the one sided shift - by Lemma <ref>, Cor.<ref> and Lemma <ref>, we conclude the existence of a continuous $\pi:Q\to\{1,2\}^\mathbf{N}$, s.t. $\pi\circ f_{p,t}=\sigma\circ\pi$, s.t. $\pi(Q)$ contains infinitely many periodic points (as well as a dense orbit). Or, in other words, $f_{p,t}$ is chaotic on $Q$ - and in particular, if $s\in\{1,2\}^\mathbf{N}$ is periodic of minimal period $k>0$, then $\pi^{-1}(s)$ contains a periodic point for $f_{p,t}$ of period $k$. Finally, since $t>0$ was chosen s.t. Prop.<ref> holds, we can replace $R_{p,t},f_{p,t},D_{\alpha,t}$ with $F_p,f_{p},D_\alpha$. That is, we have proven the existence of an invariant set $Q\subseteq\overline{D_{\alpha}}$ on which the first-return map $f_p:\overline{D_\alpha}\to\overline{D_\alpha}$ is defined, and chaotic. Additionally, $Q$ contains infinitely many periodic orbits, of all minimal periods - Th.<ref> is proven.

At this point, it must be remarked that the proof of Th.<ref> can probably be generalized to other heteroclinic parameters, which generate knots more complex than a trefoil (see Def.<ref>). However, it must also be stated that such generalizations greatly depend on the topology of the knot, and how it intersects $H_p$. Another remark which must be made is that the proof of Th.<ref> gives us more than we originally set out to. To see why, recall the curve $\rho$ from the proof of Th.<ref>, and that by Lemma <ref>, $\rho\subseteq\overline{D_{\alpha,t}}$. Provided $t$ is sufficiently small s.t. Prop.<ref> holds we have $\overline{D_\alpha}=\overline{D_{\alpha,t}}$ - this leads us to conclude section with the following restatement of Th.<ref>:

Let $p$ be a trefoil parameter, and let $I$ be the invariant set of the first return map $f_p:\overline{H_p}\to \overline{H_p}$ (wherever defined) in $\overline{H_p}\setminus\rho$. Then, the dynamics of $f_p$ on $I$ are chaotic w.r.t. Def.<ref>, and include periodic points of all minimal periods.

§  THE IMPLICATIONS:

Th.<ref> teaches us that at trefoil parameters, the dynamics of the Rössler system are essentially those of a suspended Topological Horseshoe - more precisely, it teaches us that the dynamics are complex at least like those of a suspended Smale Horseshoe map. As such, Th.<ref> can be used to derive several implications about the overall dynamics of the vector field - in particular, Th.<ref> can be used to infer information about the periodic trajectories generated by the Rössler system.

We begin with Section $4.1$ by addressing the question of persistence of symbolic dynamics under perturbations of the trefoil parameter. More precisely, let $p\in P$ be a trefoil parameter, and recall we denote by $F_p$ the vector field corresponding to it. Trefoil parameters (see Def.<ref>) in the parameter space $P$ form at most a co-dimension $2$ set - as such, Th.<ref> naturally leads us to ask the following question: given $v\in P$, does the dynamical complexity generated by $F_p$ persists in some form when $F_p$ is perturbed to $F_v$? Or, put simply, just how much of the dynamical complexity at trefoil parameters wears off on nearby vector fields in $P$? We will answer this question by proving Th.<ref>. Using similar methods to the proof of Th.<ref>, we will prove that if $p,v\in P$ and $p$ is a trefoil parameter, then provided $v$ is sufficiently close to $p$ at least some of the periodic, symbolic dynamics persist when $F_p$ is perturbed to $F_v$.

In Section $4.2$ we address the question of characterizing the minimal dynamical complexity generated by the Rössler system at trefoil parameters. Namely, given a trefoil parameter $p\in P$, we will prove that despite the absence of any hyperbolicity assumption on $F_p$, we still can associate a Template with the flow dynamics (see Th.<ref>). In other words, we will prove Th.<ref> allows us to reduce the study of the Rössler system at trefoil parameter to the study of a different, hyperbolic model - whose dynamical complexity can be described precisely with a template. As such, this model can be considered as a topological lower bound for the complexity of the flow - in particular, in Th.<ref> we will prove that at trefoil parameters $p\in P$, every knot type in the $L(0,1)$ template (see <cit.>), save perhaps for one, is realized as a periodic trajectory for $F_p$.

§.§  Symbolic dynamics, and the persistence of periodic trajectories.

From now on, let $p\in P$ be a trefoil parameter, and recall we denote by $F_p$ the vector field corresponding to $p$, and by $f_p:\overline{H_p}\to\overline{H_p}$ the first-return map. Now, let $v\in P$ be some parameter, and let $F_v$, $f_v:\overline{H_v}\to\overline{H_v}$ denote the corresponding vector field and its first return map.

Our main goal in Section $4.1$ is to prove Th.<ref>, about the persistence of periodic, symbolic dynamics of $F_p$ under perturbations in the parameter space $P$. We will do so by applying Fixed-Point Theory - and in particular, the Fixed-Point Index (see Ch.VII.5 in <cit.>). Let us first recall the Fixed-Point Index: given an open $V\subseteq\mathbf{R}^2$ and given a continuous map $f:V\to V$, assuming the set $\{x|f(x)=x\}$ is compact in $V$, the fixed point index is defined as the degree of $f(x)-x$. As proven in Ch.VII.5.8 in <cit.>, the fixed point index remains invariant under certain homotopies. As such, heuristically, when $F_p$ is smoothly deformed to $F_v$ through some curve in $P$, the deformation induces a homotopy of the first return maps $f_p,f_v$. As $f_p$ is essentially a deformed horseshoe by Th.<ref>, we would expect its fixed-point index to be non-zero - at least around some infinite collection of periodic orbits. Therefore, when $F_p$ is continuously deformed to $F_v$ we would expect at least some periodic points for $f_p$ to persist for $f_v$.

In practice, this line of reasoning simply cannot hold - if only because $f_p$ need not even be defined throughout $\overline{H_p}$. Therefore, in order to apply the fixed-point index, we must take a different approach. Recall the vector field $R_{p,t}$, introduced in Cor.<ref> - much like Th.<ref>, we will prove Th.<ref>  by studying the dynamics of $R_{p,t}$ and its first-return map $f_{p,t}:\overline{H_{p,t}}\to\overline{H_{p,t}}$ - and in particular, its continuity properties around $Q$, the invariant set from Th.<ref>. As we will prove in Lemma <ref> and Prop.<ref>, $f_p$ and $f_{p,t}$ are in fact continuous on $Q$. As we will see, these two facts will allow us to define a region $S\subseteq H_{p,t}$ on which the first-return map is defined, and compute its index.

We will begin with the following, technical Lemma:

Let $p\in P$ be a trefoil parameter, and let $Q$ denote the invariant set constructed in the proof of Th.<ref> - then, the periodic trajectories generated by $F_p$ in $Q$ are dense around $P_{In}$.

To begin, recall $f_p$ is chaotic on $Q$ (see the proof of Th.<ref>). That is, we have proven there exists a continuous $\pi:Q\to\{1,2\}^\mathbf{N}$, s.t. $\pi\circ f_p=\sigma\circ\pi$, with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift - and that $\pi(Q)$ includes every periodic symbol in $\{1,2\}^\mathbf{N}$. Furthermore. $\forall s\in\{1,2\}^\mathbf{N}$, $\pi^{-1}(s)$ forms a component of $Q$. Now, further recall we originally proved Th.<ref> for $f_{p,t}$, the first-return map of the vector field $R_{p,t}$ (see Cor.<ref>) - and we did so by proving $Q\subseteq\overline{D_{\alpha,t}}$ (see Cor.<ref>). As $F_p,R_{p,t}$ coincide on the trajectories of initial conditions in $Q$, Th.<ref> followed.

To prove the Lemma, assume this is not the case - that is, assume the closure of the periodic trajectories generated by $F_p$ in $Q$ is disjoint from $P_{In}$. By the proof of Th.<ref>, the initial conditions in $Q$ all lie in $D_{\alpha}$ - hence, when we deform $F_p$ to $R_{p,t}$ this remains true, provided we choose $t>0$ s.t. Prop.<ref> holds. By $W^{u}_{In}\cap\kern-0.7em|\kern0.7em W^{s}_{Out}$, since the closure of all periodic trajectories lies away from $P_{In}$, it must also lie away from $\overline{W^{u}_{In}\cap\kern-0.7em|\kern0.7em W^{s}_{Out}}$. We therefore conclude that when we smoothly deform $R_{p,t}$ to $G_p$ and then continuously deform $G_p$ to $G$ (see Prop.<ref> and Lemma <ref>) the same remains true for the dynamics generated by $G$.

In other words, let $Q_1$ denote the non-wandering set for the vector field $G$. Then, the previous paragraph proves the periodic trajectory $P_{In}$ is disjoint from the closure of all periodic trajectories in $Q_1$. However, by Lemma <ref>, $G$ is hyperbolic on $Q_1$, its non-wandering set - in particular, its dynamics on it are those of a suspended Smale horseshoe (see Lemma <ref>). In other words, the periodic trajectories are dense in $Q_1$ - since $P_{In}\subseteq Q_1$ (see Lemma <ref>) we have a contradiction and the Lemma follows.

In fact, we can say more - namely, we will now prove $Q$ is in fact dense in itself in a very specific way. To do so, recall that as proven in Th.<ref>, every component of $Q$ connects at least two points on $\partial H_{p,t}$ (see the discussion succeeding Lemma <ref>), which are separated by $\delta_0$. Denote by $d$ the Euclidean metric in $\overline{H_{p,t}}$ - we now prove:

Let $p\in P$ be a trefoil parameter, and let $Q$ be the invariant set given by Th.<ref>. Additionally, let $q$ be a component of $Q$, and choose any $\epsilon>0$  - then, if $H$ is a a component of $\overline{H_{p,t}}\setminus q$, $\forall x\in\partial q\cap\partial H$ there exists a point $\xi\in Q\cap H$ s.t. $d(x,\xi)<\epsilon$.

To begin, recall that by the construction of $Q$ in the proof of Th.<ref>, given any component $q\subseteq Q$, $q$ is compact and $P_{Out},P_{In}\not\in q$. Additionally, recall there exists a continuous, surjective $\pi:Q\to\{1,2\}^\mathbf{N}$ s.t. $\pi\circ f_{p}=\sigma\circ\pi$ (with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift). As proven in Th.<ref>, $\pi(q)=s$ is a symbol in $\{1,2\}^\mathbf{N}$ which is not the constant $\{1,1,1...\}$, nor pre-periodic to it.

We will prove Prop.<ref> by contradiction - to do so, assume its conclusion is incorrect. That is, assume there exists $q$, a component of $Q$, $H$, a component of $\overline{H_{p,t}}\setminus q$, an $\epsilon_0>0$ and some $x\in\partial q\cap\partial H$ s.t. $(B_{\epsilon_0}(x)\setminus q)\cap H\cap Q=\emptyset$ (see the illustration in Fig.<ref>). The idea behind the proof is as follows - first, we deform $F_p$ to the vector field $R_{p,t}$, for some $t>0$ s.t. Prop.<ref> is satisfied. As such, the same argument used in the proof of Lemma <ref> again proves the dynamics of $F_p$ and $R_{p,t}$ coincide on $Q$. Recalling $R_{p,t}$ can be smoothly deformed to the vector field $G$ (see Lemma <ref>), we will prove the assumption $x$ cannot be approximated by $y\in Q\cap H$ leads to a contradiction.

To begin, blow up $P_{In},P_{Out}$ by Hopf-bifurcations (thus smoothly deforming $R_{p,t}$ to the vector field $G_p$ - see Prop.<ref>) - as proven in Prop.<ref>, this has the effect of deforming $f_{p,t}$ to $g_p:\overline{H_{p,t}}\to\overline{H_{p,t}}$. As proven in Prop.<ref> and Lemma <ref>, there exists a $g_p$-invariant $Q'\subseteq\overline{H_{p,t}}$ and a continuous, surjective $\pi':Q'\to\{1,2\}^\mathbf{N}$ s.t. $\pi'\circ g_p=\sigma\circ\pi'$ (with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift). By the definition of $Q$ (see the discussion before Lemma <ref>), $q$ corresponds to some $D_s,s\in\{1,2\}^\mathbf{N}$ (see the discussion preceding Lemma <ref>) - therefore, by the definition of $D_s$ and because $P_{In},P_{Out}\not\in q$ (see Lemma <ref> and Cor.<ref>), we choose the deformation of $R_{p,t}$ to $G_p$ s.t. $q$ persists, unchanged - that is, when $R_{p,t}$ is smoothly deformed to $G_p$, $q$ is smoothly deformed to $q'$, a component of $Q'$ s.t. $\pi(q)=\pi'(q')$. Now, denote by $Q'_1$ the components $q_1$ of $Q''$ s.t. $\pi'(q_1)$ is not the constant $\{1,1,1,...\}$, nor pre-periodic to it - in particular, $q'\subseteq Q_1$. Furthermore, $x$ is perturbed to $x'\in q'$ - hence there exists a component $H'$ of $B_{\epsilon_0}(x')\setminus q'$ s.t. $(B_{\epsilon_0}(x')\setminus q')\cap H'\cap Q'_1=\emptyset$ (see the illustration in Fig.<ref>).

(430,290)$Q\setminus q$
(460,110)$W^u_{In}\cup W^s_{Out}$
(435,790)$Q\setminus q$

[Fig32]the intersection $B_{\epsilon_0}(x)\cap H\cap Q=\emptyset$ - we will prove by contradiction this cannot occur.

Now, consider the deformation of the vector field $G_p$ to $G$ as described in Lemma <ref>. Let us recall it induces an isotopy of $g_p:ABCD\to\overline{H_{p,t}}$ to $g:ABCD\to\overline{H_{p,t}}$ (with $g$ denoting the first return map for $G$ - see Lemma <ref>). This deformation smoothly deformation every component of $Q'$ to a component of $Q''$, a $g$-invariant set in the rectangle $ABCD$, s.t. $\exists \pi'':Q''\to\{1,2\}^\mathbf{N}$, continuous and surjective s.t. $\pi''\circ g=\sigma\circ\pi''$ (see Lemma <ref> and Cor.<ref>). In particular, since $g$ is a Smale Horseshoe map (see Lemma <ref>), given any $\omega\in\{1,2\}^\mathbf{N}$, $\pi''^{-1}(\omega)$ is homeomorphic to an interval.

(440,330)$Q'\setminus q'$
(460,110)$W^u_{In}\cup W^s_{Out}$
(320,490)$Q'\setminus q'$

[Fig32]the intersection $B_{\epsilon_0}(x')\cap H'\cap Q'$ remains empty - this will force a contradiction.

To continue, assume $x'$ is deformed to $x''$ under the isotopy from $g_p$ to $g$ - by Cor.<ref>, if $q'$ is the component of $Q'$ s.t. $x'\in q'$, then $q'$ is smoothly deformed to $q''$, a component of $Q''$ s.t. $\pi'(q')=\pi''(q'')$. Since $g$ is a Smale Horseshoe map by Lemma <ref>, we conclude every component $c\subseteq Q''$ is homeomorphic to the interval $[0,1]$, connects the $AC,BD$ sides in $ABCD$ - and there is a unique $s\in\{1,2\}^\mathbf{N}$ s.t. $c=\pi''^{-1}(s)$ (see Fig.<ref> for an illustration). Now, define $Q''_1$ as the analogous version of $Q'_1$, only w.r.t. $g$ - that is, $Q''_1$ is the set of components of $Q''$ s.t. for every component $c\subseteq Q''_1$, $\pi''(c)$ is not the constant $\{1,1,1,...\}$, nor pre-periodic to it (by Cor.<ref>, the components of $Q'_1$ are continuously deformed to $Q''_1$). Additionally, since $g$ is a Smale horseshoe map, given any $y\in Q''_1$ and $\epsilon>0$, $Q_1''$ is dense in $B_\epsilon(y)\setminus c$.

To conclude the proof, denote by $X$ the component of $Q''$ s.t. $x''\in X$ - since $\pi'(x')$ is not the constant $\{1,1,1...\}$ (nor pre-periodic to it), the same is true for $\pi''(x'')$-  hence, $X\subseteq Q''_1$. As we already have $(B_{\epsilon_0}(x')\setminus q')\cap H'\cap Q'_1=\emptyset$ and since $Q'_1$ is continuously deformed to $Q''_1$, we choose the isotopy from  $g_p$ to $g$ s.t. there exists some $\epsilon_1>0$ for which $Q''_1$ is not dense in $B_{\epsilon_1}(x')\setminus X$ (see the illustration in Fig.<ref>). Since $g$ is a Smale horseshoe map we have a contradiction and Prop.<ref> follows.

(-120,330)$Q_1''\setminus X$
(-120,180)$Q_1''\setminus X$

[Fig32]the blue lines represent $Q''_1\setminus X$, and the red circle $\partial B_{\epsilon_1}(x')$. $Q''_1\setminus X$ does not intersect the upper half of $B_{\epsilon_1}(x')\setminus X$, which implies a contradiction, since $g$ is a Smale horseshoe map.

In fact, we can even say more - using similar ideas and arguments to those used to prove Prop.<ref> we now conclude:

The set $Q\cap \partial H_{p,t}$ is dense in itself.

Now, recall Lemma <ref> and Cor.<ref>, which states the set of discontinuity points for $f_{p,t}$ in $H_{p,t}$, $Dis(f_{p,t})$ is composed of arcs, and satisfies $Dis(f_{p,t})=f^{-1}_{p,t}(\partial H_{p,t})$ - in particular, any arc $\alpha\subseteq Dis(f_{p,t})$ s.t. $\alpha\ne\delta$ is a curve with two endpoints on $\partial H_{p,t}$ (see the illustration in Fig.<ref>). By Prop.<ref>, provided $t>0$ is sufficiently small, $f_p,f_{p,t}$ have the same discontinuity curves in $\overline{D_{\alpha}}=\overline{D_{\alpha,t}}$ - hence, any component $\alpha\subseteq Dis(f_{p,t})$ s.t. $\alpha\subseteq\overline{D_{\alpha,t}}$ is also a curve of discontinuities for $f_p$, the first-return map of the original vector field $F_p$.

As stated earlier, in order to apply the fixed-point index to periodic orbits in $Q$, we must first address the question of how continuous the first-return map $f_p$ is on the components of $Q$. To do so, recall $\pi:Q\to\{1,2\}^\mathbf{N}$ denotes the continuous map s.t. $\pi\circ f_{p,t}=\sigma\circ\pi$ (with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift). Additionally recall that as proven in Th.<ref>, $Q$ is the collection of $D_s=\pi^{-1}(s)$, for $s\in\{1,2\}^\mathbf{N}$ which are not the constant $\{1,1,1...\}$ nor pre-periodic to it. Using Prop.<ref>, we first prove the following Lemma:

Assume $s\in\{1,2\}^\mathbf{N}$ is not the constant $\{1,1,1...\}$ nor pre-periodic to it. Then, there is no discontinuity curve for either $f_p,f_{p,t}$ s.t. $\alpha\subseteq D_s$. In particular, $Dis(f_{p,t})\cap Q=\emptyset$.

Like the proof of Prop.<ref>, since $Q\subseteq \overline{D_{\alpha}}$, again it would suffice to prove Lemma <ref> for $f_{p,t}$ (again, provided $t>0$ is s.t. Prop.<ref> is satisfied). We prove Lemma <ref> by contradiction. Assume the Lemma is incorrect - that is, assume the existence of a discontinuity curve $\alpha$ for $f_{p,t}$ s.t. $\alpha\subseteq D_s$. Since by Cor.<ref> $\delta\cap Q=\emptyset$, by Cor.<ref> we conclude $\overline{\alpha}$ is a curve which connects two points in $\partial H_{p,t}$, $x_1,x_2$  - since $D_s$ is compact, $x_1,x_2\in D_s$ (see the illustration in Fig.<ref>). Now, recall every component in $Q$ is compact and connects two points in $\partial H_{p,t}$ separated by $\delta_0$.

(460,110)$W^u_{In}\cup W^s_{Out}$

[Fig32]$\alpha$ is a discontinuity curve for $f_{p,t}$ and satisfies $\alpha\subseteq D_s$ (for simplicity, we sketch $\alpha=D_s$). $S$ is the closure of the region bounded between the blue curves, and $S_1,S_2$ are the two components of $S\setminus\overline{\alpha}$. $B_x$ denotes $B_r(x)\cap C$

By the definition of $Q$ (see the discussion before Lemma <ref>), we know $\pi(D_s)$ is not the constant $\{1,1,1...\}$ (nor pre-periodic to the constant $\{1,1,1...\}$). Since $\overline{\alpha}\subseteq D_s$, by Cor.<ref> we have $f_{p,t}(\overline{\alpha})\subseteq D_{\sigma(s)}$ - and since $\alpha$ is a discontinuity curve, by $Dis(f_{p,t})=f^{-1}_{p,t}(\partial H_{p,t})$ we conclude $f_{p,t}(\alpha)\subseteq\partial H_{p,t}$ (see the illustration in Fig.<ref>) (see the illustration in Fig.<ref>). By previous paragraph, $\overline{H_{p,t}}\setminus D_s$ is composed of two components, $C_1,C_2$ - by Prop.<ref>, there exist two periodic $\omega_1,\omega_2\in\{1,2\}^\mathbf{N}$, $D_{\omega_1}\subseteq C_1,D_{\omega_2}\subseteq C_2$. Now, let $S$ denote the closure of the connected component of $\overline{H_{p,t}}\setminus (D_{\omega_1}\cup D_{\omega_2})$ s.t. $S\cap D_s\ne\emptyset$ - by Lemma <ref>, Lemma <ref> and Cor.<ref>, $P_{In},P_{Out},P_0\not\in S$. Recalling $D_s$ is compact, by Lemma <ref> and Cor.<ref> we have $D_s\cap f^{-1}_{p,t}(\delta)=\emptyset$ - Therefore, by the compactness of $f^{-1}_{p,t}(\delta)$, Prop.<ref> and Cor.<ref> we choose $\omega_1,\omega_2$ as periodic of sufficiently high minimal period s.t. $P_{In},P_{Out},P_0\not\in \overline{f_{p,t}(S)}$.

Since $P_{In},P_0,P_{Out}\not\in Q$ (see Lemma <ref> and Cor.<ref>), we conclude $P_{In},P_0,P_{Out}\not\in S$. Therefore, by Prop.<ref>, since $S$ is compact in $\overline{H_{p,t}}$, when we smoothly deform $R_{p,t}$ to $G_p$, $S$ is deformed to $S'$, a compact, connected domain which is strictly interior to the rectangle $ABCD$ (see Prop.<ref> for the definition of $ABCD$). Since $g_p:\overline{H_{p,t}}\to\overline{H_{p,t}}$, the first-return map for $G_p$, is continuous on $ABCD$ (see Prop.<ref>) $g_p(S')$ is also compact and connected - and in particular, since $S$ lies away from $P_0,P_{In},P_{Out}$, $S'$ must lie away from the blown-up heteroclinic connections and the fixed-points for $G_p$. Therefore, from $P_{In},P_{Out},P_0\not\in S$, by Prop.<ref> we can choose $G_p$ s.t. it is $C^\infty$ arbitrarily close to $R_{p,t}$ on the flow-lines connecting $S',g_p(S')$. Therefore, since for any such $g_p$, $g_p(S')$ is compact and connected, $f_{p,t}(S)$ is compact and connected as well.

Now, recall $\overline{\alpha}\subseteq D_s$ - since $D_s\subseteq S$, by Cor.<ref> $S\setminus \alpha$ is composed of two components, $S_1,S_2$ (see Fig,<ref>). Without any loss of generality, since by Lemma <ref> and Cor.<ref> $f_{p,t}(\overline\alpha)$ is a closed curve on $\partial H_{p,t}$, we assume $f_{p,t}(\alpha)\subseteq\overline {f_{p,t}(S_1)}$. This implies there exists a Jordan domain, $C\subseteq S_1$ s.t. $\overline{\alpha}\subseteq\partial C$, and $f_{p,t}$ is continuous on $C$ (see the illustration in Fig.<ref> and <ref>). Now, let $R\subseteq S$ denote the domain $R=\overline{C}\cup\overline{S_2}$ (see the illustration in Fig.<ref>). The same arguments as in previous paragraph also prove $f_{p,t}(R)$ is connected and compact. Additionally, $\forall y\in\alpha$, there exists some $r>0$ s.t. $f_{p,t}(B_r(y)\cap \overline{C})$ is homeomorphic to a half-disc, s.t. $f_{p,t}(\overline{\alpha})\cap f_{p,t}(B_r(y)\cap \overline{C})$ consists of an open arc $\alpha_y\subseteq\partial H_{p,t}$, s.t. $f_{p,t}(y)$ is strictly interior to $f_{p,t}(y)$ (see the illustration in Fig.<ref>).

(460,110)$W^u_{In}\cup W^s_{Out}$

[Fig32]$\overline{f_{p,t}(C)},\overline{f_{p,t}(S_2)}$ connect at most in $f_{p,t}(x_1),f_{p,t}(x_2)$. We will prove this yields a contradiction.

To continue, since $f_{p,t}(\alpha)\subseteq\partial H_{p,t}$ and because $\alpha$ separates $\overline{C},S_2$, by the Existence and Uniqueness Theorem we have $f_{p,t}(S_2)\cap f_{p,t}(\overline{C})=\emptyset$. We now claim $\overline{f_{p,t}(S_2)}\cap f_{p,t}(\overline{C}\setminus\{x_1,x_2\})=\emptyset$. To see why, first, by previous paragraph we conclude given any $\{y_n\}_n\subseteq f_{p,t}(S_2)$ and any $x\in\alpha$, $\{y_n\}_n$ is separated from $f_{p,t}(x),\alpha_x$ by the (topological) half disc $f_{p,t}(B_r(x)\cap C)$ - therefore, $\{y_n\}_n$ cannot accumulate at $f_{p,t}(x)\in\alpha_x$. Additionally, given $x\in\partial f_{p,t}(\overline{C}\setminus\overline{\alpha})$, by the definition of $C$ we have $f^{-1}_{p,t}(x) \in S_1$ - in particular, since $x\in\partial f_{p,t}(C)\setminus f_{p,t}(\alpha)$, $f^{-1}_{p,t}(x)$ is strictly interior to $S_1$. Therefore, since no $\{x_n\}_n\subseteq\overline{S_2}$ can accumulate on $f^{-1}_{p,t}(x)$, neither can any sequence $\{y_n\}_n\subseteq f_{p,t}(S_2)$ accumulate on $x$. Therefore, since $f_{p,t}(S_2)$ cannot accumulate on $f_{p,t}(\overline{C})\setminus\{f_{p,t}(x_1),f_{p,t}(x_2)\}$, we conclude $f_{p,t}(\overline{C})$, $f_{p,t}(S_2)$ must connect with $f_{p,t}(S_1)$ at $f_{p,t}(x_1),f_{p,t}(x_2)$, or both (see the illustration in Fig.<ref>).

To conclude the proof, we now show this generates a contradiction. To see why, consider a point $x\in\alpha$, strictly interior to $\alpha$ (that is, $x\ne x_1,x_2$), and choose $\{x_n\}_n\subseteq S_2$ s.t. $x_n\to x$. Let $y_n=f_{p,t}(x_n)$, and consider the accumulation points of $\{y_n\}_n$ in $\overline{f_{p,t}(S_2)}$. If $y\in\overline{f_{p,t}(S_2)}$ is an accumulation point of $y$, as $x,\{x_n\}_n$ are all strictly interior to $R$, $y$ must lie on the trajectory of $x$. But since $f_{p,t}(x)\not\in\overline{f_{p,t}(S_2)}$, because $f_{p,t}$ is defined as a first-return map, we conclude $y\not\in\overline{f_{p,t}(S_2)}$. Since $R=\overline{C}\cup\overline{S_2}$ this proves $f_{p,t}(R)$ is not closed - which contradicts it being compact. Therefore, due to this contradiction Lemma <ref> follows.

We are now finally ready to prove $f_p$ (and by extension, $f_{p,t}$) is continuous on $Q$. Using Prop.<ref> and Lemma <ref> we prove:

Let $p\in P$ be a trefoil parameter, and let $Q$ denote the invariant set given by Th.<ref>. Let $q$ denote a component of $Q$ - then, $f_p$ is continuous on $q\cap H_{p}$.

As $Q\subseteq \overline{D_\alpha}$, similarly to the proofs of Lemmas <ref> and Prop.<ref>, provided $t>0$ is sufficiently small by Prop.<ref> it would suffice to prove Prop.<ref> for $R_{p,t},f_{p,t}$ instead of $F_p,f_p$. To this end, let $\pi:Q\to\{1,2\}^\mathbf{N}$ be as in Lemma <ref> and Prop.<ref> - that is, a continuous map s.t. $\pi\circ f_{p,t}=\sigma\circ\pi$, with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift. Now, assume Prop.<ref> is incorrect for $f_{p,t}$ - that is, there exists some component $q\subseteq Q$, and $y\in q$ interior to $H_{p,t}$, s.t. $f_{p,t}$ is discontinuous at $y$. In other words, $f_{p,t}(y)\in \partial H_{p,t}$. Since $f_{p,t}$ is discontinuous at $y$, and since $y$ is interior to $H_{p,t}$, by Lemma <ref> and Cor.<ref>, $y$ lies on some arc $\alpha\subseteq\overline{H_{p,t}}$, $f_{p,t}(\alpha)\subseteq\partial H_{p,t}$, as in Fig.<ref>. By our construction of $Q$, $\delta\cap Q=\emptyset$ (see Lemma <ref> and Lemma <ref>) - therefore, by Lemma <ref>, $\alpha$ must connect two points in $\partial H_{p,t}$ (not necessarily distinct). In particular, by Lemma <ref> no endpoint of $\alpha$ is interior to $H_{p,t}$.

(460,130)$W^u_{In}\cup W^s_{Out}$

[Fig32]the first possibility - $q\setminus\{y\}$ intersects both $C_1,C_2$. As we will prove, this is impossible.

Let us write $s=\pi(y)$. By Lemma <ref> $H_{p,t}\setminus\alpha$ is made of several components, s.t. at least two of them, $C_1,C_2$ are open and $y\in\partial C_i,i=1,2$ (see Fig.<ref> for an illustration). There are two options - either $q\setminus\{y\}\cap C_i\ne\emptyset$ for both $i=1,2$ (as illustrated in Fig.<ref>), or, say, $q\setminus\{y\}\cap C_1\ne\emptyset$, $q\setminus\{y\}\cap C_2=\emptyset$ (as illustrated in Fig.<ref>). We will first rule out $q\cap C_i\ne\emptyset$ for both $ i=1,2$. To do so, recall that given any $\omega\in\pi(Q)$, $\delta_0$ separates at least two endpoints of $q_\omega=\pi^{-1}(\omega)$ on $\partial(D_{\alpha,t}\setminus C_t)$ (see the illustration in Fig,<ref>). Hence, by Prop.<ref>, we can approximate $y$ by two sequences: $\{x_n\}_n\subseteq C_1\cap q\setminus\{y\}$ and $\{y_n\}_n\subseteq C_2\cap q\setminus\{y\}$ (see the illustration in Fig.<ref>).

Now, let us consider the symbolic dynamics for $x_n,y_n$. For points in, say, $C_1$, since $x_n\to y$, as we can choose $\{x_n\}_n$ s.t. they do not lie discontinuity curve for $f_{p,t}$, we have $f_{p,t}(x_n)\to f_{p,t}(y)$ (see Fig.<ref>) - yet, on the other hand, because $y_n\in C_2$ for all $n>0$, $f_{p,t}(y_n)\to f^k_{p,t}(y)$, for some $k>1$. By the continuity of $\pi$ on $Q$ (see Lemma <ref>), setting $s_n=\pi(x_n)$, $\omega_n=\pi(y_n)$ we conclude $s_n,\omega_n\to s$. However, because $f_{p,t}(y_n)\to f^k_{p,t}(y)$, $f_{p,t}(x_n)\to f_{p,t}(y)$, and since on $Q$ we have $\pi\circ f_{p,t}=\sigma\circ\pi$, we must also have $\sigma(\omega_n)\to \sigma^k(s)$, $\sigma(s_n)\to\sigma(s)$. As such, $\sigma^k(s)=\sigma(s)$, that is, $\sigma(s)s=\pi(f_{p,t}(y))$ is periodic of minimal period $j\leq k$.

(460,130)$W^u_{In}\cup W^s_{Out}$

[Fig32]the second possibility - $q\setminus\{y\}\subseteq C_1$. This also leads to a contradiction.

By Lemma <ref>, $\alpha\not\subseteq q$. Therefore, as we assume $q\setminus\{y\}\cap C_i\ne\emptyset,i=1,2$, by Prop.<ref>. Cor.<ref> and Th.<ref> there exists some a-periodic $\omega\in\pi(Q),\sigma(\omega)\ne\sigma^k(\omega)$ for all $k$, s.t. $q_\omega=\pi^{-1}(\omega)$ intersects $\alpha$ at, say, some $y'$. However, this implies we can choose two sequences $\{x'_n\}_n\subseteq q_\omega\cap C_1$, $\{y'_n\}_n\subseteq q_\omega\cap C_2$, $x'_n,y'_n\to y'$ - again, as $\pi(y')=\omega$, the same argument as above proves $\sigma(\omega)=\sigma^k(\omega)$ (for some $k$). This is impossible by our choice of $\omega$ - which rules out the possibility $q\setminus\{y\}\cap C_i\ne\emptyset$ for both $i=1,2$. Therefore, the only possibility is $q\setminus\{y\}\cap C_1\ne\emptyset,q\setminus\{y\}\cap C_2=\emptyset$ - as illustrated in Fig.<ref>.

However, since $q$ is a component of $Q$ it connects two points (not necessarily distinct) on $\partial H_{p,t}$ - therefore, $C_2\subseteq H$, for some components of $\overline{H_{p,t}}\setminus q$ (see Fig.<ref>). By Prop.<ref>, Cor.<ref> and Lemma <ref>, we can approximate $q$ in $H$ by components $q_n\subseteq Q\cap H,n>0$. As $q\cap \partial C_2=q\cap\alpha=\{y\}$, we conclude there exists some $n>0$ s.t. $q_n\cap C_i\ne\emptyset$ for both $i=1,2$ - hence, by previous argument we have a contradiction and Prop.<ref> follows.

Now, recall Lemma <ref>, where we proved every periodic trajectory for $F_p$ hits $H_p$ transversely at least once. As an immediate corollary from Prop.<ref> and Lemma <ref>, we immediately conclude:

Let $x\in Q$ be a periodic point that is not a fixed point for $f_p$ - then, its trajectory never hits $\partial H_p$. In other words, if $T$ is the trajectory of $x$, all its intersection points with $\overline{H_{p}}$ are transverse intersections.

We also conclude:

Let $p\in P$ be a trefoil parameter. With previous notations, for any $s\in\{1,2\}^\mathbf{N}$ s.t. $s$ is not pre-periodic to $\{1,1,1...\}$, let $q$ denote the unique component of $Q$ s.t. $\pi(q)=s$. Then, no point of $q\cap H_{p,t}$ is the accumulation point of discontinuity curves for $f_{p,t}$.

First, recall by our definition of $H_p$ that $F_p$ is transverse to it and tangent to $\partial H_p$ - that is, given $x\in H_p$, if $f_p(x)\in H_p$, the trajectory of $x$ is transverse to $H_p$ at $f_p(x)$ - this implies the discontinuities of $f_p$ in $H_p$ are precisely the points in $f^{-1}_p(\partial H_p)$. By Prop.<ref>, for every $x\in q\setminus\partial H_p$ the first-return map $f_p$ is continuous at $x$ - that is, for every such $x$ $\exists \epsilon>0$ s.t. $f_p(B_\epsilon(x))$ is interior to $H_p$. Therefore, we trivially have $\partial H_p\cap f_p(B_\epsilon(x))=\emptyset$, and the assertion follows.

(180,50)$W^u_{In}\cup W^s_{Out}$
(750,50)$W^u_{In}\cup W^s_{Out}$

[Fig32]the deformation of $H_p$ (on the left) to $H_v$ (on the right). The curve $\rho$ is deformed to $\rho_v$.

We are now ready to state and prove Th.<ref>. To do so, let us first recall that given a trefoil parameter $p\in P$, $f_p:\overline{H_p}\to\overline{H_p}$ denotes the first-return map for $F_p$ (wherever defined). In Th.<ref> we have proven there exists a continuous $\pi:Q\to\{1,2\}^\mathbf{N}$ s.t. the one-sided shift $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ satisfies $\pi\circ f_p=\sigma\circ\pi$, and $\sigma$ is chaotic on $\pi(Q)$. Moreover, we have also proven that given any  periodic $s\in\{1,2\}^\mathbf{N}$ which is both periodic of minimal period $k>0$ and distinct from $\{1,1,1...\}$, then $s\in\pi(Q)$ and $\pi^{-1}(s)$ contains a periodic point for $f_p$ of minimal period $k$. Now, further recall that by Cor.<ref> we have $Q\subseteq I$, with $I$ denoting the invariant set of $f_p$ in $\overline{H_p}\setminus\rho$. As proven in the proof of Th.<ref>, $\overline{H_p}\setminus\rho$ is composed of two subsets, $D_1,D_2$, which define the symbolic coding given by $\pi$ (see the discussion before Lemma <ref>) - see the illustration in Fig.<ref>.

Now, given any $v\in P,v\ne p$, let $f_v:\overline{H_v}\to\overline{H_v}$ denote the first-return map for the vector field $F_v$ (wherever defined - with $H_v$ as in Lemma <ref>). Since the cross-sections $H_v$ vary continuously when we vary $v\in P$ (see the discussion preceding Lemma <ref>), when $F_p$ is continuously deformed to $F_v$ the curve $\rho\subseteq \overline{H_p}$ is continuously deformed to $\rho_v\subseteq\overline{H_v}$. Hence, since $\rho$ divides $\overline{H_p}$ to $D_1,D_2$, we conclude $\rho_v$ divides $\overline{H_v}$ to $D_{1,v},D_{2,v}$ - and since $P_{In},P_{Out}\not\in\rho$ (see Lemma <ref> and Cor.<ref>), the same is true for $\rho_v$ (see the illustration in Fig.<ref>). This motivates us to define a symbolic coding on $I_v$, the invariant set of $f_v$ in $\overline{H_v}\setminus\rho_v$ - provided $I_v$ is non-empty, there exists a function $\pi_v:I_v\to\{1,2\}^\mathbf{N}$ (not necessarily continuous) s.t. $\pi_v\circ f_v=\sigma\circ\pi_v$ (again, with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift). We now prove:

Let $p$ be a trefoil parameter, and let $s\in\pi(Q)$ be periodic of minimal period $k$, s.t. $s$ is not the constant $\{1,1,1,...\}$. Then, for any $v\in P$ sufficiently close to $p$, $I_v$ contains a periodic point $x_v$ for $f_v$ of minimal period $k$, s.t. $\pi_v(x_v)=s$. Additionally, $\pi_v$ is continuous at $x_v$.

From now on throughout the remainder of the proof, let $s\in\pi(Q)$ be periodic, with minimal period $k>1$, s.t. $s$ is not the constant $\{1,1,1,...\}$. Also, let $t>0$ s.t. Prop.<ref> holds. Much like the proof of both Th.<ref> and Th.<ref>, we will prove Th.<ref> in a series of Lemmas, Propositions, and their Corollaries. The basic idea of the proof is that since we can smoothly deform $R_{p,t}$ to a hyperbolic vector field (see Lemma <ref>), we can compute the fixed point index of $f^k_{p,t}$ around periodic points of minimal period $k$ in $Q$ (for the definition of the fixed-point index, see Ch.VII in <cit.>). As $f_{p,t},f_p$ coincide on $Q$ by Prop.<ref> and Cor.<ref>, $f^k_p$ must have the same fixed-point index. As the fixed point index persists under certain homotopies, Th.<ref> will follow.

To begin, let $p\in P$ be a trefoil parameter. For $t>0$ sufficiently small, by Prop.<ref> $F_p,R_{p,t}$ coincide on the trajectories of initial conditions in $\overline{D_\alpha}=D_{\alpha,t}$ - and since in the proof of Th.<ref> we have proven $Q\subseteq D_{\alpha,t}$ (see Cor.<ref>), the conclusions of Th.<ref> must also hold for $R_{p,t}$. That is, there exists a continuous, surjective $\pi:Q\to\pi(Q)$ s.t. $\pi\circ f_{p,t}=\sigma\circ \pi$ (with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift). As proven in Th.<ref>, $\pi(Q)$ includes at least every $\omega\in\{1,2\}^\mathbf{N}$ that is not pre-periodic to the constant $\{1,1,1...\}$. To continue, fix some periodic $s\in\{1,2\}^\mathbf{N}$of minimal period $k>0$ s.t. $s$ is not the constant $\{1,1,1...\}$. Let $D_s$ denote the component of $Q$ s.t. $\pi(D_s)=s$ - as proven in Th.<ref> $D_s$ is closed and connects at least two points on $\partial H_{p,t}$, separated from one another by $\delta_0$ (see the discussion at page cod). As proven in Th.<ref> (see Lemma <ref>), $D_s$ contains at least one periodic point of minimal period $k$. We begin with the following Lemma:

Let $p$ be a trefoil parameter, and let $Per^k(s)=\{x\in D_s|f^k_{p,t}(x)=x\}$. Then, $Per^k(s)$ is compact, and lies away from both $P_{In},P_{Out}$.

Recall $H_{p,t}$ is an open topological disc. As $Per^k(s)\subseteq D_{s}$ and since $D_{s}\subseteq \overline{D_{\alpha,t}}$, because $D_{\alpha,t}$ is bounded in $H_{p,t}$ (see the proof of Prop.<ref>) we conclude $\overline{Per^k(s)}$ is compact - and since $P_{In},P_{Out}\not\in D_s$ (see Lemma <ref> and Lemma <ref>), by $\overline{Per^k(s)}\subseteq D_s$ all the points in $\overline{Per^k(s)}$ must lie away from $P_{Out},P_{In}$. Now, let $x$ be an accumulation point for $Per^k(s)$, and choose some $j$ s.t. $y=f^j_{p,t}(x)\in H_{p,t}$ (such a $j\geq 1$ exists since $H_{p,t}$ is a universal cross-section - see Cor.<ref>). Now, let $\{x_n\}_n\subseteq Per^k(s)$ be s.t. $x_n\to x$, and denote by $y_n=f^j_{p,t}(x_n)$. Since $y$ is interior to $H_{p,t}$, from $\pi\circ f_{p,t}=\sigma\circ\pi$ we have $y\in D_{\sigma^j(s)}\setminus\partial H_{p,t}$. By Prop.<ref>, $\forall i\geq0,f^i_{p,t}(y)$ is always interior to $H_{p,t}$. As such, $f^k_{p,t}$ is continuous sufficiently close to $y$ - therefore, since $\forall n>0, f^k_{p,t}(y_n)=y_n$, we also have $f^k_{p,t}(y)=y$. That is, $y$ is a fixed point for $f^k_{p,t}$ - and since $y=f^j_{p,t}(x)$, $x\in Per^k(s)$ and the Lemma follows.

We now conclude the following, immediate corollary:

There exists an open neighborhood $N_1$ of $D_s\cap\partial H_{p,t}$ in $\overline{H_{p,t}}$, s.t. $\forall x\in N_1, f^k_{p,t}(x)\ne x$.

Assume this is not the case - that is, we can approximate some $x_0\in D_s\cap\partial H_{p,t}$ with periodic points for $f_{p,t}$, all of minimal period $k$. Similarly to the argument used to prove Lemma <ref>, we conclude $x_0\in D_s$ is a periodic point for $f_{p,t}$ in $D_s\cap\partial H_{p,t}$ of period $k$. This is a contradiction to Cor.<ref>, and the assertion follows.

In order to proceed, we must introduce the following formalism. To do so, recall the vector field $R_{p,t}$ can be continuously deformed to the hyperbolic vector field $G$ (see Prop.<ref> and Lemma <ref>). Let us therefore denote such a continuous deformation by, $\Gamma(r),r\in[0,1]$, a $C^\infty$ curve of smooth vector fields in $S^3$, s.t. $\Gamma(0)=R_{p,t}$, $\Gamma(1)=G$, and the Hopf-bifurcation where we blow up the heteroclinic connections (see Prop.<ref>) occurs at $\Gamma(\frac{1}{2})=H$. We also choose this curve s.t. $\forall r>\frac{1}{2}$, $\Gamma(r)$ is a vector field satisfying the assumptions (and conclusions) of Prop.<ref> and Lemma <ref>. Now, given a vector field $\Gamma(r),r\in[0,1]$ denote its first-return map by $g_{p,r}:\overline{H_{p,t}}\to\overline{H_{p,t}},r\in[0,1]$. For simplicity, we write $g_{p,0}=f_{p,t}$ - hence, given any periodic $s\in\{1,2\}^\mathbf{N}$ which is not the constant $\{1,1,1...\}$, since $P_{In},P_{Out}\not\in D_s$ we can construct $\Gamma(r),0
\leq r\leq\frac{1}{2}$ s.t. $g^k_{p,r}|_{D_s}=f^k_{p,t}|_{D_s}$. That is, we construct $\Gamma$ s.t. $g^k_{p,r},0\leq r\leq\frac{1}{2}$ coincide on $D_s$.

As we will soon discover, the curve $\Gamma$ will play a central in the proof that the fixed-point index of $f^k_{p,t}$ is non-zero on some neighborhood of $D_s$. We will first prove:

Let $s\in\{1,2\}^\mathbf{N}$ be periodic of minimal period $k$, s.t. $s$ is not the constant $\{1,1,1,...\}$. Then, there exists a $C^\infty$ curve of vector fields $\Gamma$ as described above which satisfies the following:

* There exists an open neighborhood of $\partial H_{p,t}\cap D_s$ in $\overline{H_{p,t}}$, $N_2$, where there are no fixed points for $f^k_{p,t}$.

* $\forall 1\geq r>0$, $g^k_{p,r}$ also has no fixed points in $N_2$.

* $H_{p,t}$ is a universal cross-section for $\Gamma(r),r\in[0,1]$ - that is, if $x\in\mathbf{R}^3$ has a trajectory which does not limit to a fixed point, its trajectory eventually hits $H_{p,t}$ transversely.

Consider $N_1$ from Lemma <ref> - as proven in Lemma <ref>, there are no fixed points for $f^k_{p,t}$ in $N_1$. Now, let us construct $\Gamma$ - as both $D_s,N_1$ are disjoint from $P_{Out},P_{In}$, we can smoothly deform $R_{p,t}$ around $P_{In},P_{Out}$ s.t. the flow dynamics on some neighborhood of $D_s\cup N_1$ are unaffected. That is, we choose the $C^\infty$ sub-curve $\Gamma(0,\frac{1}{2})$ of vector fields s.t. $\forall r\in[0,\frac{1}{2}]$, $\Gamma(r),R_{p,t}$ coincide on the trajectories of initial conditions in some neighborhood $N'$ in $\mathbf{R}^3$, $(D_s\cup N_1)\subseteq N'$. This allows us to choose $\Gamma([0,\frac{1}{2}])$ s.t. $g^k_{p,r},0\leq r\leq\frac{1}{2}$ coincide on $N'\cap H_{p,t}$ - and in particular, they coincide on $D_s$. Additionally, we choose the sub-curve $\Gamma([0,\frac{1}{2}])$ s.t. the heteroclinic connection connecting $P_{In},P_{Out},\infty$ persists for every $\Gamma(r),r\in[0,\frac{1}{2}]$ - therefore, by the definition of $H_{p,t}$ (see Cor.<ref>), it remains a universal cross-section for $\Gamma(r),r\in[0,\frac{1}{2}]$,

Now, by blowing up the fixed-point and deforming $\Gamma(\frac{1}{2})$ as described in Prop.<ref>, smoothly deform $H=\Gamma(\frac{1}{2})$ to some vector field $\Gamma(\epsilon_1),1>\epsilon_1>0$ s.t. $\forall r\in[\frac{1}{2},\epsilon_1]$, $\Gamma(r)$ satisfies the assumptions (and conclusions) of Prop.<ref>. Let us recall this smooth deformation removes whatever discontinuities $g_{p,r},r\in[0,\frac{1}{2}]$ has in $\overline{H_{p,t}}$, thus generating a first return map $g_{p,r},r>\frac{1}{2}$, which is continuous sufficiently far away from the fixed points of $\Gamma(r)$ (see the proof of Prop.<ref> for the details). Additionally, we choose this deformation s.t. the heteroclinic connection which connects $\infty,P_{In},P_{Out}$ persists for every $r\in[0,\epsilon_1]$. Therefore, recalling the definition of $H_{p,t}$ (see Cor.<ref>), we conclude $\Gamma$ can be chosen s.t. $H_{p,t}$ is always a universal cross-section for every $\Gamma(r),r\in[0,\epsilon_1]$ .

To proceed, we now prove the existence of some $\frac{1}{2}<\epsilon$ s.t. for every $\epsilon\geq r>\frac{1}{2}$, $g^k_{p,r}$ has no fixed points in some neighborhood $N_2$ of $D_s\cap\partial H_{p,t}$. That is, we will prove there exists some $\epsilon>0$ and some fixed neighborhood $N_2$ of $D_s\cap\partial H_{p,t}$ in $\overline{H_{p,t}}$, independent of $r\in[0,\epsilon]$, s.t. the Hopf bifurcation at $\Gamma(\frac{1}{2})=H$ does not generate any new fixed points in $N_2$ for $g^k_{p,r},\epsilon>r>\frac{1}{2}$. See the illustration in Fig.<ref>. To do so, consider some sequence $\{(x_n,r_n)\}_n\subseteq \overline{H_{p,t}}\times[0,1]$ s.t. $\forall n>0, g^k_{p,r_n}(x_n)=x_n$, $r_n\to \frac{1}{2}$. We will now rule out the possibility $\{x_n\}_n$ accumulates on $\partial D_s\cap\partial H_{p,t}$. To do so assume this is not the case, and let $x\in\partial D_s\cap\partial H_{p,t}$ be an accumulation point for the sequence $\{x_n\}_n$, $g^k_{p,r_n}(x_n)=x_n$ as described above - without any loss of generality, we may well assume $x_n\to x$. As $H_{p,t}$ is a universal cross-section for $\Gamma(\frac{1}{2})=H$, and since there are no fixed points for the vector field $H$ in $D_s$, there exists some $j$ s.t. $g^j_{p,\frac{1}{2}}(x)=y\in H_{p,t}$ - that is, the flow line (w.r.t. $H$) emanating from $x$ eventually hits the cross-section $H_{p,t}$ transversely. Therefore, since $\Gamma(0,\epsilon_1)$ is a smooth curve (and in particular, $C^\infty$), for all sufficiently large $n>0$, $y_n=g^j_{p,r_n}(x_n)$ also lies inside $H_{p,t}$ - in particular, the sequence $\{y_n\}_n$ satisfies $y_n\to y$.

(460,110)$W^u_{In}\cup W^s_{Out}$

[Fig32]the neighborhood $N_2$ - as proven in Prop.<ref>, $\forall r\in[0,\epsilon],x\in N_2$, $g^k_{p,r}(x)\ne x$.

Since by construction $\Gamma(r_n)\to \Gamma(\frac{1}{2})=H$ in the $C^\infty$ metric, by $g^k_{p,r_n}(y_n)=y_n$ for all $n>0$, we must also have $g^k_{p,\frac{1}{2}}(y)=y$. In other words, $y$ is a fixed point for $g^k_{p,\frac{1}{2}}$ - which implies $x$ is also a fixed point for $g^k_{p,\frac{1}{2}}$. Since $x\in \partial D_s$ and since we constructed $\Gamma([0,\frac{1}{2}])$ s.t. $H=\Gamma(\frac{1}{2}),R_{p,t}=\Gamma(0)$ coincide on $N'$, this is a contradiction to Cor.<ref>. Therefore, we conclude there exists some open set $N_2$ in $\overline{H_{p,t}}$, $(D_s\cap\partial H_{p,t})\subseteq N_2$ s.t. for every $r\in[\frac{1}{2},\epsilon],\epsilon\leq\epsilon_1$, $g^k_{p,r}$ has no fixed points in $N_2$.

Now, to conclude the proof, apply Lemma <ref> to the vector field $\Gamma(\epsilon)$ to create the vector field $G=\Gamma(1)$ - the process described in Lemma <ref> can only decrease the number of fixed points for $g^k_{p,r},1\geq r>\epsilon$ in $N_2$- as such, if $\forall r\in[0,\epsilon]$ $g^k_{p,r}$ has no fixed points in $N_2$, the same is true for $g^k_{p,r},r\in[\frac{1}{2},1]$. Therefore, $\forall r\in[0,1]$, $g^k_{p,r}$ has no fixed points in $N_2$. Finally, recalling the process described in Lemma <ref>, we deform $\Gamma(\epsilon)$ to $G=\Gamma(1)$ s.t. the heteroclinic connection through $\infty$ persists for all $\Gamma(r),r\in[0,\epsilon]$ - therefore, $H_{p,t}$ remains universal cross-section for every $\Gamma(r),r\in[\epsilon,1]$. Because we already constructed $\Gamma$ s.t. $H_{p,t}$ is a universal cross-section for $\Gamma(r),r\in[0,\epsilon]$ the conclusion follows.

Now, recall we denote by $Per^k(s)=\{x\in D_s|f^k_{p,t}(x)=x\}$. Having constructed a curve $\Gamma$ and $N_2$ as in Prop.<ref>, we are almost ready to construct a homotopy between $f^k_{p,t}$ and $g^k_{p,1}$ around $D_s$. To do so, we first prove:

Let $p\in P$ be a trefoil parameter, let $s\in\{1,2\}^\mathbf{N}$ be a periodic symbol of minimal period $k>0$ s.t. $s$ is not the constant $\{1,1,1...\}$, and let $\Gamma$ denote the curve from Lemma <ref>. With previous notations, there exist two open topological discs, $S$, $V_p$, in $H_{p,t}$ s.t.:

* $Per^k(s)\subseteq V_p$, $V_p\subseteq S$, and $\overline{D_s}\subseteq\overline{S}$.

* $f^i_{p,t},0\leq i\leq k$ is continuous on $V_p$ - and in particular, $\forall 0\leq j\leq k$, we have $f^j_{p,t}(V_p)\cap \partial H_{p,t}=\emptyset$, $V_p\cap f^{-j}_{p,t}(\rho)=\emptyset$.

* $\forall j,i\in\{0,...,k-1\}$, $f^i_{p,t}(V_p)\cap f^j_{p,t}(V_p)=\emptyset$.

* There are no fixed points for $f^k_{p,t}$ in $\partial V_p$.

* There exists annulus $S_0\subseteq S$ s.t. $\partial S$ is a component of $\partial S_0$, and $f^k_{p,t}$ has no fixed points in $\overline{S_0}$. Additionally, $\partial V_p\subseteq S_0$ (see the illustration in Fig.<ref>).

* With the notations of Lemma <ref> - we can choose $\Gamma$ s.t. for every $r_1,r_2\in[0,\frac{1}{2}]$, $g^k_{p,r_1}|_{\overline{S}}=g^k_{p,r_2}|_{\overline{S}}$ - that is, for every such $r_1,r_2$, the maps $g^k_{p,r_1},g^k_{p,r_2}$ coincide on $S$. In particular, since $f^k_{p,t}$ is continuous on $V_p$, so is $g^k_{p,r},r\in[0,\frac{1}{2}]$.

To begin, recall the components of $Q$ are denoted by $D_\omega,\omega\in\{1,2\}^\mathbf{N}$, $\pi(D_\omega)=\omega$. Further recall the discontinuity curve $\delta$ for $f_{p,t}$ (see Lemma <ref>). For any $d\geq 0$, $f^{-d}_{p,t}(\delta)$ is a compact collection of curves in $\overline{H_{p,t}}$ - therefore, so is $\cup_{k+1\geq d\geq 0}f^{-d}_{p,t}(\delta)$. Additionally, recall $D_s$ is the (unique) component of $Q$ s.t. $\pi(D_s)=s$, for some periodic $s\in\{1,2\}^\mathbf{N}$ of minimal period $k$ - by the proof of Th.<ref>, $\forall i\leq0$, $f^i_{p,t}(D_s)\cap\delta=\emptyset$ hence $D_s$ must lie away from  $\cup_{k+1\geq d\geq 0}f^{-d}_{p,t}(\delta)$. Additionally, recall there exist only a finite number of periodic $\omega\in\{1,2\}^\mathbf{N}$ of minimal period $j\leq k$ - therefore, denoting the union of such $D_\omega$ by $L(k)$, we conclude $D_s\subseteq L(k)$, that $L(k)$ is compact, and by the proof of Th.<ref>, $L(k)$ lies away from both $\rho,\delta$. Now, choose two periodic symbols $\omega_1,\omega_2\in\{1,2\}^\mathbf{N}$ s.t. both are periodic of minimal period greater then $k$, and consider the components of $\overline{H_{p,t}}\setminus (D_{\omega_1}\cup D_{\omega_2})$ - as $D_{\omega_i},i=1,2$ both connect at least two distinct points of $\partial H_{p,t}$ separated by $\delta_0$, there exists a component $S$ of $\overline{H_{p,t}}\setminus (D_{\omega_1}\cup D_{\omega_2})$  s.t. $D_s\subseteq S$ (see the illustration in Fig.<ref>).

In particular, we choose $\omega_1,\omega_2$ with sufficiently large minimal periods s.t. the following holds:

* $D_s=L(k)\cap S$. That is, if $\omega\in\{1,2\}^\mathbf{N}$ is periodic and $D_\omega\cap S\ne\emptyset$, then $D_\omega\subseteq S$ and the minimal period of $\omega$ is at least $k$ - and if it is $k$, then $\omega=s$.

* As every component of $Q$ is separated from $\delta$ by infinitely many other components of $Q$, we conclude $\delta\cap S=\emptyset$ - see the illustration in Fig.<ref>.

* By the compactness of $\cup_{k+1\geq d\geq 0}f^{-d}_{p,t}(\delta)$ we choose the minimal periods of $\omega_1,\omega_2$ to be sufficiently large s.t. s.t. $\forall 0\leq j\leq k+1$, $f^{-j}_{p,t}(\delta)\cap S=\emptyset$.

(460,110)$W^u_{In}\cup W^s_{Out}$

[Fig32]$S$, trapped between $D_{\omega_1},D_{\omega_2}$ and satisfying $\partial S\cap\partial H_{p,t}\subseteq N_2$.

By construction, $\partial S$ is composed from subsets of $D_{\omega_1},D_{\omega_2}$, and two arcs $I_1,I_2$ on $\partial H_{p,t}\cap\partial D_{\alpha,t}$. In particular, $I_1,I_2$ are separated from one another in $\partial H_{p,t}\cap\partial D_{\alpha,t}$ by $\delta_0$ - see the illustration in Fig<ref>. To continue, recall the set $N_2$ constructed in Lemma <ref> - by Prop.<ref> we can choose $D_{\omega_1},D_{\omega_2}$ sufficiently close to $D_s$ s.t. both $I_1,I_2$ lie in $N_2$ (see the illustration in Fig.<ref>). Now, further recall we originally chose $\omega_1,\omega_2$ s.t. their minimal periods are greater than $k$ - which implies $\forall 0\leq j\leq k$, $f^j_{p,t}(D_{\omega_i})\cap D_{\omega_i}=\emptyset,i=1,2$. Additionally, by Lemma <ref>, $\forall x\in N_2,f^k_{p,t}(x)\ne x$ - from which we conclude $\forall x\in S$, $f^k_{p,t}(x)\ne x$. In other words, $f^k_{p,t}$ has no fixed points in $\partial S$. Using a similar argument to the proof of Lemma <ref>, we conclude the set of fixed points of $f^k_{p,t}$ in $\overline{H_{p,t}}$ is compact - hence, since $\forall x\in\partial S, f^k_{p,t}(x)\ne x$, there exists some annulus $S_0$, $S_0\subseteq S,\partial S\subseteq\partial S_0$, s.t. s.t. $f^k_{p,t}$ has no fixed points in $S_0$ - see the illustration in Fig.<ref>.

Now, let us recall $Dis(f_{p,t})$ is the set of discontinuity points of $f_{p,t}$ in $H_{p,t}$ - by the proof of Lemma <ref>,  $Dis(f_{p,t})=f^{-1}_{p,t}(\partial H_{p,t})\cap H_{p,t}$. Now, recall $\delta\cap S=\emptyset$, and that by Prop.<ref> we have $D_{\omega_i}\cap Dis(f_{p,t})=\emptyset,i=1,2$. Therefore, if $S\cap Dis(f_{p,t})\ne\emptyset$ the only possibility is that there exists a component $\alpha$ of $Dis(f_{p,t})$ s.t. $\alpha\ne \delta, \alpha\subseteq S$. By Lemma <ref>, $\alpha$ has two endpoints in $\partial H_{p,t}$ (by the Existence and Uniqueness Theorem, these endpoints are distinct). Now, let us consider $\overline{Dis(f_{p,t})\cap S}$. By Cor.<ref>, $\overline{Dis(f_{p,t})\cap S}$ cannot accumulate on $D_s\cap H_{p,t}$ - hence, by Prop.<ref>, provided the minimal periods of $\omega_1,\omega_2$ are sufficiently large, any component $\alpha\subseteq Dis(f_{p,t})\cap S$ must lie in $N_2\cap S_0$ - that is, $\alpha\subseteq N_2\cap S_0$. Additionally, since by Prop.<ref> $f^j_{p,t},j\in\{1,...,k\}$ is continuous on $D_s\cap H_{p,t}$, the same argument now yields that the components of  $f^{-j}_{p,t}(Dis(f_{p,t})),j\in\{1,...,k\}$ also cannot accumulate on $D_s\cap H_{p,t}$ - hence, provided the minimal periods of $\omega_1,\omega_2$ are sufficiently large, $\forall0\leq j\leq k$ we have $f^{-j}_{p,t}(Dis(f_{p,t}))\subseteq N_2\cap S_0$.

To summarize this discussion, we conclude that provided the minimal periods of $\omega_1,
\omega_2$ are sufficiently large , we have $S\cap(\cup_{0\leq j\leq k}f_{p,t}^{-j}(Dis(f_{p,t})))\subseteq N_2\cap S_0$. Therefore, we choose $V_p$ to be some topological disc in $S$, s.t. $Per^k(s)\subseteq V_p$, $\partial V_p\subseteq S_0$, $\partial V_p\cap(\cup_{0\leq j\leq k}f_{p,t}^{-j}(Dis(f_{p,t})))=\emptyset$. This can be done, as $Per^k(s)\cap S_0=\emptyset$, and because we can choose $S$ s.t. $\cup_{0\leq j\leq k}f_{p,t}^{-j}(Dis(f_{p,t}))\subseteq S_0$. In other words, by our choice of $V_p$, $f^j_{p,t},0\leq j\leq k$ is continuous on $\overline{V_p}$.  Additionally, by construction, as we choose $V_p$ s.t. $\partial V_p\subseteq S_0$, there are no fixed points for $f^k_{p,t}$ in $\partial V_p$. Finally, recall that $D_{\sigma^i(s)}\cap D_{\sigma^i(s)}=\emptyset$ $\forall i\ne j$, $i,j\in\{0,...,k-1\}$ - therefore, since $f^j_{p,t}(D_s)\subseteq D_{\sigma^j(s)}$, $f^i_{p,t}(D_s)\subseteq D_{\sigma^i(s)}$ by increasing the minimal period of $\omega_1,\omega_2$ we can choose $V_p$ s.t. for every $i\ne j$, $i,j\in\{0,...,k-1\}$, $f^j_{p,t}(V_p)\cap f^i_{p,t}(V_p)=\emptyset$

(460,110)$W^u_{In}\cup W^s_{Out}$

[Fig32]the annulus $S_0$, trapped between $\partial S$ and the blue loop.

All in all, to conclude the proof of Prop.<ref> it remains to prove that given the curve $\Gamma$ from Lemma <ref>, $g^k_{p,r_1},g^k_{p,r_2}$ coincide on $\overline{S}$ (recall $g_{p,0}=f_{p,t}$). To do so, recall that $P_{In},P_{Out}\not\in Q$ - and as a consequence, $P_{In}, P_{Out}\not\in\overline{S}$. As such, we can construct $\Gamma$ (see the discussion before Lemma <ref>) s.t. the first-return maps of $\Gamma(r)$, $g^k_{p,r}:\overline{H_{p,t}}\to\overline{H_{p,t}}$, are unaffected on $\overline{S}$ when $\Gamma(r)$ is smoothly varied in $r\in[0,\frac{1}{2}]$ - that is, $\forall r_1,r_2\in[0,\frac{1}{2}],g^k_{p,r_1}|_{\overline{S}}=g^k_{p,r_2}|_{\overline{S}}$. In particular, since $f^k_{p,t}=g^k_{p,0}$ is continuous on $V_p$, the same is true for $g^k_{p,r},r\in[0,\frac{1}{2}]$. Prop.<ref> is now proven.

Let $V_p,S,S_0$ denote the sets constructed in Prop.<ref> - as proven in Prop.<ref>, $f^k_{p,t}$ is continuous on $V_p$. We are now ready to prove the fixed-point index of $f^k_{p,t}:V_p\to H_{p,t}$ is $1$. To this end, let us first recall the definition of the fixed-point index - given a topological disc $V$, provided the set $Fix(V)=\{x\in V|f(x)=x\}$ is compact, the fixed-point index of a continuous $f:V\to \mathbf{R}^2$ is the degree of $f(x)-x$ (see Ch.VII.5 in  <cit.>). Given a homotopy of continuous maps $f_t:V\to\mathbf{R}^2,t\in[0,1]$, denote by $F_t=\{x\in V|f_t(x)=x\}$, and assume $\cup_{t\in[0,1]} F_t$ is compact in $V\times[0,1]$ - under these assumptions, if the fixed-point index of $f_0$ is $1$, the same is true for $f_1$ (see Ch.VII.5.8 in <cit.>). Or, put simply, if $f_0$ has a fixed point and its fixed-point index is non-zero, $f_1$ also has at least one fixed point. We now prove:

Let $p\in P$ be a trefoil parameter, and let $s\in\{1,2\}^\mathbf{N}$ be periodic of minimal period $k$, s.t. $s$ is not the constant $\{1,1,1,...\}$. Additionally, let $V_p,S,S_0$ be as in Prop.<ref>. Then, the fixed-point index of $f^k_{p,t}$ on $V_p$ is $1$.

Throughout the proof of Prop.<ref>, we always assume $s\in\{1,2\}^\mathbf{N}$ is periodic of minimal period $k$, s.t. $s$ is not the constant $\{1,1,1,...\}$. Additionally, let $\Gamma$ be as in Lemma <ref>, and Prop. <ref>. and let $S,V_p$ be as in Prop.<ref>. We will prove Prop.<ref> as a sequence of three lemmas. The idea behind the proof is as follows - by the existence of $\Gamma$, we can continuously deform the dynamics of $f^k_{p,t}$ on $V_p$ to those of $g^k_{p,1}$ on some topological disc, $V_{p,1}$ (see Lemma <ref> and Lemma <ref>). Since $g_{p,1}$ is a Smale Horseshoe map it is hyperbolic on its invariant set - which we will use to prove the fixed-point index of $g^k_{p,1}$ on $V_{p,1}$ is $1$ (see Lemma <ref>). Using Th.VII.5.8 in <cit.>, we will then conclude the fixed-point index of $f^k_{p,t}$ on $V_p$ is also $1$ and Prop.<ref> will follow.

Now, recall the vector fields $\Gamma(r),1\geq r>\frac{1}{2}$ all satisfy the assumptions (and conclusions) of Prop.<ref>. We begin with the following Lemma:

With previous notations, there exists an $\epsilon\in(\frac{1}{2},1]$ s.t. the fixed point index of $f^k_{p,t}$ on $V_p$ and the fixed point index of $g^k_{p,\epsilon}$ on $V_p$ are equal.

To begin, recall we constructed $S$ in Prop.<ref> s.t. $\forall r\in[0,\frac{1}{2}]$ $g^k_{p,r}$ coincide on $S$. Since $V_p\subseteq S$, we conclude that for all $r_1,r_2\in[0,\frac{1}{2}]$, $g^k_{p,r_1},g^k_{p,r_2}$ coincide on $V_p$. Now, recall we proved in Prop.<ref> that $f^k_{p,t}=g^k_{p,0}$ is continuous on $V_p$ - therefore, we immediately conclude $g^k_{p,r}:V_p\to\overline{H_{p,t}}$$,r\in[0,\frac{1}{2}]$ is a homotopy of continuous maps. Now, for every $r\in[0,1]$, define $Fix(r)=\{x\in V_{p}|g^k_{p}(x)=x\}$ - we immediately conclude $\cup_{r\in[0,\frac{1}{2}]}Fix(r)\times\{r\}$ is compact in $V_p\times[0,\frac{1}{2}]$. As $g^k_{p,r},r\in[0,\frac{1}{2}]$ are a homotopy of continuous functions on $V_p$, by Th.VII.5.8 in <cit.> we conclude the fixed-point index for $g^k_{p,\frac{1}{2}}$ on $V_p$ is the same as the fixed point index for $f^k_{p,t}$.

Since $S_0\subseteq S$, by Prop.<ref> for every $r\in[0,\frac{1}{2}]$ the function $g^k_{p,r}$ has no fixed points in $S_0\setminus V_p$. We now claim this property persists when we continuously deform the vector field $\Gamma(r_1),r_1\leq\frac{1}{2}$ to the vector field $\Gamma(r_2),r_2>\frac{1}{2}$ - that is, we claim there exists some $\epsilon>\frac{1}{2}$ s.t. $\forall r\in[0,\epsilon]$, $g^k_{p,r}$ has no fixed points in $\overline{S_0}\setminus V_p$. To prove this assertion, assume this is not the case - that is, assume there exists a sequence $\{(x_n,r_n)\}_n,r_n>\frac{1}{2}$, $x_n\in S_0\setminus V_p$ s.t. $r_n\to\frac{1}{2}$, and $g^k_{p,r_n}(x_n)=x_n$. By the pre-compactness of $V_p$, $\{x_n\}_n$ has an accumulation point $x\in\overline{S_0\setminus V_p}$ - as $H_{p,t}$ is a universal cross section for the vector field $\Gamma(\frac{1}{2})=H$ (see Lemma <ref>), a similar argument to the one used in the proof of Lemma <ref> proves $g^k_{p,\frac{1}{2}}(x)=x$. This is a contradiction, since by Prop.<ref> we have $f^k_{p,t}|_{\overline{S_0}}=g^k_{p,\frac{1}{2}}|_{\overline{S_0}}$, and $f^k_{p,t}$ has no fixed points in $\overline{S_0}$.

Therefore, there exists some $\epsilon>\frac{1}{2}$ s.t. $\forall r\in[\frac{1}{2},\epsilon]$, $g^k_{p,r}$ has no fixed points in $S_0\setminus V_p$. Since $\cup_{r\in[\frac{1}{2},\epsilon]}Fix(r)\times\{r\}$ does not accumulate on $V_p\times[\frac{1}{2},\epsilon]$, we conclude $\cup_{r\in[\frac{1}{2},\epsilon]}Fix(r)\times\{r\}$ is compact in $V_p\times[\frac{1}{2},\epsilon]$ - and combined with previous paragraphs, we conclude $\cup_{r\in[0,\epsilon]}Fix(r)\times\{r\}$ is also compact in $V_p\times[0,\epsilon]$.

Now, let us recall that $f_{p,t},f^2_{p,t},...,f^k_{p,t}$ are all continuous on $V_p$ (see Prop.<ref>) - that is, given any $0\leq j\leq k-1$, the first-return point for the trajectory of every $x\in V_p$ is transverse to $H_{p,t}$. Since by Prop.<ref> $g^k_{p,r_1},g^k_{p,r_2}, r_1,r_2\in[0,\frac{1}{2}]$ coincide on $V_p$, we conclude that for any $\epsilon$ sufficiently close to $\frac{1}{2}$, $g^k_{p,r},r\in[\frac{1}{2},\epsilon]$ is also continuous on $V_p$. All in all, provided $\epsilon>0$ is sufficiently close to $\frac{1}{2}$, $g^k_{p,r}:V_p\to\overline{H_{p,t}},r\in[0,\epsilon]$ is a homotopy of continuous maps. Since $\cup_{r\in[0,\epsilon]}Fix(r)\times\{r\}$ is compact in $V_p\times[0,\epsilon]$ and since $f^k_{p,t},g^k_{p,\epsilon}$ are homotopic on $V_p$, applying again Th.VII.5.8 in <cit.> we conclude their fixed point indices are equal.

Having proven Lemma <ref>, we will now compute the fixed point index of $g^k_{p,\epsilon}$ on some larger set. To do so, we must first recall several facts. First, recall we choose $\Gamma$ s.t. $\forall r>\frac{1}{2}$, the vector field $\Gamma(r)$ satisfies the assumptions (and conclusions) of Prop.<ref> and Lemma <ref>. That is, $\forall r\in(\frac{1}{2},1]$ there exists a $g_{p,r}$-invariant $Q_r\subseteq\overline{H_{p,t}}$ on which $g_{p,r}$ is continuous, and a continuous, surjective $\pi_r:Q_r\to\{1,2\}^\mathbf{N}$ s.t. $\pi_r\circ g_{p,r}=\sigma\circ\pi_r$ (with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift). By Lemma <ref>, $\forall r\in(\frac{1}{2},1]$ $L^r_\omega=\pi^{-1}_r(\omega)$ is a compact set, s.t. $L^r_\omega\cap\partial H_{p,t}$ have at least two components. Now, by the definition of $D_s,D_{\omega_1},D_{\omega_2}$ (see the discussion before Lemma <ref>), when $R_{p,t}$ is continuously deformed to $\Gamma(r),r>\frac{1}{2}$, $D_{\omega_1},D_{\omega_2}$ and $D_s$ are continuously deformed to $L^r_{\omega_1},L^r_{\omega_2},L^r_s$ - and by Cor.<ref>, given $r_1,r_2\in(\frac{1}{2},1]$, when we vary $\Gamma(r_1)$ is smoothly deformed to $\Gamma(r_2)$ (along $\Gamma$) $L^{r_1}_{\omega_1},L^{r_1}_{\omega_2},L^{r_1}_s$ are smoothly deformed to $L^{r_2}_{\omega_1},L^{r_2}_{\omega_2},L^{r_2}_s$.

Therefore, since $S$ is a component of $H_{p,t}\setminus(D_{\omega_1}\cup D_{\omega_2})$, when we smoothly deform $\Gamma(0)=R_{p,t}$ to $\Gamma(r),1\geq r>\frac{1}{2}$, $S$ is continuously deformed to a topological disc $S_r,r>\frac{1}{2}$ (see the illustration in Fig.<ref>). Analogously to $S$, $\forall r>0$ $S_r$ is the component of $\overline{H_{p,t}}\setminus(L^r_{\omega_1}\cup L^r_{\omega_2})$ which contains $L^r_s$ (see the illustration in Fig.). Now, let $\omega_1,...,\omega_n\in\{1,2\}^\mathbf{N}$ be all the periodic symbols of minimal period at most $k$. By the construction of $S$ (see Prop.<ref>), $(\cup_{i=1}^n D_{\omega_i})\cap S=D_s$ - which proves $\forall r\in(\frac{1}{2},1]$, $(\cup_{i=1}^n L^r_{\omega_i})\cap S_r=L^r_s$. In particular, any periodic point $x$ in $Q_r\cap S_r$ is of minimal period at least $k$ - and provided its minimal period is precisely $k$, then $x\in L^r_s$.

(180,30)$W^u_{In}\cup W^s_{Out}$
(790,40)$W^u_{In}\cup W^s_{Out}$

[Fig32]the deformation of $S$ to $S_r$. The set $D_s$ is deformed to $L^r_s$.

We now prove:

We can choose $\epsilon\in(\frac{1}{2},1]$ s.t. both the conclusion of Lemma <ref> holds, and all the fixed points of $g^k_{p,\epsilon}$ in $S_\epsilon$ lie in $V_p$.

It would suffice to prove that for any $r>\frac{1}{2}$ sufficiently close to $\frac{1}{2}$ there are no fixed points for $g^k_{p,r}$ in $S_r\setminus V_p$. To do so, assume this is not the case - that is, assume there exists a sequence of $r_n\to\frac{1}{2}$ s.t. $\forall n>0$, $\exists x_n\in S_{r_n}\setminus V_p$ s.t. $g^k_{p,r_n}(x_n)=x_n$. Let $(x,\frac{1}{2})$ denote an accumulation point of the sequence $\{(x_n,r_n)\}_n$, $(x_n,r_n)\in S_{r_n}\setminus V_p\times\{r_n\}$ (there exists at least one by the pre-compactness of $S_{r_n}\setminus V_p,n>0$ and $S\setminus V_p$ in $\overline{H_{p,t}}$). As such, $x\in\overline{S\setminus V_p}$ - since $\Gamma$ is a $C^\infty$ curve of vector fields, a similar argument to the one used in the proof of Lemma <ref> proves $g^k_{p,\frac{1}{2}}(x)=x$. However, by Prop.<ref>, $g^k_{p,\frac{1}{2}},f^k_{p,t}$ coincide on $S$ - and since by Prop.<ref>all the fixed points of $f^k_{p,t}$ in $S$ lie in $V_p$ we have a contradiction and Lemma <ref> follows.

In other words, Lemma <ref> proves that provided $\epsilon>\frac{1}{2}$ is sufficiently close to $\frac{1}{2}$, we may well assume all the fixed points of $g^k_{p,\epsilon}$ in $S_\epsilon$ in fact lie in $V_p$. We now prove:

Let $\epsilon$ be sufficiently small s.t. both <ref> and Lemma <ref> hold. Then, with previous notations, the fixed-point index of $g^k_{p,r}:S_r\to\overline{H_{p,t}},r\in[\epsilon,1]$ is $1$.

First recall that with the notation of Lemma <ref>, both $L^r_{\omega_1},L^r_{\omega_2}$ belong to the forward-invariant subset of $g_{p,r}$ in the rectangle $ABCD$ (see Lemma <ref> for the details). This implies $g_{p,r},...,g^k_{p,r}$ are all continuous on $S_r,1\geq r>\frac{1}{2}$ - and in particular, since $\Gamma(r)$ satisfies the assumptions of Prop.<ref>, $\forall 1\leq j\leq k$ we have $g^j_{p,r}(\partial H_{p,t}\cap\partial S_r)\subseteq H_{p,t}$. As such,$g^j_{p,r}(\partial H_{p,t}\cap\partial S_r), 1\leq j\leq k$ is strictly interior to the cross-section $H_{p,t}$ - which proves that for every $1\geq r>\frac{1}{2}$, $g^k_{p,r}$ has no fixed points in $\partial S_r\cap \partial H_{p,t}$. Now, recall we constructed $S$ s.t. the minimal period of $\omega_1,\omega_2$ is greater than $k$ (see Prop.<ref>) - hence, given $1\geq r>\frac{1}{2}$ there can be no fixed points for $g^k_{p,r}$ in either $L^r_{\omega_1},L^r_{\omega_2}$. All in all, for every $1\geq r>\frac{1}{2}$, $\partial S_r$ contains no fixed points for $g^k_{p,r}$.

Now, denote by $F(r)=\{x\in S_r|g^k_{p,r}(x)=x\}$, and recall $\epsilon>\frac{1}{2}$. Since we have just proven that $\forall r\in(\frac{1}{2},1]$, $F(r)\cap \partial S_r=\emptyset$, a similar argument to the one used in the proof of Lemma <ref> now proves $\cup_{r\in[\epsilon,1]}F(r)\times\{r\}$ is compact in $\cup_{r\in[\epsilon,1]}S_r\times\{r\}$. Therefore, applying again Th.VII.5.8 from <cit.>, we conclude the fixed point index for $g^k_{p,r}$ on $S_r$ is constant when $r\in[\epsilon,1]$ is varied.

However, as we have proven earlier, given any $\omega\in\{1,2\}^\mathbf{N}$ of minimal period $j\leq k$, denoting by $L^r_\omega$ the corresponding component of $Q_r$, then $L^r_\omega\cap S_r\ne\emptyset$ if and only if $\omega=s$ - that is, $j=k$, and $L^r_\omega$ must coincide with $L^r_s$ (see the discussion before Lemma <ref>). By the definition of $\Gamma$, the vector field $\Gamma(1)$ is generated from the vector field $\Gamma(\epsilon)$ by applying the algorithm in Lemma <ref> (see the discussion preceding Lemma <ref>). That is, the dynamics of $\Gamma(\epsilon)$ are smoothly deformed to those of $\Gamma(1)$, whose dynamics correspond to a suspended Smale Horseshoe, hyperbolic on its invariant set - which immediately proves $L^1_s$ contains precisely one fixed point, $\{x\}$. Since $x$ is a hyperbolic periodic point for $g_{p,1}$ of minimal period $k$, interior to $S_1$, the degree of $g^k_{p,1}(x)-x$ is $1$ - which proves the fixed-point index of $g^k_{p,1}$ on $S_1$ is $1$. Therefore, by previous paragraph the fixed-point index of $g^k_{p,\epsilon}$ on $S_\epsilon$ is also $1$ and Lemma <ref> follows.

We now conclude the proof of Prop.<ref>. To do so, assume $\epsilon$ is such that Lemma <ref>, Lemma <ref> and Lemma <ref> all hold. By Lemma <ref>, the fixed-point index of $g^k_{p,\epsilon}$ on $S_\epsilon$ is $1$ - however, with previous notations, Lemma <ref> implies $F(\epsilon)\subseteq V_p$. Since $F(\epsilon)$ is compact (as $F(\epsilon)$ is closed and $V_p$ is bounded), and since $V_p$ is open, there are no fixed points of $g^k_{p,\epsilon}$ in $S_\epsilon\setminus V_p$. Therefore, by Th.VII.5.5 in <cit.> the fixed point index of $g^k_{p,\epsilon}$ in $S_\epsilon\setminus V_p$ is $0$ - which, by Th.VII.5.6 in <cit.> proves the fixed-point index of $g^k_{p,\epsilon}$ in $V_p$ is $1$. Therefore, by Lemma <ref> we conclude the fixed-point index of $f^k_{p,t}$ on $V_p$ is also $1$ and Prop.<ref> follows.

We now conclude the proof of Th.<ref>. To do so, let $p\in P$ be a trefoil parameter, and let us recall $f_p:\overline{H_p}\to\overline{H_p}$ is the first-return map of $F_p$, the original vector field. Now, choose a periodic $s\in\{1,2\}^\mathbf{N}$ of minimal period $k$ that is not the constant $\{1,1,1,...\}$. In the notation of Prop.<ref>, let us recall both $D_{\omega_1},D_{\omega_2}$ are subsets of $Q$. However, as we have proven in Th.<ref>, $Q\subseteq \overline{D_{\alpha,t}}$ - therefore, since we chose $t>0$ s.t. Prop.<ref> is satisfied, by Prop.<ref> and Prop.<ref> we conclude $f^k_{p,t},f^k_p$ coincide on $V_p$. As such, since by Prop.<ref> $f^k_{p,t}$ is continuous and has fixed-point index $1$ on $V_p$, the same is true for $f^k_p$. Additionally,  by the construction of $V_p$ in Prop.<ref>, given any $x\in V_p$, $\{x,f_{p,t}(x),...,f^k_{p,t}(x)\}$ are all transverse intersection points between the trajectory of $x$ and the cross section $H_{p,t}$ - as such, $\{x,f_p(x),...,f^k_p(x)\}$ are also all transverse intersection points between the trajectory of $x$ and $H_p$.

Now, recall every vector field $F_v,v\in P$ generates a cross-section $H_{v}$, and that these cross-sections vary smoothly when we smoothly deform $F_p$ to $F_v$ in $P$ (see Lemma <ref>). Therefore, denote by $V_v$ the deformation of $V_p$ when $p\in P$ is smoothly deformed to $v\in P$. Now, given $v\in P$, denote by $f_v:\overline{H_v}\to\overline{H_v}$ the first-return map of $F_v$ (wherever defined). By previous paragraph, provided $v\in P$ is sufficiently close to $p$, $f^k_v:V_v\to\overline{H_v}$ is continuous, and homotopic to $f^k_p:V_p\to\overline{H_p}$.

To continue, recall the curve $\rho$ from Cor.<ref> - as we have proven in Prop.<ref>, we can choose $V_p$ s.t. $V_p\cap f^{-j}_{p,t}(\rho)=\emptyset$ for all $0\leq j\leq k$. As a consequence, we have  $f^j_{p,t}(V_p)\cap\rho=\emptyset$ for all $0\leq j\leq k$ - hence by previous paragraph we also have $f^j_{p}(V_p)\cap\rho=\emptyset$ for all $0\leq j\leq k$. With the notations of page conti2, recall $\rho$ is continuously deformed to $\rho_v$ when $F_p$ is smoothly deformed to $F_v$ in $P$ (see the illustration in Fig.<ref>) - therefore, provided $v$ is sufficiently close to $p$, $\forall 0\leq j\leq k$ we have $f^j_v(V_v)\cap\rho_v=\emptyset$. Furthermore, recall that by Prop.<ref> for every $i\ne j,i,j\in\{0,...,k-1\}$ we have $f^j_{p,t}(V_p)\cap f^i_{p,t}(V_p)=\emptyset$. Therefore, a similar argument also proves that provided $v\in P$ is sufficiently close to $p$, given any $i,j\in\{0,...,k-1\}$, $i\ne j$, we also have $f^i_v(V_v)\cap f^j_v(V_v)=\emptyset$.

Now, for every $v\in P$ s.t. $f^k_v$ is continuous on $V_v$, set $Fix(v)=\{x\in V_v|f^k_v(x)=x\}$. Similarly to the proof of Lemma <ref>, let $\{(x_n,v_n)\}_n$ be a sequence in $\cup_{v\in P}Fix(v)\times\{v\}$. We now claim that when $v_n\to p$, $\{(x_n,v_n)\}_n$ does not have any accumulation points in $\partial V_p\times\{p\}$. To see why, assume this was not the case, and recall $F_p$ can be smoothly deformed to $F_v$. Now, since $V_p\subseteq \overline{D_\alpha}$, by Remark <ref> we conclude that a similar argument to the one used in the proof of Lemma <ref> proves the existence of $x_p\in\partial V_p$ s.t. $f^k_p(x_p)=x_p$. By Lemma <ref> block, $\partial V_p\subseteq S_0$, with $S_0$ being an annulus in $S$ where $f^k_{p,t}$ has no fixed points. Since $S\subseteq\overline{D_{\alpha,t}}$, by Prop.<ref>, $f^k_p,f^k_{p,t}$ coincide on $S$ - hence $f^k_{p,t}(x_p)=x_p$ as well - this is impossible, and we have a contradiction. Therefore, there exists $O_1\subseteq P$, a neighborhood of $p$, s.t. no sequence in $\cup_{v\in O_1}Fix(v)\times\{v\}$ accumulates in $\partial V_p\times\{p\}$. As a consequence, provided $O_1$ is sufficiently small $\forall v\in O_1$ we have $Fix(v)\cap\partial V_v=\emptyset$, and in particular, $\cup_{v\in O_1}Fix(v)\times\{v\}$ is compact in $\cup_{v\in O_1}V_v\times\{v\}$.

Summarizing our findings, we conclude the existence of a bounded neighborhood of the trefoil parameter $p$ in the parameter space $P$, denoted by $O\subseteq O_1\subseteq P$, s.t. the following conditions hold:

* Given any $v\in\overline{O}$, $x\in V_v$, the points $\{x,f_v(x),...,f^k_v(x)\}$ are all transverse intersection points between the trajectory of $x$ and the cross-section $H_v$.

* $\forall v\in \overline{O}$, $f^k_v:V_v\to \overline{H_v}$ is continuous, and homotopic to $f^k_p:V_p\to\overline{H_p}$.

* $\forall v\in \overline{O}$ and for every $0\leq j\leq k$, $f^j_v(V_v)\cap\rho=\emptyset$.

* $\forall v\in\overline{O}$ and $\forall i\ne j$, $0\leq i,j\leq k-1$, $f^j_v(V_v)\cap f^i_v(V_v)=\emptyset$.

* The set $\cup_{v\in \overline{O}}Fix(v)\times\{v\}$ is compact in $\cup_{v\in \overline{O}}V_v\times\{v\}$.

Therefore, because the fixed-point index of $f^k_p$ on $V_p$ is $1$, by Th.VII.5.8 in <cit.> given any $v\in \overline{O}$ the fixed-point index of $f^k_v:V_v\to\overline{H_v}$ is also $1$ - that is, there exists some $x_v\in V_v$ s.t. $f^k_v(x_v)=x_v$. And because we chose $v$ sufficiently close to $p$ s.t. $f^i_v(V_v)\cap f^j_v(V_v)=\emptyset$ for every $i,j\in\{0,...,k-1\},i\ne j$, the minimal period of $x_v$ is $k$. Additionally, since $f^j_p(V_p)\cap\rho=\emptyset, 0\leq j\leq k$, provided $v\in P$ is sufficiently close to $p$, if $\pi(D_s)=s$, recalling the discussion preceding Th.<ref> we must also have $\pi_v(x_v)=s$. Finally, the arguments above also yield the trajectory of $x_v$ is transverse to $H_v$ at $\{x_v,f_v(x_v),...,f^{k-1}_v(x_v)\}$ - therefore, recalling the definition of $\pi_v$ , given any $\{x_n\}_n\subseteq I_v,x_n\to x_v$, we also have $\pi_v(x_n)\to\pi_v(x_v)=s$. In other words, $\pi_v$ is continuous at $x_v$. The proof of Th.<ref> is now complete.

In <cit.>, the fixed-point index was applied to prove the existence of infinitely many periodic trajectories in the Rössler System (albeit in a non-heteroclinic setting).

In fact, the arguments in Th.<ref> (and in particular, Prop.<ref>) prove more than we originally set out to - namely, they prove every periodic trajectory in $D_s$ can be continuously deformed to a globally continuable trajectory (see <cit.>,<cit.>) - hence, there are also strong restrictions on how the periodic trajectories in $D_s$ can terminate in $P$.

Now, let $p\in P$ be a trefoil parameter. As must be remarked, the proof of Th.<ref> and Th.<ref>, are both independent of the assumption that $\nu_{Out}<1\lor\nu_{In}<1$ (see the discussion in page eq:9) - that is, Th.<ref> and Th.<ref> are both independent of the assumption that at least one of the fixed points for $F_p$ satisfies the Shilnikov condition. Plugging in this assumption, we conclude this subsection with the following theorem:

Let $p\in P$ be a trefoil parameter. Then, $p$ is an accumulation point in the space of vector fields for infinitely many period-doubling and saddle-node bifurcation sets.

For completeness, let us recall sections 2,3 of <cit.>. In that paper, the setting is that of a smooth, two-parameter family of vector fields $\{X_{(a_1,a_2)}\}_{(a_1,a_2)\in O}$, with $(a_1,a_2)$ varying smoothly in some open set $O\subseteq\mathbf{R}^2$. Assume there exists a curve $\gamma\subseteq O$ corresponding to the existence of homoclinic trajectories. Also assume the saddle indices along $\gamma$ are all strictly lesser then $1$ - that is, by Shilnikov's Theorem the dynamics around these homoclinic trajectories all generate infinitely many suspended Smale horseshoes.

As proven in <cit.>, under these assumptions one can construct a function $B$ from the phase space to $O$. The analysis of $B$ allows one to find a family of spirals, $\{\delta_o\}_{o\in\gamma}$, s.t. $\delta_o$ is a curve in $O$, spiralling towards $o$. Each $\delta_o$ corresponds to the existence of a periodic trajectory, which undergoes a cascade of period-doubling and saddle-node bifurcations as $\delta_o$ accumulates on $o$. Moreover, this family of spirals varies continuously when $o\in\gamma$ varies - and provided the saddle index along $\gamma$ is bounded, there exists some $c>0$ s.t. $\forall o\in\gamma$, $diam(\delta_o)>c$ (with that diameter taken w.r.t. the Euclidean metric in $O$). When the function $B$ was analyzed, its critical values corresponded to curves of period-doubling and saddle-node bifurcations - and these critical values accumulated around every point in $\gamma$ (see Sections $2,3$ in <cit.> for more details). That is, for any $\epsilon<0$ and any $o\in\gamma$ there exist $\Gamma_1,\Gamma_2$, a saddle node bifurcation curve and a period doubling bifurcation curve (respectively) s.t. $d(\Gamma_i,o)<\epsilon,i=1,2$.

Now, back to the Rössler System - without any loss of generality, assume $\nu_{In}<1$ (by assumption, we have $\nu_{In}<1\lor\nu_{Out}<1$). Now, let us approximate $F_p$ by a curve $\gamma_{In}$ in the space of $C^\infty$ vector fields on $\mathbf{R}^3$, s.t. $\forall \omega\in\gamma_{In}$, the vector field corresponding to $\omega$ generates a homoclinic curve to $P_{In}$ - moreover, we construct these approximations by perturbing $F_p$ in some small neighborhood of $P_{Out}$, s.t. the local dynamic of $F_p$ around $P_{In}$ are unaffected by this perturbation. In other words, we construct $\gamma_{In}$ s.t. the saddle index $\nu_{In}$ remains constant $\forall \omega\in\gamma_{In}$. As $\nu_{In}<1$ by assumption, the discussion above implies the period-doubling and saddle node bifurcations sets are dense around $\gamma_{In}$ (in the space of $C^\infty$ vector fields on $\mathbf{R}^3$). Since $F_p\in\overline{\gamma_{In}}$, the conclusion follows.

Assume both $\nu_{In},\nu_{Out}<1$. In that case, the proof of Th.<ref> essentially proves the bifurcations around trefoil parameters in the space of $C^\infty$ vector fields on $\mathbf{R}^3$ are doubly as complicated when compared to homoclinic bifurcations.

§.§  Knots and the complexity of chaos at trefoil parameters

Let $p\in P$ be a trefoil parameter. Having studied the persistence of periodic trajectories, we now turn our gaze to studying the complexity of the dynamics generated at trefoil parameters. Namely, in this section we will prove the dynamics generated by the vector field $F_p$ can be reduced to those of a simpler model, capturing all its essential dynamics - we will refer to this model as a topological lower bound for the flow. In more detail, using Th.<ref>, Lemma <ref> and their corollaries, we will prove the flow dynamics of $F_p$ on the periodic trajectories in $Q$ can be described by those of a suspended Horseshoe - in particular, in Th.<ref> we will prove that save perhaps for one knot-type, every knot type in the $L(0,1)$ template is also generated by $F_p$, as some periodic trajectory for the flow. As such, Th.<ref> would give a precise description for the minimal dynamical complexity generated by $F_p$.

To begin, unless stated otherwise, from now on $p\in P$ would always denote a trefoil parameter for the Rössler system. As stated in previous paragraph, or main goal in this subsection is to prove Th.<ref> - that is, our goal is to associate a template with the flow dynamics of $F_p$ (for the definition of a Template, see the discussion in page TEMP1). Due to the Birman-Williams Theorem (see page TEMP2), Template theory, in general, can only be applied when the dynamics on the non-wandering set are hyperbolic (see <cit.>). Therefore, recalling Cor.<ref> provided we can prove $f_p$ is hyperbolic on its invariant set in $\overline{H_{p,t}}\setminus\rho$, we could associate a template with $F_p$. In practice, however, this line of reasoning cannot hold. This is due to the following Lemma:

Let $p$ denote the trefoil parameter, and let $Q$ denote the invariant set from Th.<ref>. Then, $F_p$ does not satisfy any dominated splitting condition on the trajectories of initial conditions in $Q$.

Let us first recall the notion of Dominated Splitting - let $M$ be a Riemannian manifold, $\phi_t:M\rightarrow M$ be a smooth flow. Also, given a hyperbolic point $x\in M$ (w.r.t. $\phi_t$) let us denote the components of the decomposition of the tangent space by $T_x\mathbf{R}^3=E(x)\oplus F(x)$ - with $E(x),F(x)$ denoting the respective stable and unstable parts of the decomposition.  A compact, $\phi_t$-invariant set $\Lambda$ is said to satisfy a dominated splitting condition if the following conditions are satisfied:

* The tangent bundle satisfies $T\Lambda=E\oplus F$, s.t. $E=\cup_{x\in\Lambda}E(x)\times\{x\}$, $F=\cup_{x\in\Lambda}F(x)\times\{x\}$.

* $\exists c>0,0<\lambda<1$, s.t. $\forall t>0,x\in\Lambda$, $(||D\phi_t|_{E(x)}||)(||D\phi_{-t}|_{F(\phi_t(x))}||)<c\lambda^t$.

* $\forall x\in \Lambda,t>0$, $E(x),F(x)$ vary continuously to $E(\phi_t(x)),F(\phi_t(x))$.

Now, let $p$ be a trefoil parameter and let $\Lambda$ be the closure of the bounded, invariant set given by Th.<ref> - by definition, $\Lambda$ is also $F_p$-invariant. By Lemma <ref>, $\Lambda$ is inseparable from $P_{In}$ - as such, since the two dimensional manifolds $W^u_{In},W^s_{Out}$ intersect transversely (see page As4), we conclude $\Lambda$ is also inseparable from $P_{Out}$. Now, recall $P_{In},P_{Out}$ are hyperbolic saddle-foci of opposing indices (see the discussion at page eq:9), connected by a bounded heteroclinic connection $\Theta$ (see Def.<ref>) - with $\Theta$ being a component of both one-dimensional invariant manifolds $W^s_{In},W^u_{Out}$. As such, by previous discussion and the compactness of $\Lambda$, we conclude $\Theta,P_{In},P_{Out}\subseteq\Lambda$. Now, assume we can decompose $T\Lambda$ to $T\Lambda=E\oplus F$ as in the definition above - with previous notations, this implies $F(P_{Out})$ is one-dimensional, while $F(P_{In})$ is two dimensional. Hence, for $x\in\Theta$, $F(x)$ cannot vary continuously, and we have a contradiction. This contradiction proves $F_p$ cannot satisfy any Dominated Splitting condition on $\Lambda$, and Lemma <ref> follows.

Hyperbolic invariant sets all satisfy some Dominated Splitting condition. Therefore, Lemma <ref> teaches us the dynamics around the bounded heteroclinic connection cannot be hyperbolic - in particular, we cannot associate a template with $F_p$ by directly analyzing it. In order to bypass this obstacle, we will first associate a template with $G$, the vector field constructed in Lemma <ref>. As we will prove, this would suffice to prove Th.<ref>. To do so, we first prove the flow $G$ generates is unique, at least up to orbital equivalence. Later on in this section we will see how this fact, combined with Prop.<ref> and Th.<ref> allows us to pull back these results to $F_p$, the original flow. To do so, we must first introduce the following definition:

Let $\phi$ be a flow on a $3$-manifold $M$, with a hyperbolic non-wandering set in $M$, $K\subseteq M$. A plug for $\phi$ is a $3$-manifold $N$ with boundary, satisfying the following properties:

* There exists a flow $\psi$ on $N$, transverse to $\partial N$ and hyperbolic on its invariant set in $N$.

* $\phi,\psi$ are orbitally equivalent on their respective non-wandering sets in $M,N$.

A model is a plug with a minimal total genus.

As far as models are concerned, recall the following theorem, proven in <cit.>:

Given $\phi,M,K$ as above, there exists a model - unique up to orbital equivalence of the flows.

Applying Th.<ref>, we now prove:

The flow generated by $G$ is unique, up to orbital equivalence on its hyperbolic invariant set.

By Lemma <ref>, $G$ generates a smooth flow in $S^3$, hyperbolic on its non-wandering set. Also recall $G$ has precisely two fixed points in $S^3$, $P'_{In}$, a sink, and $P'_{Out}$, a source. Let $B_{In},B_{Out}$ denote open balls centered at $P'_{In},P'_{Out}$ respectively s.t. $G$ is transversal to the respective boundaries of both, $S_{In},S_{Out}$. Also, by the construction of $G$ in Lemma <ref>, we can choose $B_{In},B_{Out}$ s.t. both balls lie in some positive distance from the non-wandering set. This implies $S^3\setminus(B_{In}\cup B_{Out})$ has a total genus $0$, hence, the flow generated by $G$ on $S^3\setminus(B_{In}\cup B_{Out})$ is a model. From Th.<ref>, the flow generated by $G$ is unique up to orbital equivalence in $S^3\setminus(B_{In}\cup B_{Out})$.

(350,270)$\partial B_2$
(860,570)$\partial B_1$

[Fig34]a horseshoe template, $T$ suspended around the trefoil.

Having proven Prop.<ref>, we can now find a template for the periodic trajectories of $G$ in $S^3\setminus(B_{In}\cup B_{Out})$. To do so, let us first suspend a horseshoe map around the heteroclinic trefoil knot. This generates a flow, hyperbolic on its invariant set, with a template $T$ depicted in Fig.<ref> .

[Fig35]$h:S\to S$, a Smale Horseshoe map. $F$ in Fig.<ref> denoting the fold-line of $h$ in $S$, and $B$ the branch line

Let $H$ denote the vector field on $S^3$ described above, whose periodic trajectories all lie on $T$. Assume $\partial B_1$ denotes a sphere on which $H$ points outwards, and $\partial B_2$ another sphere on which $H$ points inwards. Let $B$ be the branch line depicted above. Therefore, by the template $T$, we conclude there exists a cross-section $S$ along with a first-return map for $H$, $h:S\rightarrow S$, which is a Smale horseshoe map, like the one depicted Fig.<ref>. Because $h([a,b]),h([c,d])\subseteq B_2$, $h$ acts on its invariant set in $S\setminus(B_1\cup B_2)$ as a horseshoe map. Now, recall the first-return map generated by $G$, $g:R\rightarrow R$, depicted in Fig.<ref>, with $R$ denoting the rectangle $ABCD$ inside the cross-section.

[Fig36]the horseshoe map $g:R\to R$.

Comparing $h:S\rightarrow S$ with $g:R\rightarrow R$, we see $H,G$ suspend a horseshoe in the exact same way, and more importantly - that both are models for the same flow (see the discussion before Th.<ref>). We therefore conclude:

Every periodic trajectory generated by $G$ is a knot type on the template $T$ (see Fig.<ref>). Conversely, all the knot types in $T$ are realized as periodic trajectories for $G$.

Recalling Lemma <ref>, recall the vector field $G$ was generated by performing a Hopf-bifurcation at both $P_{In},P_{Out}$ - with $G$ pointing outwards on $\partial B_{Out}$ and inwards on $\partial B_{In}$ (see Prop.<ref> and Lemma <ref> for details). Conversely, $H$ points outwards on $\partial B_1$ and inwards on $\partial B_2$. Hence, both $G,H$ are horseshoes suspended the same way around the trefoil on $S^3\setminus(B_{In}\cup B_{Out})$,$S^3\setminus(B_1\cup B_2)$ respectively. In addition, both are hyperbolic on their non-wandering sets - therefore, both are models for the same flow. As such, by Th.<ref> they are orbitally equivalent.

By its definition, $T$ encodes all the knots generated by $H$ (see <cit.> - the Birman-Williams Theorem). Therefore, by previous paragraph the knot type of every periodic trajectory generated by $G$ can be found on $T$. Conversely, every knot in $T$ manifests as a periodic trajectory for $H$ - hence, the same is true for $G$. This concludes the proof.

Now, consider the template $T$ sketched on the left in Fig.<ref> below. By deforming that template isotopically, we derive $L(0,1)$ template (see <cit.> for more details on $L(0,1)$), depicted on the right:

[Fig37]the two templates, $T$ and $L(0,1)$.

In other words, every knot type on $T$ also lies in $L(0,1)$ and vice versa - hence, by Prop.<ref> and the discussion above we immediately conclude $L(0,1)$ encodes all the dynamical complexity generated by $G$. Or, put more precisely:

Every knot type on $L(0,1)$ is generated by $G$ - and conversely, every knot type generated by $G$ also appears in $L(0,1)$.

Having proven the $L(0,1)$ template describes the dynamical complexity generated by $G$, we are ready to prove Th.<ref> - that is, we will now generalize Cor.<ref> to $F_p$, the original vector field. To continue, recall $f_p:\overline{H_p\rm}\rightarrow\overline{H_p\rm}$ is the first return map generated by $F_p$ (wherever it is defined) on the cross-section $H_p$ (see Lemma <ref>). We now prove:

Let $p\in P$ be trefoil parameter. Then, save perhaps for one knot type, $F_p$ generates every knot-type present in the $L(0,1)$ template. In particular, $F_p$ generates infinitely many torus knots.

Much like the proof of Th.<ref>, we will first prove the theorem for the vector field $R_{p,t}$ (see Cor.<ref>) and then pull the result back to $F_p$. Let $t>0$ be sufficiently small s.t. Prop.<ref> holds, and recall $f_{p,t}:\overline{H_{p,t}}\to\overline{H_{p,t}}$ is the first-return map generated by $R_{p,t}$ (see Cor.<ref>). Let $g_p:\overline{H_{p,t}}\to\overline{H_{p,t}}$ denote the first-return map generated by the vector field $G_p$ (see Prop.<ref>)- as proven in Prop.<ref>, there exists a $g_p$-invariant $Q\subseteq \overline{H_{p,t}}$ and a continuous $\xi:Q\to\{1,2\}^\mathbf{N}$ s.t. $\xi\circ g_p=\sigma\circ\xi$, with $\sigma:\{1,2\}^\mathbf{N}\to\{1,2\}^\mathbf{N}$ denoting the one-sided shift. By Lemma <ref>, given a periodic $s\in\{1,2\}^\mathbf{N}$ of minimal period $k>0$, $\xi^{-1}(s)$ contains a periodic point $x_p$ for $g_p$, of minimal period $k$. Let $T_p$ denote the periodic trajectory for $G_p$ s.t. $x_p\in T_p$ - as $G_p$ can be smoothly deformed to $G$, from the proof of Lemma <ref> and Prop.<ref>, the knot-type of $T_p$ is in $L(0,1)$. By Lemma <ref> and Prop.<ref>, the knot type of $T_p$ is independent of the vector field $G_p$.

Now, assume $s$ is not the constant $\{1,1,1...\}$, and let $G_{p,n}$ denote a sequence of smooth vector fields on $S^3$, s.t. $G_{p,n}\to R_{p,t}$ in the $C^\infty$ metric. By previous paragraph, each such vector field generates a periodic trajectory $T_{p,n}$ corresponding to the symbol s - and by the proof of Prop.<ref>, we may well assume $T_{p,n}\to T$, a periodic trajectory for the vector field $R_{p,t}$, s.t. $T\cap Q\ne\emptyset$. Recalling $f_{p,t}:\overline{H_{p,t}}\to\overline{H_{p,t}}$ is the first-return map for $R_{p,t}$, by Lemma <ref>, the minimal period of the orbit $T\cap\overline{H_{p,t}}$ is $k$. Additionally, by Lemma <ref>, $T$ is transverse to $\overline{H_{p,t}}$ throughout $T\cap\overline{H_{p,t}}$. As such, since $T_{p,n}$ all have the same knot type, we conclude $T$ has the same knot type as well. By previous paragraph, the knot type of $T$ is in $L(0,1)$, and by Lemma <ref>, we conclude $T$ intersects $H_{p,t}$ at $\overline{D_{\alpha,t}}$. Since we chose $t>0$ sufficiently small s.t. Prop.<ref> holds, $F_p$ and $R_{p,t}$ coincide around $T$ - which prove $F_p$ generates every knot-type in $L(0,1)$ (save perhaps for one). Finally, since $L(0,1)$ includes infinitely many Torus knots (see Sect.6 in <cit.>), Th.<ref> follows.

From now on, we refer to $G$ (see Lemma <ref>) as the geometric model for $F_p$. As such, Th.<ref> essentially proves the Rössler flow at trefoil parameters (that is, the flow generated by the vector field $F_p$) is a distortion of its geometric model - a horseshoe map, suspended around the trefoil. Moreover, the complexity of that suspension can be encoded as the $L(0,1)$ template. As $L(0,1)$ is a well-studied template, a lot can be said about the knots generated by it (see, for example, <cit.>). All in all, from Th.<ref>, Th.<ref>, Prop.<ref> and Lemma <ref> we conclude:

Every knot type generated by the geometric model (save perhaps for one) is also generated by $F_p$.

As proven in Th.<ref>, the periodic symbolic dynamics persist under sufficiently small perturbations of $F_p$ in the parameter space $P$. As must be stated, the proof Th.<ref> does not teach us whether periodic trajectories persist under perturbations of $F_p$ in $P$ or not - if, however, they do, then as an immediate consequence of Lemma <ref> so do their knot-types. Therefore, by Cor.<ref> and Th.<ref> both motivate us to conjecture the following:

There exists a neighborhood of $p$ in $P$, $O\subseteq P$, s.t. for every $v\in O$ one can also find a topological lower bound for the complexity of $F_v$. That is, given any $v\in O$ and its corresponding vector field $F_v$, there exists an $F_v$-invariant set $I_v$ and a template $T_v$ which describes all the knot types in $I_v$ (save perhaps for one). Moreover, $T_v$ is a sub-template of $L(0,1)$.

If correct, Conjecture 1 implies the topological lower bounds around the trefoil parameter $p$ all depend on the dynamics of $F_p$. As such, if true, Conjecture <ref> should allow us to study the bifurcations around trefoil parameters by reducing them to the study of a smooth family of hyperbolic vector fields.

To conclude this paper, its must also be stated that $L(0,1)$ is not a universal template - that is, it does not contain every knot type. Motivated by this fact (and by the discussion before Cor.<ref>), we conjecture that Th.<ref> can be generalized to other heteroclinic parameters in $P$ (see Def.<ref>) - provided they are complex at least like the trefoil, and intersect $H_p$ transversely. For example, consider the heteroclinic knot in Fig.<ref>. If Th.<ref> can be generalized to such a heteroclinic knot, the template corresponding to the flow would have to include a horseshoe on $3$ symbols. Therefore, in this spirit, we make the following conjecture:

Let $p\in P$ be a heteroclinic parameter which generates a heteroclinic knot, $K$ (see Def.<ref>). Then, under the assumptions in pages eq:9 and As4, the more $K$ is linked with itself, the more knot types are in the template associated with $F_p$ - in other words, the more complex $K$ is, the more complex are the dynamics of $F_p$.

If true, both Conjecture <ref> and Conjecture <ref> provide an analytic explanation to several well-known numerical observations. The first is that as the parameters $(a,b,c)$ are varied, the number of symbols in the first-return map of the Rössler attractor increases - i.e., the symbolic dynamics change from two symbols into three, four, and so on (for example, see <cit.>,<cit.>,<cit.>). The second observation is that the same is true for the associated templates - whose topology also change with the variation of $(a,b,c)$, thus forcing the creation of more symbols in the first return map of the attractor (see <cit.>,<cit.>).

[Fig39]a heteroclinic knot more complex than a trefoil.

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New Jersey: World Scientific

titleTopology from the Differentiable viewpoint

titleFlots de smale en dimension 3: pr´esentations finies de voisinages invariants d’ensembles selles

journaltitleChaos, Solitons, and Fractals
titleA new proof for existence of horseshoe in the Rössler system

titleThree-Dimensional Flows

journaltitleInternational Journal of Bifurcation and Chaos
titleThe Structure of Infinite Periodic and Chaotic Hub Cascades in Phase Diagrams of Simple Autonomous Flows

journaltitleJ. Phys. A: Math. Theor.
titleGlobal dynamics of the Rössler system with conserved quantities

journaltitlePhys. Rev. Lett.
titleTopological Changes in Periodicity Hubs of Dissipative Systems
volume108, 214102

World Scientific

chapter33 - Symbolic Dynamics and Spiral Structures due to the Saddle Focus Bifurcations
titleChaos, CNN, Memristors and beyond – a festschrift for Leon Chua

titleUnbounded dynamics in dissipative flows: Rössler model

journaltitleJournal of Physics A: Mathematical and Theoretical, IOP Publishing
titleTemplates and subtemplates of Rössler attractors from a bifurcation diagram

titlePeriodic solutions and invariant torus in the Rössler system
volume33 4512

titleHomoclinic chaos in the Rössler model

titleAround Smale's 14th Problem