# Topological lower bounds for the Rössler System

Eran Igra

###### Abstract.

The Rössler System is one of the best known chaotic dynamical systems,
exhibiting a plethora of complex phenomena - and yet, only a few studies
tackled its complexity analytically. Building on previous work by the author,
in this paper we characterize the dynamical complexity for the Rössler System
at parameter values at which the flow satisfies a certain heteroclinic
condition. This will allow us to characterize the knot type of infinitely many
periodic trajectories for the flow - and reduce the Rössler system to a
simpler hyperbolic flow, capturing its essential dynamics.

###### Key words and phrases:

Keywords \- The Rössler Attractor, Heteroclinic bifurcations, Topological
Dynamics, Template Theory

##  1\. Introduction

In 1976, Otto E. Rössler introduced the following system of Ordinary
Differential Equations, depending on parameters $A,B,C\in\mathbf{R}^{3}$ [4]:

$\begin{cases}\dot{X}=-Y-Z\\\ \dot{Y}=X+AY\\\ \dot{Z}=B+Z(X-C)\end{cases}$ (1)

Inspired by the Lorenz attractor (see [1]), Otto E. Rössler attempted to find
the simplest non-linear flow exhibiting chaotic dynamics. This is realized by
the vector field above, as it has precisely one non-linearity, $XZ$ in the
$\dot{Z}$ component - and as observed by Rössler, the vector field appears to
generate a chaotic attractor for $(A,B,C)=(0.2,0.2,5.7)$. In more detail, at
these parameter values Rössler observed the first return map of the flow has
the shape of a horseshoe (i.e., numerically), which is known to be chaotic
(see [3]).

\begin{overpic}[width=173.44534pt]{images/attractor.jpg} \end{overpic} Figure
1. The Rössler attractor at $(A,B,C)=(0.2,0.2,5.7)$

Since its introduction in 1976, the Rössler system was the focus of many
numerical studies - despite the simplicity of the vector field, the flow gives
rise to many non-linear phenomena (see, for example: [27], [21], [19], [23],
[22]). One particular feature is that varying the parameters $A,B,C$ often
leads to a change in the topological complexity of the periodic trajectories
on the attractor - see, for example, [10],[25]. In more detail, it was
observed that varying the $A,B,C$ parameters changes the Template
corresponding to the periodic trajectories on the attractor - namely, more
twists and half twists are added (for the definition of a template, see
Def.1.3 bellow).

However, in contrast to the vast corpus of numerical studies, analytical
results on the Rössler system are sparse. For example, in [13], the existence
of periodic trajectories at some parameters was established; in [26] the
existence of an invariant Torus (and its breakdown) at some parameters was
proven; and in [20] the dynamics of the flow at $\infty$ were analyzed. As far
as chaotic dynamics go, their existence at $(A,B,C)=(0.2,0.2.5.7)$ was proven
with rigorous numerical methods - see both [12],[17]. Additionally, in a
recent paper, the author proved analytically the existence of a certain
criterion that, when satisfied, implies the existence of complex dynamics for
the Rössler system. In more detail, it was proven that provided the parameters
$(A,B,C)$ are such that the vector field generates a very specific
heteroclinic knot $T$ in $S^{3}$, the topology of $S^{3}\setminus T$ forces
the flow to behave essentially like a suspended Smale Horseshoe map. As such,
it follows the corresponding vector field generates infinitely many periodic
trajectories in $S^{3}\setminus T$ (see Th.3.1 in [29], or the discussion at
the next section).

The connection between the topology of a given surface $S$ and the possible
discrete-time dynamics it forces on a homeomorphism $f:S\to S$ is a well-known
theme in two-dimensional topological dynamics (see [8] for a survey). In
particular, provided $S$ is of negative Euler characteristic, by the Thurston-
Nielsen Classification Theorem (see, for example, [5]) one can find a
dynamically-minimal homeomorphism $g:S\to S$ s.t. $f$ is both isotopic to $g$,
and $g$ is a factor map for $f$ (see, for example, [9] and [8]).
Heuristically, since first-return maps for flows are two dimensional dynamical
systems, one would expect similar ideas to hold for three-dimensional flows -
in practice, however, the problem of generalizing these results to three-
dimensional flows is far from solved (see [15] for a survey of this problem,
and [18] for a possible physical interpretation). It is precisely this problem
that our results address - motivated by the results of both [29] and [28], we
prove that for the Rössler system at the heteroclinic parameters described
above one can reduce the flow to a simpler model - as well as classify its
dynamical complexity using Template Theory (see [11], or Th.6 in the next
section). As these results are proven using mostly topological methods, they
can be considered as a steps toward generalizing the notion of a dynamically
minimal map to three dimensional flows.

To introduce our results, let us first make one definition - roughly speaking,
we say a parameter $(A,B,C)\in P$ is a trefoil parameter provided the
corresponding Rössler system generates a heteroclinic knot whose knot-type is
the trefoil knot (see Def.1.2). The main result of this paper is the following
theorem, which we prove in Section $2$ (see Th.3.1):

######  Theorem 1.

Let $(A,B,C)=p$ be a trefoil parameter for the Rössler system, and let $F_{p}$
denote the corresponding vector field for the Rössler system. Then, there
exists a smooth vector field $G$, hyperbolic on its non-wandering set, s.t.
the following holds:

  * •

The dynamics of $F_{p}$ can be smoothly deformed in $\mathbf{R}^{3}$ to those
of $G$.

  * •

Given any periodic trajectory $T_{1}$ for $G$ (save perhaps for one), there
exists an isotopy of $\mathbf{R}^{3}$ which deforms $T_{1}$ to $T_{0}$, a
periodic trajectory for $F_{p}$. Consequentially, every knot type generated by
$G$ as a periodic trajectory is also generated by $F_{p}$ as a periodic
trajectory (save perhaps for one).

That is, Th.1 essentially tells us the vector field $G$ is an analogue for the
dynamically minimal maps from two-dimensional dynamics in the following sense:
even though the dynamics of $F_{p}$ are possibly more complicated from those
of $G$, the dynamics of $G$ form a lower bound, in a topological sense, to the
possible dynamical complexity of $F_{p}$. As such, the vector field $G$ can be
considered as a Topological Lower Bound for the dynamical complexity of
$F_{p}$ \- we often refer to it throughout this paper as a dynamically minimal
flow. The proof of Th.1, as we will discover, in Section $2$, is mostly
constructive - in particular, it is based on smoothly deforming the vector
field $F_{p}$ to $G$ by analyzing its first-return map and blowing up the
heteroclinic connections generated by $R_{p}$. Therefore, in Section $3$ we
apply Th.1 to classify the knot types generated as periodic trajectories for
$G$ \- thus, in essence, classifying the minimal dynamical complexity
generated by $F_{p}$. Namely, using Template Theory and in particular, the
$L(0,1)$ Template (see [11], [7] or the discussion at the next Section), we
prove the following Theorem (see Th.3.1 and Prop.3.1):

######  Theorem 2.

Let $p$ be a trefoil parameter for the Rössler system, and let $G,F_{p}$ be as
in Th.1. Then, the following holds:

  * •

The vector field $G$ is unique up to orbital equivalence.

  * •

Let $T$ be a knot-type realized by $G$ as a periodic trajectory - then, $T$ is
a knot-type on the $L(0,1)$ Template. Conversely, any knot-type on the
$L(0,1)$ Template is realized as a periodic trajectory for $G$.

  * •

Consequentially, every knot type on the $L(0,1)$ Template, save perhaps for
one, is also realized as a periodic trajectory for $F_{p}$. Moreover, $F_{p}$
generates infinitely many Torus Knots.

As must be remarked, there had been several numerical studies on the knot
types in the Rössler system - as such, the analytical results of Th.2 should
be compared with the numerical results of [10] and [25], where the topology of
the periodic trajectories on the attractor was studied with the aid of
Template Theory. Additionally, it should also be compared with the results of
[24], where the topology of the numerically observed knot-types on the Rössler
attractor were studied analytically.

## Preliminaries

Let $X$ be a metric space, with a metric $d$. For any $A\subseteq X$, we will
denote by $\overline{A}$ the closure of $A$, and by $\partial A$ the boundary
of $A$. Given any $x\in X,r>0$, we denote the ball of radius $r$ around $x$ by
$B_{r}(x)$. Also, given $A,B,C,D\subseteq X$, we say that $C$ connects $A,B$
if $C$ is connected and $A\cap C$,$B\cap C\neq\emptyset$. We say that $D$
separates $A,B$ if given any connected $C$ which connects $A,B$, we also have
$D\cap C\neq\emptyset$ \- see the illustration in Fig.2. Moreover, we say a
curve $\gamma:[0,1]\to\mathbf{R}^{3}$ is simple or not self intersecting if
for every $x,y\in[0,1]$, $x\neq y$, $\gamma(x)\neq\gamma(y)$. Given a
topological disc $D\subseteq\mathbf{R}^{2}$, we say $D$ is a Jordan domain if
its boundary $\partial D$ can be parameterized by some $\gamma[0,1]$ s.t.
$\gamma(0)=\gamma(1)$ and for every other $x,y\in(0,1)$,
$\gamma(x)\neq\gamma(y)$.

\begin{overpic}[width=130.08731pt]{images/sepa.png} \put(-40.0,300.0){$D$}
\put(670.0,390.0){$C$} \put(400.0,230.0){$A$} \put(690.0,590.0){$B$}
\end{overpic} Figure 2. The set $C$ connects $A$ and $B$, while $D$ separates
them.

From now on given $(a,b,c)\in\mathbf{R}^{3}$, we switch to the more convenient
form of the Rössler system, given by the following system of ODEs:

$\begin{cases}\dot{x}=-y-z\\\ \dot{y}=x+ay\\\ \dot{z}=bx+z(x-c)\end{cases}$
(2)

Denote this vector field corresponding to $(a,b,c)\in\mathbf{R}^{3}$ by
$F_{a,b,c}$. This definition is slightly different from the one presented in
Eq.1 \- however, setting $p_{1}=\frac{-C+\sqrt{C^{2}-4AB}}{2A}$, it is easy to
see that whenever $C^{2}-4AB>0$, $(X,Y,Z)=(x-ap_{1},y+p_{1},z-p_{1})$ defines
a change of coordinates between the vector fields in Eq.1 and Eq.2.

Since the vector field in Eq.2 depends on three parameters, $(a,b,c)$, to
continue we must specify the region in the parameter space in which we prove
our results. As observed in several numerical studies (see, for example,
[21],[27],[19] and [22]), many interesting bifurcation phenomena occur at a
very specific open region in the parameter space $\mathbf{R}^{3}$. In that
parameter space the $a,b,c$ parameters always satisfy $a,b\in(0,1)$, $c>1$ \-
and moreover, the vector field $F_{p}$ always generates precisely two fixed
points in $\mathbf{R}^{3}$, $P_{In},P_{Out}$, both are saddle-foci (of
opposing indices). Additionally, in these studies the eigenvalues of the
linearization at $P_{In}$ satisfy a resonance condition known as the Shilnikov
condition (see [2] or below for a definition), which is stable in the
parameter space. Therefore, to make everything clear, let us write down the
assumptions we impose on our parameter space $P$ in which we will prove our
results:

  * •

Assumption $1$ \- $\forall p\in P,p=(a,b,c)$ the parameters satisfy
$a,b\in(0,1)$ and $c>1$. As can be seen, for every choice of such $p$, the
vector field $F_{p}$ generates precisely two fixed points - $P_{In}=(0,0,0)$
and $P_{Out}=(c-ab,b-\frac{c}{a},\frac{c}{a}-b)$.

  * •

Assumption 2 \- We assume that for every $p\in P$ the fixed points
$P_{In},P_{Out}$ are both saddle-foci of opposing indices. In more detail, we
always assume that $P_{In}$ has a one-dimensional stable manifold,
$W^{s}_{In}$, and a two-dimensional unstable manifold, $W^{u}_{In}$.
Conversely, we assume $P_{Out}$ has a one-dimensional unstable manifold,
$W^{u}_{Out}$, and a two-dimensional stable manifold, $W^{s}_{Out}$.

  * •

Assumption 3 \- For every $p\in P$, let $\gamma_{In}<0$ and $\rho_{In}\pm
i\omega_{In}$, $\rho_{In}>0$ denote the eigenvalues of $J_{p}(P_{In})$, the
linearization of $P_{In}$, and set $\nu_{In}=|\frac{\rho_{In}}{\gamma_{In}}|$.
Conversely, let $\gamma_{Out}>0$, $\rho_{Out}\pm i\omega_{Out}$ s.t.
$\rho_{Out}<0$ denote the eigenvalues of $J_{p}(P_{Out})$ and define
$\nu_{Out}=|\frac{\rho_{Out}}{\gamma_{Out}}|$. We will refer to
$\nu_{In},\nu_{Out}$ as the respective saddle indices at $P_{In},P_{Out}$, and
we will always assume $(\nu_{In}<1)\lor(\nu_{Out}<1)$ \- that is, for every
$p\in P$ at least one of the fixed points satisfies the Shilnikov condition
(see [2] for more details).

As must be remarked, the parameter space $P$ we are considering is not only
open in the parameter space, but in fact but also includes the region
considered in numerical studies. Additionally, given any parameter $p\in P$ we
have the following result, proven in [29] (see Th.2.1 in [29]):

######  Theorem 3.

For every $p\in P$, $F_{p}$ satisfies the following:

  * •

$F_{p}$ can be extended to a continuous vector field on $S^{3}$, smooth in
$\mathbf{R}^{3}$, s.t. it has precisely three fixed points in $S^{3}$: the
saddle-foci $P_{In},P_{Out}$ \- and a degenerate fixed point at $\infty$ of
index $0$.

  * •

There exists an unbounded separatrix $\Gamma_{In}\subseteq W^{s}_{In}$, s.t.
$\overline{\Gamma_{In}}$ connects $P_{In},\infty$ (in $S^{3}$).

  * •

There exists an unbounded separatrix $\Gamma_{Out}\subseteq W^{u}_{Out}$, s.t.
$\overline{\Gamma_{Out}}$ connects $P_{Out},\infty$ (in $S^{3}$).

  * •

For every sufficiently large $r$ there exists a smooth vector field on
$S^{3}$, $R_{p}$, s.t. $F_{p},R_{p}$ coincide on the ball $B_{r}(P_{In})$.
Moreover, $R_{p}$ has precisely two fixed points in $\mathbf{R}^{3}$ \-
$P_{In},P_{Out}$, connected by an unbounded heteroclinic connection $\Gamma$
(in $S^{3}$), connecting $P_{In},P_{Out}$.

For an illustration, see Fig.4 and 11. We will also need several other results
from [29] \- to recall these results, begin by considering the cross-section
$Y=\\{\dot{y}=0\\}=\\{(x,-\frac{x}{a},z)|x,z\in\mathbf{R}\\}$, and set
$l_{p}=\\{(x,-\frac{x}{a},\frac{x}{a})|x\in\mathbf{R}\\}$. Because the normal
vector to $Y$ is $N=(1,a,0)$, it follows by direct computation that
$l_{p}=\\{v\in Y|F_{p}(v)\bullet N=0\\}$, and $P_{In},P_{Out}\in l_{p}$. In
particular, it follows $Y\setminus l_{p}$ constitutes of two components, both
half planes, parameterized as follows:

  * •

$U_{p}=\\{(x,-\frac{x}{a},z)|x\in\mathbf{R},\frac{x}{a}<z\\}$ \- that is, the
upper half plane (see the illustration in Fig.3). On $U_{p}$ the vector field
$F_{p}$ points into the half-space $\\{\dot{y}<0\\}$.

  * •

$L_{p}=\\{(x,-\frac{x}{a},z)|x\in\mathbf{R},\frac{x}{a}>z\\}$ \- that is, the
lower half plane (see the illustration in Fig.3). On $U_{p}$ the vector field
$F_{p}$ points into the half-space $\\{\dot{y}>0\\}$.

In particular, both $U_{p},L_{p}$ vary smoothly in $p$. As proven in both
Cor.2.1.1 and Lemma 2.1 in [29], we have:

######  Corollary 1.0.1.

For any $p\in P$, the cross-section $U_{p}$ satisfies the following:

  * •

The two-dimensional $W^{u}_{In},W^{s}_{Out}$ are transverse to $U_{p}$ at
$P_{In},P_{Out}$ (respectively).

  * •

Given any $s\in\mathbf{R}^{3}$, denote its trajectory by $\gamma_{s}(t)$,
$t\in\mathbf{R},\gamma_{s}(0)=0$. Provided $\Gamma(s)=\\{\gamma_{s}(t)|t>0\\}$
is bounded and $\Gamma(s)$ does not limit to a fixed point, there exists some
$t_{0}>0$ s.t. $\gamma_{s}(t_{0})$ is a transverse intersection point between
$U_{p},\Gamma(s)$.

\begin{overpic}[width=238.49231pt]{images/cross.png}\put(630.0,280.0){$W^{s}_{Out}$}
\put(750.0,430.0){$W^{u}_{Out}$} \put(690.0,390.0){$P_{Out}$}
\put(390.0,130.0){$W^{u}_{In}$} \put(410.0,220.0){$P_{In}$}
\put(440.0,420.0){$U_{p}$} \put(570.0,180.0){$L_{p}$}
\put(250.0,110.0){$W^{s}_{In}$} \end{overpic} Figure 3. The local dynamics
around the fixed points. The green and red flow lines are the one-dimensional
separatrices in $W^{s}_{In},W^{u}_{Out}$.

As stated in the introduction, in [29] the author had proven a criterion for
the existence of complex dynamics for the Rössler system. Since the proofs of
both Th.3.1 and 2.1 are heavily based on that criterion, let us introduce it.
To do so, let us begin with the following definition:

######  Definition 1.1.

Let $p\in P$ be a parameter s.t. there exists a bounded heteroclinic
trajectory for $F_{p}$ in $W^{s}_{In}\cap W^{u}_{Out}$, connecting
$P_{In},P_{Out}$. In that case we say $p$ is a heteroclinic parameter \- see
the illustration at Fig.4.

\begin{overpic}[width=216.81pt]{images/trefoil2.jpg}
\put(440.0,210.0){$P_{In}$} \put(340.0,70.0){$\Gamma_{In}$}
\put(400.0,350.0){$\Theta$} \put(660.0,370.0){$P_{Out}$}
\put(780.0,430.0){$\Gamma_{Out}$} \end{overpic} Figure 4. A heteroclinic
trefoil knot (see Def.1.2). $\Theta$ denotes the bounded heteroclinic
connection, while $\Gamma_{In},\Gamma_{Out}$ denote the unbounded ones given
by Th.3.

Heteroclinic parameters can come in many forms - therefore, we can
differentiate between different heteroclinic parameters as follows - let $p\in
P$ be a heteroclinic parameter, and consider the set $\Lambda=W^{s}_{In}\cup
W^{u}_{Out}\cup\\{P_{In},P_{Out},\infty\\}$. By Th.3, it immediately follows
$\Lambda$ is a knot in $\mathbf{R}^{3}$. Motivated by this observation, from
now on we consider a very specific type of heteroclinic parameters, trefoil
parameters, defined below:

######  Definition 1.2.

Let $p\in P$ be a heteroclinic parameter and $\Lambda$ be as above. Let
$\Theta$ denote the bounded heteroclinic connection - that is, the bounded
component of $\Lambda\setminus\\{P_{Out},P_{In}\\}$. We say $p$ is a trefoil
parameter provided the following three conditions are satisfied:

  * •

$\Theta$ is as in Fig.4.

  * •

We have $W^{u}_{In}=W^{s}_{Out}$ (w.r.t. $F_{p}$) - that is, the two-
dimensional manifolds coincide for $F_{p}$.

  * •

$\Theta\cap\overline{U_{p}}=\\{P_{0}\\}$ is a point of transverse intersection
(i.e., $P_{0}$ is interior to $U_{p}$).

See Fig.4 and Fig.5 for illustrations.

######  Remark 1.1.

It is not immediate from the definition above that given a trefoil parameter
$p\in P$, $\Lambda$ indeed forms a trefoil knot. However, as proven in
Cor.2.1.5 in [29], this is indeed the case.

\begin{overpic}[width=108.405pt]{images/INTER11.png} \put(80.0,70.0){$P_{In}$}
\put(395.0,70.0){ $L$} \put(410.0,500.0){$P_{0}$}
\put(210.0,300.0){$D_{\alpha}$} \put(-30.0,400.0){$l$}
\put(330.0,750.0){$\delta$} \put(390.0,970.0){$\delta_{0}$}
\put(830.0,780.0){$P_{Out}$} \end{overpic} Figure 5. The geography of the
cross-section $U_{p}$ for a trefoil parameter $p\in P$, sketched as a disc for
simplicity. The green arc denotes
$L=W^{u}_{In}\cap\overline{U_{p}}=W^{s}_{Out}\cap\overline{U_{p}}=W\cap\overline{U_{p}}$.
The set $D_{\alpha}$ corresponds to $B_{\alpha}\cap\overline{U_{p}}$.

As must be remarked, the existence of parameters $p\in P$ at which $\Lambda$
forms a trefoil knot was observed numerically (see Fig.$5.B1$ in [27]).
Moreover, in the same study it was numerically observed that in many
parameters the two-dimensional $W^{s}_{Out}$ is inseparable from the fixed-
point $P_{In}$. Therefore, inspired by the numerics, we consider trefoil
parameters as an idealized form of the dynamics observed in [27] (for more
details and references, see the discussion at the end of Section $2$ in [29]).
The reason we are interested in trefoil parameters is, as stated earlier,
their existence is a criterion for the existence of complex dynamics in $P$.

\begin{overpic}[width=108.405pt]{images/INTER12.png} \put(80.0,70.0){$P_{In}$}
\put(395.0,70.0){ $L$} \put(410.0,500.0){$P_{0}$} \put(210.0,300.0){$H_{p}$}
\put(-60.0,400.0){$l_{1}$} \put(180.0,750.0){$f^{-1}_{p}(l)$}
\put(390.0,970.0){$\delta_{0}$} \put(830.0,780.0){$P_{Out}$} \end{overpic}
Figure 6. The cross-section $H_{p}\subseteq D_{\alpha}\subseteq U_{p}$ for a
trefoil parameter $p\in P$ \- again, sketched as a disc for simplicity. The
green arc denotes
$L=W^{u}_{In}\cap\overline{U_{p}}=W^{s}_{Out}\cap\overline{U_{p}}=W\cap\overline{U_{p}}$,
while the red curve is $f^{-1}_{p}(l)$. $l_{1}$ is the sub-arc of $l$
connecting $P_{In},P_{Out}$.

To state these results, we must first recall several facts - to do so, let
$p\in P$ be a trefoil parameter, and consider the two dimensional
$W=W^{s}_{Out}=W^{u}_{In}$ \- it is immediate that given a trefoil parameter
$p\in P$, there exists a three-dimensional bounded topological ball,
$B_{\alpha}$, satisfying $\partial B_{\alpha}=\\{P_{In},P_{Out}\\}\cup W$ (see
the illustration in Fig.7). In Lemma 3.2, Cor.3.0.1, Prop.3.1 and Prop.3.2 in
[29] we have proven the following catalogue of results:

######  Theorem 4.

Let $p\in P$ be a trefoil parameter, and let $B_{\alpha},l_{p},U_{p}$ be as
above (w.r.t. the trefoil parameter $p$). Additionally, set
$l=\\{l_{p}(x)|x\in(0,c-ab)\\}$, $L=W\cap U_{p}$, $D_{\alpha}=B_{\alpha}\cap
U_{p}$, and let $P_{0}$ be as in Def.1.2. Then, we have the following:

  * •

$L\cap l_{p}=\emptyset$.

  * •

$D_{\alpha}$ is a Jordan domain on $U_{p}$, bounded by $\partial
D_{\alpha}=L\cup l\cup\\{P_{In},P_{Out}\\}$ (see the illustration in Fig.5).

  * •

$P_{0}$ is strictly interior to $D_{\alpha}$.

  * •

The first-return map
$f_{p}:\overline{D_{\alpha}}\setminus\\{P_{0}\\}\to\overline{D_{\alpha}}\setminus\\{P_{0}\\}$
is well-defined.

  * •

The discontinuities of $f_{p}$ in $D_{\alpha}\setminus\\{P_{0}\\}$ are given
by $f^{-1}_{p}(l)$. Moreover, there exists a curve $\delta\subseteq
D_{\alpha}\setminus\\{P_{0}\\}$ connecting $P_{0}$ and some unique
$\delta_{0}\in l$ (see the illustration in Fig.5).

\begin{overpic}[width=130.08731pt]{images/boundedtref.png}
\put(460.0,240.0){$\Theta$} \put(260.0,130.0){$P_{In}$}
\put(160.0,50.0){$\Gamma_{In}$} \put(400.0,480.0){$B_{\alpha}$}
\put(660.0,430.0){$P_{Out}$} \put(860.0,370.0){$\Gamma_{Out}$} \end{overpic}
Figure 7. The separatrix $\Theta$ trapped inside $B_{\alpha}$, sketched as a
the blue sphere.

Using the results listed in Th.4 (among others), it was proven in [29] by the
author that the set $f^{-1}_{p}(l)$ forms a curve in $\overline{D_{\alpha}}$,
connecting $P_{0}$ with $P_{Out}$ (see the illustration in Fig.6). Now, denote
by $l_{1}$ the sub-arc of $l$ connecting $P_{In},\delta_{0}$ (see the
illustration in Fig.6), and let $H_{p}$ denote the (open) topological disc in
$D_{\alpha}$ bounded by $l_{1},f^{-1}_{p}(l)$, and $L$ (see the illustration
in Fig.6). We are finally ready to state the following results proven in [29],
where the connection between trefoil parameters and complex dynamics for the
Rössler system was proven:

######  Theorem 5.

Let $p\in P$ be a trefoil parameter, and denote by
$\sigma:\\{1,2\\}^{\mathbf{N}}\to\\{1,2\\}^{\mathbf{N}}$ the one-sided shift.
Then, there exists an $f_{p}$-invariant set $Q\subseteq\overline{H_{p}}$ s.t.
the following is satisfied:

  * •

$f_{p}$ is continuous on $\overline{H_{p}}$. Additionally, for every $n>0$,
$f^{n}_{p}$ is continuous on $Q$.

  * •

There exists a continuous $\pi:Q\to\\{1,2\\}^{\mathbf{N}}$ s.t. $\pi\circ
f_{p}=\sigma\circ\pi$.

  * •

Given any periodic $s\in\\{1,2\\}^{\mathbf{N}}$ of minimal period $k$ s.t. $s$
is not the constant $\\{1,1,1...\\}$, $\pi^{-1}(s)=D_{s}$ is nonempty,
connected, and connects the components of $\partial
H_{p}\setminus(L\cup\delta)$ (with $\delta_{0},L$ as in Th.4).

  * •

Given any periodic $s\in\\{1,2\\}^{\mathbf{N}}$ of minimal period $k$ s.t. $s$
is not the constant $\\{1,1,1...\\}$, $D_{s}$ includes $x_{s}$, a periodic
point for $f_{p}$ of minimal period $k$.

  * •

The periodic trajectories in $Q$ lie uniformly away from $P_{In},P_{Out}$ (see
Def.1.2) - that is, there exists an open neighborhood
$N\subseteq\mathbf{R}^{3}$ of $P_{In},P_{Out}$ s.t. given any periodic
$s\in\\{1,2\\}^{\mathbf{N}}$ which is not the constant $\\{1,1,1...\\}$,
$D_{s}\cap\overline{N}=\emptyset$.

For a proof, see Th.3.1 in [29]. In other words, Th.5 implies the dynamics of
the vector field $F_{p}$ \- that is, the dynamics of the Rössler system at
trefoil parameters - are complex at least like those of a Smale Horseshoe,
suspended around the heteroclinic trefoil knot. However, unlike Smale
Horseshoe maps, by Prop.4.1 in [29] it follows the dynamics of $F_{p}$ are
probably not hyperbolic.

\begin{overpic}[width=108.405pt]{images/INTER13.png} \put(80.0,70.0){$P_{In}$}
\put(395.0,70.0){ $L$} \put(410.0,520.0){$P_{0}$}
\put(110.0,350.0){$f^{2}_{p}(\delta)$} \put(-200.0,400.0){$f_{p}(\delta)$}
\put(150.0,750.0){$f_{p}(H_{p})$} \put(-50.0,700.0){$l$}
\put(390.0,970.0){$\delta_{0}$} \put(830.0,780.0){$P_{Out}$} \end{overpic}
Figure 8. The image of $f_{p}(H_{p})$ in $D_{\alpha}$, represented by the
banana-shaped shaded region (for a proof, see Stages $II$ and $III$ in the
proof of Th.3.1 in [29]). $H_{p}$ is the simply connected domain bounded by
$l$, the green curve $L$ and the red line $f^{-1}_{p}(l)$ ($\delta$ is the red
sub-curve of $f^{-1}_{p}(l)$ connecting $P_{0},\delta_{0}$). As can be seen,
$P_{0}$ lies away from $f_{p}(H_{p})$ \- hence, so does $Q$.

In addition to the results from [29] listed above, to prove Th.3.1, we will
also need to several notions from Knot Theory - in particular, we will need
the notion of a Template. Therefore, we conclude this section with a brief
introduction to Template Theory. Following [11], we define a template as
follows:

######  Definition 1.3.

A Template is a compact branched surface with a boundary endowed with a smooth
expansive semiflow, built locally from two types of charts - joining and
splitting (see the illustrations in Fig.9 and Fig.10). Additionally, the
gluing maps between charts all respect the semiflow, and act linearily on the
edges (see Fig.9 and Fig.10 for illustrations).

\begin{overpic}[width=130.08731pt]{images/TEMP.png} \end{overpic} Figure 9. A
joining chart on the left, and a splitting chart on the right.

The connection between Templates and three-dimensional Flows is given by the
Birman-Williams Theorem (see [6]). Let $M$ be a three-manifold, and let
$\phi_{t}$ be flow with a hyperbolic chain recurrent set $\Omega\subseteq M$.
Moreover, let $P(\Omega)$ denote the periodic trajectories for $\phi_{t}$ in
$\Omega$. Then, under these assumptions, we have the following result:

######  Theorem 6.

The Birman-Williams Theorem \- Given $M,\phi_{t},\Omega$ as above, there
exists a unique Template $T$, embedded in $M$ and endowed with a smooth
semiflow $\psi_{t}$. Moreover, letting $P(T)$ denote the periodic trajectories
of $\psi_{t}$ in $T$, there exists an injective map $f:P(\Omega)\to P(T)$ s.t.
$P(T)\setminus f(P(\Omega))$ contains at most two periodic trajectories for
$\psi_{t}$.

For a proof, see either [6] or Th.2.2.4 in [11]. The Birman-Williams Theorem
essentially states that given $M,\phi_{t},\Omega$ as above, we can
characterize the knot types in $\Omega$ (that is, $P(\Omega)$) by studying the
periodic trajectories on the Template $T$ (for more information, see [11] and
the references therein). In particular, the Template $T$ encodes all the knot-
types which exist in $P(\Omega)$ \- see Fig.10 for an illustration and [11]
for more details. As such, the Birman-Williams Theorem will play a major role
in the proof of Th.3.1.

\begin{overpic}[width=86.72267pt]{images/LORTEMP.png} \end{overpic} Figure 10.
A Lorenz template (see Ch.2 in [11]).

##  2\. Constructing a dynamically minimal flow

As stated at the Introduction, in this section we prove Th.2.1 \- namely, in
this section we prove the dynamics of the Rössler system at trefoil parameters
can be reduced to those of a hyperbolic flow. In more detail, we prove that
given any trefoil parameter $p\in P$, the vector field $F_{p}$ can be smoothly
deformed in $\mathbf{R}^{3}$ to $G$, a hyperbolic vector field on
$\mathbf{R}^{3}$, s.t. the periodic trajectories given by Th.5 are all
(isotopically) deformed to periodic trajectories of $G$. As will be made clear
by the proof, the dynamics of $G$ are simpler then the original dynamics of
$F_{p}$ \- thus, it truly is an analogue for the dynamically minimal maps
mentioned at the Introduction.

Due to the length of the argument and the amount of details involved, this
section will be devoted in its entirety to the proof of Th.2.1 \- as such, the
proof of its corollaries (and in particular, Th.3.1) will be deferred to the
next section. Despite the lengthy argument, the main idea of the proof is
quite simple. The proof is based on the following idea: since the first-return
map of the Rössler system at trefoil parameters is essentially a distorted
horseshoe (see Th.5), by carefully blowing up the fixed points and contracting
the components of the set $Q$ (see Th.5) we smoothly deform the vector field
$F_{p}$ to $G$ \- a vector field whose first-return map is a Smale Horseshoe
map (see [3]). As will be made evident, these deformations do not create any
new periodic dynamics (save for two extra periodic trajectories - Step $III$
of the proof) - nor do they destroy any periodic dynamics in $Q$. As such, all
the periodic (and a-periodic) dynamics for $F_{p}$ persist when we deform it
to $G$, thus completing the proof of Th.2.1. Therefore, we now prove:

######  Theorem 2.1.

Let $p\in P$ be a trefoil parameter. Then, there exists a smooth curve of
$C^{\infty}$ vector fields $\Gamma(t),t\in[0,1]$ s.t. $\Gamma(0)=F_{p}$, and
$\Gamma(1)=G$ s.t. the following conditions are satisfied:

  * •

For every $t\geq\frac{1}{4}$, $\Gamma(t)$ is extends to a smooth vector field
on $S^{3}$.

  * •

The vector field $G$ is hyperbolic on its non-wandering set - in particular,
it generates a universal cross-section $U_{p,1}$ which is a topological disc,
s.t. the first-return map $g_{p,1}:\overline{U_{p,1}}\to\overline{U_{p,1}}$
for $G$ includes a Smale Horseshoe map.

  * •

Given any periodic trajectory $T_{1}$ for $G$ (save perhaps for one), there
exists $T_{0}$, a periodic trajectory for $F_{p}$, s.t. $T_{1}$ can be
deformed to $T_{0}$ by an isotopy of $\mathbf{R}^{3}$. Moreover, if
$T_{1},T_{2}$ are two distinct periodic trajectories for $G$, they cannot both
be deformed to the same $T_{0}$.

  * •

$F_{p}$ generates every knot type generated by $G$ as a periodic trajectory
(save perhaps for one).

###### Proof.

As stated above, we prove Th.2.1 by constructing the curve $\Gamma$
explicitly. For clarity, we do so in several steps:

  * •

First, we construct $\Gamma(t),t\in[0,\frac{1}{4}]$, s.t. $\Gamma(0)=F_{p}$,
$\Gamma(\frac{1}{4})=R_{p}$, the vector field given by Th.3. As we will see,
this deformation will induce a smooth deformation on the cross-section $U_{p}$
(see the discussion after Th.3), s.t. $U_{p}$ is deformed to $U_{p,t}$ \-
consequentially, $\Gamma(t)$ will also induce a continuous deformation of the
first-return map $f_{p}:\overline{U_{p}}\to\overline{U_{p}}$ to
$g_{p,t}:\overline{U_{p,t}}\to\overline{U_{p,t}},t\geq 0$ (wherever defined -
with $g_{p,0}=f_{p}$, $U_{p,0}=U_{p}$).

  * •

Second, we construct $\Gamma(t),t\in(\frac{1}{4},\frac{1}{2}]$, and we do so
by explicitly deforming the vector field $R_{p}$ and its first-return map. In
particular, we will construct the deformation s.t.
$\Gamma(t),t\in[0,\frac{1}{2}]$ is unchanged around the fixed-points
$P_{In},P_{Out}$ \- and moreover, we will construct it s.t. no periodic
trajectories in the set $Q$ for $f_{p}=g_{p,0}$ are destroyed when $t$ is
varied between $0,\frac{1}{2}$.

  * •

Third, we deform $\Gamma(\frac{1}{2})$ to $\Gamma(\frac{3}{4})$ by blowing up
the fixed points $P_{In},P_{Out}$ by Hopf-bifurcations. As we will see, this
deformation leaves intact the chaotic dynamics of $F_{p}$ on $Q$. As we will
see,
$g_{p,\frac{3}{4}}:\overline{U_{p,\frac{3}{4}}}\to\overline{U_{p,\frac{3}{4}}}$
remains chaotic - and in particular, its dynamics can be semi-conjugated to a
Smale Horseshoe (i.e., it is a Topological Horseshoe - see [14]).

  * •

Finally, we deform $\Gamma(\frac{3}{4})$ to $G=\Gamma(1)$, by ”straightening”
the non-wandering set of $\Gamma(\frac{3}{4})$ to a Smale Horseshoe. As we
will see, given any periodic trajectory $T_{0}$ for $F_{p}=\Gamma(0)$ in $Q$,
the deformation $\Gamma(t),t\in[0,1]$ induces an isotopy of $\mathbf{R}^{3}$
which deforms $T_{0}$ to $T_{1}$, some periodic trajectory for $G$.

###  2.1. Step $I$ \- constructing $\Gamma(t)$, $t\in[0,\frac{1}{4}]$.

To begin, recall the bounded topological ball $B_{\alpha}$, and that
$\overline{U_{p}}\cap\overline{B_{\alpha}}=\overline{F_{\alpha}}$, a bounded
topological disc on the cross-section $\overline{U_{p}}$ (see Th.4). Also
recall $U_{p}$ is a half-plane on the cross-section
$\\{\dot{y}=0\\}=\\{(x,-\frac{x}{a},z)|x,z\in\mathbf{R}\\}$, s.t. $U_{p}$ is
the maximal set on $\\{\dot{y}=0\\}$ on which $F_{p}$ points into
$\\{\dot{y}<0\\}$. Additionally, recall the vector field $R_{p}$ from Th.3,
and recall we assume $p\in P$ is a trefoil parameter. As a consequence, by the
boundedness of $B_{\alpha}$ we can smoothly deform $F_{p}$ to $R_{p}$ without
destroying any of the dynamics of $F_{p}$ inside $B_{\alpha}$ \- or, in other
words, there exists a smooth curve of $C^{\infty}$ vector fields,
$\Gamma(t),t\in[0,\frac{1}{4}]$ s.t. $\Gamma(0)=F_{p}$,
$\Gamma(\frac{1}{4})=R_{p}$, and a positive $r>0$ (which can be chosen to be
arbitrarily large) s.t. the following holds:

  * •

$\overline{B_{\alpha}}\subseteq B_{r}(0)$ \- in other words,
$W^{u}_{In},W^{s}_{Out}$ lie inside $B_{r}(0)$.

  * •

For every $t_{1},t_{2}\in[0,\frac{1}{4}]$, $\Gamma(t_{1}),\Gamma(t_{2})$
coincide on $B_{r}(0)$.

Consequentially, it follows that for every $t_{1},t_{2}\in[0,\frac{1}{4}]$, we
have
$\Gamma(t_{1})|_{\overline{B_{\alpha}}}=\Gamma(t_{2})|_{\overline{B_{\alpha}}}$
\- i.e., the dynamics on $B_{\alpha}$ remain unchanged when we vary
$\Gamma(t),t\in[0,\frac{1}{4}]$. In particular, since $p\in P$ is a trefoil
parameter, by Def.1.2 it follows $W^{s}_{In}=W^{u}_{Out}=W$, and in particular
$\partial B_{\alpha}=\\{P_{In},P_{Out}\\}\cup W$ \- therefore, by the
discussion above it follows the dynamics of $\Gamma(t),t\in[0,\frac{1}{4}]$
coincide on $W$ \- that is, the invariant manifolds $W^{s}_{In},W^{u}_{Out}$
are not affected when we vary $\Gamma(t),t\in[0,\frac{1}{4}]$. All in all, we
summarize our results as follows:

  * •

There exists a smooth curve of vector fields $\Gamma(t),t\in[0,\frac{1}{4}]$
s.t. $\Gamma(0)=F_{p}$, $\Gamma(\frac{1}{4})=R_{p}$. In particular, by Th.3,
$R_{p}$ can be extended to a smooth vector field on $S^{3}$, with precisely
two fixed points in $S^{3}$ \- the saddle-foci $P_{In},P_{Out}$, connected by
an unbounded heteroclinic connection $\Gamma$ (see the illustration in
Fig.11).

  * •

By choosing $r>0$ sufficiently large, we ensure that for every
$t\in[0,\frac{1}{4}]$ $\Gamma(t)$ has precisely two fixed points in $S^{3}$,
$P_{In},P_{Out}$, both saddle-foci, which persist, unchanged, when $t$ is
varied in $[0,\frac{1}{4}]$. Moreover, the set $\overline{B_{\alpha}}$ also
persists, unchanged - and so does $\partial
B_{\alpha}=W\cup\\{P_{In},P_{Out}\\}$.

We are almost done constructing $\Gamma(t),t\in[0,\frac{1}{4}]$. As stated
earlier, in the later stages of the proof we will construct some segments in
the curve $\Gamma$ by considering the dynamics of the first-return map -
however, in order to do so we must first deform $F_{p}$ in a way which makes
the first return map well-defined for $\Gamma(t)$ to begin with. In other
words, we will now prove there exists a cross-section $U_{p,t}$, varying
continuously with $t$ and some $t_{0}\in(0,\frac{1}{4}]$ s.t. for $t\geq
t_{0}$ and every initial condition $x\in\mathbf{R}^{3}$ (save perhaps for one
flow line), the forward trajectory of $x$ w.r.t. the vector field $\Gamma(t)$
hits $U_{p,t}$ in some finite time.

To do so, consider the cross-section $U_{p}$ introduced after Th.3. Since
$U_{p}$ is a topological disc, we can construct the deformation s.t. for every
$t\in(0,\frac{1}{4}]$, $U_{p}$ is smoothly deformed to
$U_{p,t}\subseteq\\{\dot{y}=0\\}$ (with $\dot{y}$ taken w.r.t. $\Gamma(t)$).
Moreover, we choose $\Gamma(t),t\in[0,\frac{1}{4}]$ s.t. $\overline{U_{p,t}}$
is always homeomorphic to a closed disc (with $U_{p}=U_{p,0}$), and denote by
$g_{p,t}:\overline{U_{p,t}}\to\overline{U_{p,t}}$ the first-return map for
$\Gamma(t)$ (wherever defined - with $f_{p}=g_{p,0}$). As a consequence, we
choose $\Gamma(t),t\in[0,\frac{1}{4}]$ s.t. $U_{p,t}$ satisfies the following:

  * •

$U_{p,t}$ is the maximal set on $\\{\dot{y}=0\\}$ (w.r.t. $\Gamma(t)$) on
which $\Gamma(t)$ points into $\\{\dot{y}>0\\}$.

  * •

By $D_{\alpha}=B_{\alpha}\cap U_{p}$, for every $t\in[0,\frac{1}{4}]$ we have
$D_{\alpha}\subseteq U_{p,t}$. Consequentially, for every
$t\in[0,\frac{1}{4}]$ we have
$g_{p,t}|_{\overline{D_{\alpha}}}=f_{p}|_{\overline{D_{\alpha}}}$.

\begin{overpic}[width=238.49231pt]{images/TRANS.png}\put(930.0,280.0){$\Gamma$}
\put(840.0,410.0){$P_{Out}$} \put(340.0,350.0){$P_{Out}$}
\put(390.0,80.0){$\infty$} \put(90.0,200.0){$P_{In}$}
\put(600.0,230.0){$P_{In}$} \put(970.0,120.0){$\infty$}
\put(420.0,180.0){$\Gamma_{Out}$} \put(250.0,110.0){$\Gamma_{In}$}
\end{overpic} Figure 11. The deformation of $F_{p}$ to $R_{p}$ \-
$\Gamma_{Out},\Gamma_{In}$ are connected to a heteroclinic trajectory $\Gamma$
(thus destroying the fixed point at $\infty$).

Finally, recall $\Gamma(\frac{1}{4})=R_{p}$ generates a heteroclinic
trajectory passing through $\infty$ and connecting $P_{In},P_{Out}$ (see
Th.3), and consider the cross-section $U_{p,\frac{1}{4}}$. Since
$g_{p,\frac{1}{4}}|_{\overline{D_{\alpha}}}=f_{p}|_{\overline{D_{\alpha}}}$,
we conclude the bounded heteroclinic connection $\Theta$ is also generated by
$R_{p}$ (see Def.1.2). Moreover, recall that in the proof of Th.3 in [29], we
constructed the unbounded heteroclinic connection $\Gamma$ s.t.
$\Gamma\cap\\{\dot{y}=0\\}=\emptyset$ \- as such, since $R_{p},F_{p}$ coincide
on $\overline{B_{\alpha}}$, it follows
$\Gamma\cup\Theta\cap\overline{U_{p,t}}=\\{P_{0}\\}$, and by Cor.4, $P_{0}$ is
strictly interior to $U_{p,t}$ (see the illustration in Fig.11). We now use
these facts to prove $g_{p,\frac{1}{4}}$ is well-defined, and that
$U_{p,\frac{1}{4}}$ is a universal cross-section for
$R_{p}=\Gamma(\frac{1}{4})$ \- namely, we prove:

######  Lemma 2.1.

Let $p\in P$ be a trefoil parameter (see Def.1.2), and recall we denote by
$W^{s}_{In}$ the stable, one-dimensional manifold of the saddle-focus
$P_{In}$. Then, $R_{p}$, $g_{p,\frac{1}{4}}$ and $U_{p,\frac{1}{4}}$ satisfy
the following:

  * •

Given any initial condition $s\in\mathbf{R}^{3}\setminus W^{s}_{In}$ which is
not a fixed point, the trajectory of $s$ (w.r.t. $\Gamma(\frac{1}{4})=R_{p}$)
eventually hits $U_{p,\frac{1}{4}}$ transversely.

  * •

The first return map,
$g_{p,\frac{1}{4}}:\overline{U_{p,\frac{1}{4}}\rm}\setminus\\{P_{0}\\}\rightarrow\overline{U_{p,\frac{1}{4}}\rm}\setminus\\{P_{0}\\}$
is well-defined.

###### Proof.

We first recall and note several facts about $R_{p}$ and $U_{p,\frac{1}{4}}$.
First, recall that by Th.3, $R_{p}$ is a smooth vector field defined
throughout $S^{3}$, with precisely two fixed points - $P_{In},P_{Out}$. Also
recall $R_{p}$ generates an unbounded heteroclinic connection $\Gamma$
connecting $P_{In},P_{Out}$ s.t. $\Gamma\cap\\{\dot{y}=0\\}=\emptyset$. Now,
by the construction of $\Gamma(t),t\in[0,\frac{1}{4}]$ above, because $p\in P$
is a trefoil parameter (i.e., we have $W=W^{u}_{In}=W^{s}_{Out}$) by
$\overline{B_{\alpha}}\subseteq B_{r}(0)$ it follows $D=U_{p}\cap
B_{r}(0)=U_{p,\frac{1}{4}}\cap B_{r}(0)$ \- i.e., $F_{p},R_{p}$ coincide on
$D$. Therefore, provided $r>0$ is chosen sufficiently large, since Cor.1.0.1
implies $W$ is transverse to $U_{p}=U_{p,0}$ at both $P_{In},P_{Out}$, the
same is true for $R_{p},U_{p,\frac{1}{4}}$ \- i.e., $W$ is transverse to
$U_{p,\frac{1}{4}}$ at the fixed points $P_{In},P_{Out}$. Finally, given
$s\in\mathbf{R}^{3}$, denote by $\gamma_{s}$ its solution curve and
parameterize it s.t. $\gamma_{s}(0)=s$, and recall the topological disc
$U_{p,\frac{1}{4}}$ is the maximal set on $\\{\dot{y}=0\\}$ on which $R_{p}$
points into $\\{\dot{y}<0\\}$. As such, we immediately conclude that if
$\dot{y}$ changes sign along $\gamma_{s}$ at least twice (w.r.t. $R_{p}$),
then $\gamma_{s}$ must intersect $U_{p,\frac{1}{4}}$ transversely.

Having made the remarks above, we now prove that given
$s\in\mathbf{R}^{3}\setminus W^{s}_{In}$ which is not a fixed point, its
forward trajectory hits $U_{p,\frac{1}{4}}$ transversely infinitely many
times. By the discussion above, it would suffice to prove the sign of
$\dot{y}$ changes infinitely many times along $\gamma_{s}$. With previous
notations, since $\Gamma\subseteq W^{s}_{In}$ we conclude the forward
trajectory of $s$ cannot blow up to $\infty$ (in $S^{3}$) - which implies that
provided $\gamma_{s}$ does not limit to either saddle foci $P_{In},P_{Out}$ in
forward time, the sign of $\dot{y}$ along $\gamma_{s}$ must change infinitely
many times. Or, in other words, $\gamma_{s}$ intersects transversely with
$U_{p,\frac{1}{4}}$ infinitely many times. Therefore, we have just proven that
if $s\in\mathbf{R}^{3}\setminus W$, its forward trajectory hits $U_{p,t}$
transversely infinitely many times. Now, recall $P_{Out}$ is a saddle focus,
and that the two-dimensional stable manifold $W^{s}_{Out}=W$ is transverse to
$U_{p,\frac{1}{4}}$ at $P_{Out}$ (see Cor.1.0.1 and the illustration in
Fig.3). Therefore, given $s\in W$, its forward trajectory eventually flows to
some neighborhood of $P_{Out}$ where it hits $U_{p,\frac{1}{4}}$ infinitely
many times. All in all, we conclude that given any
$s\in\mathbf{R}^{3}\setminus W^{s}_{In}$ which is not a fixed point, its
trajectory hits $U_{p,\frac{1}{4}}$ transversely infinitely many times.

Now, recall $W^{s}_{In}\cap\overline{U_{p,\frac{1}{4}}}=\\{P_{0}\\}$ (see the
illustration in Fig.11) - by the discussion above it now follows that given
any $s\in\overline{U_{p,\frac{1}{4}}}\setminus\\{P_{0}\\}$, there exists some
future time $t>0$ s.t. $\gamma_{s}(t)\in\overline{U_{p,\frac{1}{4}}}$.
Moreover, since $W^{s}_{In}$ is invariant under the flow, by
$W^{s}_{In}\cap\overline{U_{p,\frac{1}{4}}}=\\{P_{0}\\}$ it follows
$\gamma_{s}(t)\neq P_{0}$. Therefore, by the definition of the first-return
map
$g_{p,\frac{1}{4}}:\overline{U_{p,\frac{1}{4}}}\setminus\\{P_{0}\\}\to\overline{U_{p,\frac{1}{4}}}\setminus\\{P_{0}\\}$
we conclude $g_{p,\frac{1}{4}}$ is well-defined on
$\overline{U_{p,\frac{1}{4}}}\setminus\\{P_{0}\\}$ and Cor.2.1 follows. ∎

###  2.2. Step $II$ \- extending $\Gamma$ to $(\frac{1}{4},\frac{1}{2}]$.

To continue, let us recall we proved in Step $I$ the dynamics of $\Gamma(t)$
remain unchanged on $B_{\alpha}$. We now construct
$\Gamma(t),t\in(\frac{1}{4},\frac{1}{2}]$ by deforming the dynamics inside the
set $B_{\alpha}$, in a very specific way. To begin, recall the cross section
$H_{p}\subseteq D_{\alpha}$ and its image inside $D_{\alpha}$. $H_{p}$ is an
(open) topological disc, bounded by the curves $l=\partial U_{p}\cap
B_{\alpha}$, $L=\partial B_{\alpha}$, and by $f^{-1}_{p}(l)$ (see the
illustration in Fig.6). Since we have proven in Stage $I$ that for every
$t\in[0,\frac{1}{4}]$ we have
$\overline{D_{\alpha}}\subseteq\overline{U_{p,t}}$, and because
$f_{p}|_{\overline{D_{\alpha}}}=g_{p,t}|_{\overline{D_{\alpha}}}$ for every
$t\in[0,\frac{1}{4}]$, we smoothly deform $\Gamma(\frac{1}{2})$ to
$\Gamma(\frac{1}{2})$ s.t. the following two conditions are satisfied:

  * •

For every $t\in[\frac{1}{4},\frac{1}{2}]$, $\Gamma(t)$ generates a
heteroclinic trajectory $\Gamma$ through $\infty$. In particular, $\Gamma(t)$
is always extendable to a smooth vector field on $S^{3}$.

  * •

For every $t\in[\frac{1}{4},\frac{1}{2}]$, $B_{\alpha}$ persists as a
topological disc, while $\Gamma(t)$ always generates two fixed points:
$P_{In},P_{Out}$, both saddle-foci.

  * •

The cross-section $U_{p,t}$ varies smoothly in $t\in[0,\frac{1}{2}]$, and in
particular - for every $t\geq\frac{1}{4}$ the conclusion of Lemma 2.1 still
hold: i.e., for every $t\geq\frac{1}{4}$ $U_{p,t}$ is a universal cross-
section, and the first-return map for $\Gamma(t)$,
$g_{p,t}:\overline{U_{p,t}}\setminus\\{P_{0}\\}\to\overline{U_{p,t}}\setminus\\{P_{0}\\}$
is well-defined.

  * •

For every $t\in[0,\frac{1}{2}]$, $\Gamma(t)$ always generates the heteroclinic
connection $\Theta$, and moreover, $\Theta\cup\Gamma\cup\\{P_{In},P_{Out}\\}$
always form a trefoil knot in $S^{3}$. Additionally, for every
$t\in[\frac{1}{4},\frac{1}{2}]$, $\Theta\cap\overline{U_{p,t}}=\\{P_{0}\\}$,
with $P_{0}$ being interior to $D_{\alpha}$.

However, that is not all - we will now add another condition on the curve
$\Gamma(t),t\in(\frac{1}{4},\frac{1}{2}]$, by means of the first-return map.
To begin, recall that by Stage $I$ for every $t_{1},t_{2}\in[0,\frac{1}{4}]$
we have
$g_{p,t_{1}}|_{\overline{D_{\alpha}}}=g_{p,t_{2}}|_{\overline{D_{\alpha}}}$
(with $g_{p,0}=f_{p}$ \- see the illustration in Fig.8), by
$\overline{H_{p}}\subseteq\overline{D_{\alpha}}$ we conclude
$g_{p,t_{1}}|_{\overline{H_{p}}}=g_{p,t_{2}}|_{\overline{H_{p}}}$ as well. As
such, since $f_{p}=g_{p,0}$ is continuous on $\overline{H_{p}}$ (see Th.5),
the same is true for $g_{p,t}$, $t\in[0,\frac{1}{4}]$. Therefore, we choose
$\Gamma(t),t\in[\frac{1}{4},\frac{1}{2}]$ s.t. there exists an isotopy of
$g_{p,t}:\overline{H_{p}}\to\overline{D_{\alpha}}$,
$t\in[\frac{1}{4},\frac{1}{2}]$ s.t.
$g_{p,\frac{1}{2}}(\overline{H_{p}})\subseteq\overline{H_{p}}$ (see the
illustration in Fig.12). In other words, we deform $\Gamma(\frac{1}{4})=R_{p}$
by pushing the image of $H_{p}$ under $g_{p,\frac{1}{4}}$ inside
$\overline{H_{p}}$. In particular, since this isotopy was constructed only by
squeezing the bundle of flow lines emanating from $H_{p}$, it follows no new
periodic trajectories in $H_{p}$ are created (or destroyed) when we deform
$F_{p}=\Gamma(0)$ to $\Gamma(\frac{1}{2})$.

Now, recall the invariant set $Q\subseteq\overline{H_{p}}$ given by Th.5. By
Th.5, the periodic trajectories in $Q$ all lie uniformly away from the fixed
points $P_{In},P_{Out}$ (see Th.5, and the illustration in Fig.8) - that is,
there exists an open neighborhood $N\subseteq\mathbf{R}^{3}$ s.t. the periodic
trajectories in $Q$ all lie outside of $N$. Therefore, by the discussion
above, we conclude the following:

  * •

The periodic trajectories in $Q$ all persist when we vary
$t\in[0,\frac{1}{2}]$ \- also, for every $\Gamma(t),t\in[0,\frac{1}{2}]$, the
said periodic trajectories always lie uniformly away from $P_{0}$, i.e., they
all lie away from $\Theta\cap\overline{U_{p}}$ (see Def.1.2 uniformly away
from the fixed-points $P_{In},P_{Out}$ (see the illustration in Fig.12).

  * •

Let $T_{0}$ denote any periodic trajectory for $F_{p}=\Gamma(0)$ in the set
$Q$. Then, when we smoothly deform $F_{p}$ to $\Gamma(\frac{1}{2})$, $T_{0}$
is deformed to $T_{\frac{1}{2}}$, a periodic trajectory for
$\Gamma(\frac{1}{2})$, by some isotopy of $\mathbf{R}^{3}$. In addition,
$T_{\frac{1}{2}}$ intersects $U_{p,\frac{1}{2}}$ precisely at $H_{p}$, and its
knot-type is the same as $T_{0}$’s.

  * •

Moreover, with the notations above, the minimal periods of
$T_{0}\cap\overline{H_{p}}$ w.r.t. $f_{p}=g_{p,0}$ and the minimal period of
$T_{\frac{1}{2}}\cap\overline{H_{p}}$ w.r.t. $g_{p,\frac{1}{2}}$ are the same.

In other words, by the construction of the curve
$\Gamma(t),t\in[0,\frac{1}{2}]$ described above it follows no periodic
dynamics in $H_{p}$ are either lost or changed qualitatively - both for the
flow and the first return map. In particular, all the knot-types generated by
$F_{p}$ in $Q\subseteq H_{p}$ are also generated by $\Gamma(\frac{1}{2})$ in
$H_{p}$.

\begin{overpic}[width=216.81pt]{images/isotop1.png} \put(0.0,35.0){$P_{In}$}
\put(600.0,35.0){$P_{In}$} \put(710.0,230.0){$\delta$}
\put(750.0,185.0){$P_{0}$} \put(150.0,175.0){$P_{0}$}
\put(120.0,340.0){$\delta_{0}$} \put(330.0,300.0){$P_{Out}$}
\put(930.0,300.0){$P_{Out}$} \put(730.0,340.0){$\delta_{0}$}
\put(120.0,230.0){$\delta$} \end{overpic} Figure 12. The deformation varying
$\Gamma(\frac{1}{4})$ to $\Gamma(\frac{1}{2})$, obtained by pushing
$g_{\frac{1}{4}}(\overline{H_{p}})$ inside $\overline{H_{p}}$ (recall $H_{p}$
is the subdomain on $U_{p}$ bounded by $L$, the green arc, $f^{-1}_{p}(l)$ the
read arc, and the arc connecting $P_{In},P_{Out}$).

###  2.3. Step $III$ \- extending $\Gamma$ to $(\frac{1}{2},\frac{3}{4}]$.

Before extending the curve $\Gamma$ to $(\frac{1}{2},\frac{3}{4}]$, we must
recall several facts. To begin, recall the unbounded heteroclinic connection
$\Gamma$ for $R_{p}=\Gamma(\frac{1}{4})$ (see Th.3), and the heteroclinic
connection $\Theta$ (see Def.1.2). Now, recall the one-dimensional manifolds
$W^{s}_{In},W^{u}_{Out}$ (see the discussion in page 2). Since $p\in P$ is a
trefoil parameter, it follows that w.r.t. $F_{p}=\Gamma(0)$ we have
$\Theta\subseteq W^{u}_{Out}\cap W^{s}_{In}$ \- therefore, by the discussion
above and by the construction of $\Gamma(t),t\in[0,\frac{1}{2}]$, it follows
that for every $t\in[0,\frac{1}{2}]$, $\Theta$ persists as a separatrix (and
as a heteroclinic connection) in $W^{s}_{In}\cap W^{u}_{Out}$. Moreover, since
$\Gamma$ is an unbounded heteroclinic connection for
$R_{p}=\Gamma(\frac{1}{4})$, a similar argument implies $\Gamma$ persists as a
separatrix in $W^{s}_{In}\cap W^{u}_{Out}$ for
$\Gamma(t),t\in[\frac{1}{4},\frac{1}{2}]$. Therefore, it that for every
$\frac{1}{2}\geq t\geq\frac{1}{4}$ the one-dimensional invariant manifolds
$W^{s}_{In}$, $W^{u}_{In}$ coincide, and contain heteroclinic trajectories
which flow from $P_{Out}$ to $P_{In}$.

We can no extend $\Gamma$ to $(\frac{1}{2},\frac{3}{4}]$ \- begin by blowing
up both fixed points $P_{In},P_{Out}$ by Hopf- bifurcations, s.t. both
$P_{Out}$, $P_{In}$ become hyperbolic periodic trajectories for the vector
field $\Gamma(\frac{3}{4})$ \- moreover, we construct this Hopf-deformation
$\Gamma(\frac{3}{4})$ s.t. the two-dimensional invariant manifolds of
$P_{In},P_{Out}$ still satisfy $W^{u}_{In}=W^{s}_{Out}$ \- see the
illustration in Fig.13. As must be noted, this deformation has the effect of
blowing up the two separatrices $\Gamma,\Theta$ as well - and moreover, it
generates two fixed points, $P^{\prime}_{In},P^{\prime}_{Out}$ \- an attractor
and a repeller, respectively, with immediate respective basins of attraction
$D_{In},D_{Out}$ in $U_{p,\frac{3}{4}}$ (see the illustration in Fig.13). In
particular, this bifurcation blows
$P_{0}=\Theta\cap\overline{U_{p,\frac{1}{2}}}$ to a disc.

Additionally, we also construct the curve
$\Gamma(t),t\in(\frac{1}{2},\frac{3}{4}]$ s.t. for every $t$ the cross section
$U_{p,t}$ is varied smoothly and remains a universal cross-section for the
first-return map
$g_{p,t}:\overline{U_{p,t}}\setminus\\{P_{0}\\}\to\overline{U_{p,t}}\setminus\\{P_{0}\\}$,
$t\in(\frac{1}{2},\frac{3}{4}]$ \- moreover, since the trajectory of any
initial condition on $\Theta$ (w.r.t. $\Gamma(t),t\in[0,\frac{1}{2}]$ flows to
$P_{In}$, we construct $\Gamma(\frac{3}{4})$ s.t.
$g_{p,\frac{3}{4}}(P_{0})\subseteq D_{In}$ \- see the illustration in Fig.13.
In particular, we deform $\Gamma(\frac{1}{2})$ to $\Gamma(\frac{3}{4})$ s.t.
the curve $\delta$ (see the illustration in Fig.8 and Fig.6) blows up to an
open set, $\Delta$, connecting $P_{0},\partial U_{p,\frac{3}{4}}$ s.t.
$g_{p,\frac{3}{4}}(\Delta)\subseteq D_{In}$ as well (see the illustration in
Fig.13).

\begin{overpic}[width=260.17464pt]{images/isotop2.png}
\put(0.0,35.0){$P_{In}$} \put(650.0,35.0){$P^{\prime}_{In}$}
\put(730.0,45.0){$P_{In}=B$} \put(580.0,0.0){$D_{In}$} \put(590.0,150.0){$A$}
\put(750.0,215.0){$P_{0}$} \put(150.0,175.0){$P_{0}$}
\put(120.0,340.0){$\delta_{0}$} \put(330.0,300.0){$P_{Out}$}
\put(870.0,320.0){$P^{\prime}_{Out}$} \put(920.0,210.0){$P_{Out}$}
\put(930.0,360.0){$D_{Out}$} \put(770.0,345.0){$D$} \put(710.0,330.0){$C$}
\put(120.0,230.0){$\delta$} \end{overpic} Figure 13. The deformation varying
$\Gamma(\frac{1}{2})$ to $\Gamma(\frac{3}{4})$, obtained by blowing up the
fixed-points by Hopf bifurcations and choosing the sides $AB,CD$. The red disc
corresponds to $g_{p,\frac{3}{4}}(P_{0})$.

Now, recall that for every $t\in[0,\frac{1}{2}]$, $g_{p,t}$ is continuous on
$H_{p}$ \- and that $g_{p,\frac{1}{2}}(H_{p})\subseteq H_{p}$. Therefore, it
follows that when we deform $\Gamma(\frac{1}{2})$ to $\Gamma(\frac{3}{4})$ we
can do so s.t. $H_{p}$ is deformed to $H^{\prime}_{p}\subseteq
U_{p,\frac{3}{4}}$, a topological disc s.t. $g_{p,\frac{3}{4}}$ is continuous
on $H^{\prime}_{p}$ \- moreover, $\partial\Delta\subseteq\partial
H^{\prime}_{p}$ and $D_{In}\cap H^{\prime}_{p}=\emptyset$, $D_{Out}\cap
H^{\prime}_{p}=\emptyset$(see the illustration in Fig.13). Now, let us define
a topological rectangle $ABCD\subseteq\overline{H^{\prime}_{p}}$ s.t. the $AB$
side corresponds to an arc on $\partial D_{In}$ connecting $A$ and $P_{In}=B$
(see the illustration in Fig.13) while the $CD$ side corresponds to some curve
in $\overline{\Delta}$ which connects two points, $C,D$ with $P_{0}$ (see the
illustration in Fig.13. Now, since the vector field $\Gamma(\frac{3}{4})$ was
constructed out of blowing the fixed-points $P_{In},P_{Out}$ as described
above, it follows $g^{-1}_{p,\frac{3}{4}}(P_{0})\subseteq D_{Out}$, i.e.,
given an initial condition $s\in H^{\prime}_{p}$, $g_{p,\frac{3}{4}}(s)\not\in
H^{\prime}_{p}$. As a consequence, $g_{p}(H^{\prime}_{p})$ can be imposed on
$H^{\prime}_{p}$ as appears in Fig.13.

Consequentially, it follows that given any curve $\gamma\subseteq ABCD$ s.t.
$\gamma$ connects the $CD$ and $AB$ sides in $ABCD$,
$g_{p,\frac{3}{4}}(\gamma)$ includes at least two curves in $ABCD$,
$\gamma_{1},\gamma_{2}$, both of which connect the $AB,CD$ sides in $ABCD$.
Therefore, by The Topological Horseshoe Theorem in [14] there exists an
invariant $Q^{\prime}\subseteq ABCD$ and a continuous, surjective
$\pi^{\prime}:Q^{\prime}\to\\{1,2\\}^{\mathbf{N}}$ s.t. $\pi^{\prime}\circ
g_{p,\frac{3}{4}}=\sigma\circ\pi^{\prime}$ (with
$\sigma:\\{1,2\\}^{\mathbf{N}}\to\\{1,2\\}^{\mathbf{N}}$ denoting the one-
sided shift). In fact, we can say more - applying a similar algorithm to the
one used to construct the invariant set of the Smale Horseshoe in [3], it is
immediate we can construct $Q^{\prime}$ s.t. for every
$s\in\\{1,2\\}^{\mathbf{N}}$, $\pi^{\prime-1}(s)=R_{s}$ is a connected,
compact set connecting the $AC,BD$ sides in $ABCD$.

Now, recall the set $Q$ given by Th.5 \- in particular, all its periodic
dynamics lie away from the fixed points $P_{In},P_{Out}$. That is, with the
notations of Th.5 given any periodic $s\in\\{1,2\\}^{\mathbf{N}}$ which is not
the constant $\\{1,1,1...\\}$, $\pi^{-1}(s)=D_{s}$ is a connected, compact set
- and moreover, there exists an open neighborhood $N\subseteq\mathbf{R}^{3}$,
$P_{In},P_{Out}\in N$, independent of $s$, s.t.
$D_{s}\cap\overline{N}=\emptyset$ (see Th.5). Moreover, by Th.5, $D_{s}$
connects the components of $\partial H_{p}\setminus(L\cup\delta)$, i.e., it
connects the arcs on $\partial H_{p}$ which, when we deform $F_{p}=\Gamma(0)$
to $\Gamma(\frac{3}{4})$ along $\Gamma$, are deformed to the $AC,BD$ sides in
$ABCD$.

Therefore, given $s\in\\{1,2\\}^{\mathbf{N}}$ which is not the constant
$\\{1,1,1...\\}$, when we smoothly deform $F_{p}=\Gamma(0)$ to
$\Gamma(\frac{3}{4})$ along $\Gamma$ it follows $D_{s}$ is continuously
deformed to $R_{s}$. In particular, since we deform $F_{p}$ to
$\Gamma(\frac{3}{4})$ by blowing up the fixed points by Hopf-bifurcations, by
the existence of $N$ (and because $P_{In},P_{Out}\in N$ ) it follows we can
construct the curve $\Gamma(t),t\in[\frac{1}{2},\frac{3}{4}]$ s.t.
$\Gamma(\frac{1}{2})|_{\mathbf{R}^{3}\setminus
N}=\Gamma(\frac{3}{4})|_{\mathbf{R}^{3}\setminus N}$. As such, since we
constructed $\Gamma(t),t\in[0,\frac{1}{2}]$ s.t. no new periodic trajectories
are generated in $Q$ when we vary $\Gamma(t),t\in[0,\frac{1}{2}]$. it follows
that when we deform $D_{s}$ to $R_{s}$, no new periodic trajectories are
generated - or, in other words, every periodic trajectory $T_{\frac{3}{4}}$
which intersects $R_{s}$ can be deformed by some isotopy of $\mathbf{R}^{3}$
to $T_{0}$, a periodic trajectory for $F_{p}=\Gamma(0)$ which intersects
$D_{s}$.

Now, let us also recall that given any periodic trajectory $T_{0}$ in $Q$
satisfies $T_{0}\cap\overline{N}=\emptyset$. Additionally, recall we
constructed $\Gamma(t),t\in[0,\frac{1}{2}]$ s.t. when we deform $F_{p}$ to
$\Gamma(\frac{1}{2})$, $T_{0}$ is isotopically deformed to $T_{\frac{1}{2}}$
\- and moreover, $T_{\frac{1}{2}}\cap\overline{N}=\emptyset$. Now, since
$T_{0}$ is a periodic trajectory in $Q$ it follows there exists a finite
number of periodic symbols $s_{1},s_{2},...,s_{k}$ s.t. $T_{0}\cap
Q\subseteq\cup_{i=1}^{k}D_{s_{i}}$ \- therefore, since $D_{s_{i}},i=1,...,k$
is deformed continuously to $R_{s_{i}}$ it foll Therefore, the previous
paragraph implies that when we deform $F_{p}$ to $\Gamma(\frac{3}{4})$ the
trajectory $T_{0}$ is smoothly deformed by some isotopy of $\mathbf{R}^{3}$ to
some periodic trajectory $T_{\frac{3}{4}}$, s.t. $T_{\frac{3}{4}}\cap
Q^{\prime}\subseteq\cup^{k}_{i=1}R_{s_{i}}$. Therefore, all in all we
summarize our results as follows:

######  Corollary 2.1.1.

Given $s\in\\{1,2\\}^{\mathbf{N}}$ that is not the constant $\\{1,1,1...\\}$
and given a periodic trajectory $T_{\frac{3}{4}}$ s.t. $T_{\frac{3}{4}}\cap
R_{s}$, $R_{s}=\pi^{\prime-1}(s)$, there exists an isotopy of $\mathbf{R}^{3}$
s.t. $T_{\frac{3}{4}}$ is deformed by the said isotopy to some $T_{0}$, a
periodic trajectory for $F_{p}$ satisfying $T_{0}\cap D_{s}\neq\emptyset$,
$D_{s}=\pi^{-1}(s)$. Consequentially, every knot type generated by $F_{p}$ as
a periodic trajectory is also generated by $\Gamma(\frac{3}{4})$ as a periodic
trajectory.

###  2.4. Step $IV$ \- extending $\Gamma$ to $(\frac{3}{4},1]$.

Having constructed the vector field $\Gamma(\frac{3}{4})$, we now smoothly
deform it in $S^{3}$ to $\Gamma(1)=G$, thus concluding the proof of Th.2.1. To
begin, let us recall we proved in Stage $III$ that $g_{p,\frac{3}{4}}(ABCD)$
can be imposed on the rectangle $ABCD$ as appears in Fig.14. As such, it is
immediate that we can construct an invariant set $B^{\prime}\subseteq
Q^{\prime}$ s.t. there exists a continuous function
$\omega:B^{\prime}\to\\{1,2\\}^{\mathbf{Z}}$ satisfying $\omega\circ
g_{p,\frac{3}{4}}=\xi\circ\omega$ \- with
$\xi:\\{1,2\\}^{\mathbf{Z}}\to\\{1,2\\}^{\mathbf{Z}}$ denoting the double-
sided shift. In particular, given any $s\in\\{1,2\\}^{\mathbf{Z}}$ it follows
$b_{s}=\xi^{-1}(s)$ is a connected, compact set, generated by intersecting
infinitely many Jordan domains. As such, $b_{s}$ is homeomorphic to a convex
set. Therefore, when $s$ is periodic of minimal period $k$, it follows
$g^{k}_{p,\frac{3}{4}}(b_{s})\subseteq b_{s}$,
$g^{i}_{p,\frac{3}{4}}(b_{s})\cap b_{s}=\emptyset,i\in\\{1,...,k-1\\}$ \-
hence, by the Brouwer Fixed Point Theorem there exists a periodic point
$x_{s}\in b_{s}$ of minimal period $k$. Additionally, since $b_{s}\subseteq
Q^{\prime}$, with previous notations it follows $b_{s}\subseteq R_{\nu}$ for
some periodic $\nu\in\\{1,2\\}^{\mathbf{N}}$ \- and therefore, $x_{s}\in
T_{\frac{3}{4}}$, some periodic trajectory for $\Gamma(\frac{3}{4})$ s.t.
$T_{\frac{3}{4}}\cap R_{\nu}\neq\emptyset$.

\begin{overpic}[width=195.12767pt]{images/rectanglemap.png}
\put(50.0,70.0){$A$} \put(200.0,5.0){$g_{p}(D)$} \put(340.0,35.0){$g_{p}(C)$}
\put(860.0,70.0){$g_{p}(B)=B$} \put(710.0,70.0){$g_{p}(A)$}
\put(910.0,470.0){$D$} \put(50.0,470.0){$C$} \end{overpic} Figure 14.
$g_{p}=g_{p,\frac{3}{4}}$ re-drawn as a rectangle map.

To continue, let $I$ denote the maximal invariant set of $g_{p,\frac{3}{4}}$
in the rectangle $ABCD$ \- that is, $I$ denotes the set in $ABCD$ s.t. for
every $n\in\mathbf{Z}$ and every $x\in I$ we have $g^{n}_{p,\frac{3}{4}}(x)\in
ABCD$. Now, smoothly deform the vector field $\Gamma(\frac{3}{4})$ as follows:
begin by removing from $I$ all the components of $I\setminus B^{\prime}$.
Following that, contract every component in $B^{\prime}$ to a singleton - we
can do so because since by previous discussion, every component in
$B^{\prime}$ is homeomorphic to a convex set, thus contractible. In
particular, when $s\in\\{1,2\\}^{\mathbf{Z}}$ is periodic (of any minimal
period), we crush the component $b_{s}$ to some periodic point $x_{s}\in
b_{s}$ (by previous paragraph, $b_{s}$ exists). This induces a smooth
deformation of $\Gamma(\frac{3}{4})$ to $\Gamma(1)=G$, as described in Fig.15.
Moreover, we do so s.t. the cross-section $U_{p,\frac{3}{4}}$ is smoothly
deformed to $U_{p,1}$, a universal cross-section for $G$ and its first-return
map $g_{p,1}$. Furthermore, recall $\Gamma(\frac{3}{4})$ has two fixed points
in $S^{3}$ \- namely, $P^{\prime}_{Out}$, a repeller, and $P^{\prime}_{In}$,
an attractor - and in particular, both admit immediate basin or repelling and
attraction in the cross-section $U_{p,\frac{3}{4}}$: $D_{Out}$ and $D_{In}$,
respectively (see the illustration in Fig.13). We construct the deformation of
$\Gamma(\frac{3}{4})$ to $G=\Gamma(1)$ s.t.
$P^{\prime}_{In},P^{\prime}_{Out}$, $D_{In},D_{Out}$ all persist for every
$\Gamma(t),t\in[\frac{3}{4},1]$.

\begin{overpic}[width=260.17464pt]{images/isotop2.jpg} \put(-10.0,35.0){$A$}
\put(-10.0,230.0){$C$} \put(550.0,230.0){$C$} \put(400.0,230.0){$D$}
\put(970.0,230.0){$D$} \put(400.0,35.0){$B$} \put(650.0,-15.0){$g_{p,1}(CD)$}
\put(820.0,25.0){$g_{p,1}(AB)$} \put(970.0,35.0){$B$} \put(550.0,35.0){$A$}
\put(50.0,5.0){$g_{p,\frac{3}{4}}(CD)$}
\put(230.0,25.0){$g_{p,\frac{3}{4}}(AB)$} \end{overpic} Figure 15. The
deformation varying $\Gamma(\frac{3}{4})$ to $\Gamma(1)$, which deforms the
first-return map to a Smale Horseshoe.

From the discussion above we conclude the deformation described above deforms
$\Gamma(\frac{3}{4})$ to a vector field $G$, s.t. $G$ generates a first-return
map $g_{p,1}:ABCD\to ABCD$ conjugate to a Smale Horseshoe map - and moreover,
its invariant set, $I$, is made of a Cantor set, generated by crushing every
component of $B^{\prime}$ to a singleton. Therefore, given any
$x\in\overline{U_{p,1}}\setminus I$, there exists some $n_{1}\leq 0$ and some
$n_{2}\geq 0$ s.t. for $n\leq n_{1}$ $g^{n}_{p,1}(x)\in D_{Out}$, while for
$n\geq n_{2}$, $g^{n_{2}}_{p,1}(x)\in D_{In}$. In other words, the trajectory
of every initial condition $x\not\in I$ tends to the repeller
$P^{\prime}_{Out}$ in negative time, and to the attractor $P^{\prime}_{In}$ in
positive time. Moreover, since $g_{p,1}$ is conjugate to a Smale Horseshoe
map, it follows $g_{p,1}$ is hyperbolic on initial conditions in $I$ \- and
consequentially, it follows the vector field $G$ is hyperbolic on it non-
wandering set. Moreover, there exists a continuous, surjective
$\zeta:I\to\\{1,2\\}^{\mathbf{Z}}$ s.t. $\zeta\circ g_{p,1}=\xi\circ\zeta$ (as
before, $\xi:\\{1,2\\}^{\mathbf{Z}}\to\\{1,2\\}^{\mathbf{Z}}$ denotes the
double-sided shift).

Therefore, all in all, we have proven the curve $\Gamma(t),t\in[0,1]$
satisfies the following:

  * •

$\Gamma(0)=F_{0}$, $\Gamma(1)=G$. Moreover, $\Gamma$ is a smooth curve of
$C^{\infty}$ vector fields on $\mathbf{R}^{3}$.

  * •

For every $t\geq\frac{1}{4}$, $\Gamma(t)$ is extendable to a smooth vector
field in $S^{3}$.

  * •

The vector field $G$ is hyperbolic on its non-wandering set - in particular,
it generates a universal cross-section $U_{p,1}$ which is a topological disc,
s.t. the first-return map $g_{p,1}:\overline{U_{p,1}}\to\overline{U_{p,1}}$
for $G$ includes a Smale Horseshoe map.

Therefore, to conclude the proof of Th.2.1 it remains to prove every knot type
generated by $G$ as a periodic trajectory is also generated by $F_{p}$ as a
periodic trajectory. To do so, let us choose some periodic
$s\in\\{1,2\\}^{\mathbf{Z}}$, s.t. $s$ is not the constant
$\\{...,1,1,1,...\\}$ \- by construction, there exists a unique periodic
trajectory $T_{1}$ s.t. $T_{1}\cap\zeta^{-1}(s)\neq\emptyset$. As such, by the
construction of $\Gamma(t),t\in[\frac{3}{4},1]$, it follows $T_{1}$ can be
deformed by an isotopy of $\mathbf{R}^{3}$ to a periodic trajectory
$T_{\frac{3}{4}}$ for $\Gamma(\frac{3}{4})$, s.t. (with previous notations),
$T_{\frac{3}{4}}\cap b_{s}\neq\emptyset$. However, this implies there exists a
periodic symbol $\nu\in\\{1,2\\}^{\mathbf{N}}$, s.t. $\nu$ is not the constant
$\\{1,1,1...\\}$ s.t. $T_{\frac{3}{4}}\cap R_{\nu}\neq\emptyset$, i.e.,
$T_{\frac{3}{4}}\cap Q^{\prime}\neq\emptyset$. Therefore, by Cor.2.1.1 there
exists an isotopy of $\mathbf{R}^{3}$ which deforms $T_{\frac{3}{4}}$ to
$T_{0}$, a periodic trajectory for $F_{p}$ \- and consequentially, it follows
$T_{0},T_{\frac{3}{4}}$ have the same knot type. Now,
$s\in\\{1,2\\}^{\mathbf{Z}}$ was chosen arbitrarily - with the only
restriction being that $s$ is not the constant $\\{...,1,1,1,...\\}=\mu$.
However, since $g_{p,1}$ is conjugate to a Smale Horseshoe map it follows
$\zeta^{-1}(\mu)$ includes precisely one periodic orbit (of minimal period
$1$) for $g_{p,1}$ \- therefore, there exists precisely one periodic
trajectory $T$ for $G$ s.t. $T\cap\zeta^{-1}(\mu)\neq\emptyset$.

Consequentially, given any periodic trajectory $T_{1}$ for $G$ s.t. $T_{1}\neq
T$, it follows $T_{1}$ can be deformed by an isotopy of $\mathbf{R}^{3}$ to
$T_{0}$, a periodic trajectory of $F_{p}$ \- and moreover, $T_{1},T_{0}$ have
the same knot-type. As such, it follows that every knot type generated by $G$
as a periodic trajectory (save perhaps for one) is also realized as a periodic
trajectory for $F_{p}$ \- and Th.2.1 now follows. ∎

######  Remark 2.1.

As remarked earlier, the existence of Topological Horseshoes in the Rössler
system was proven in [17] with rigorous numerical methods (albeit in non-
heteroclinic parameters).

##  3\. Classifying the periodic trajectories

Having proven Th.2.1, in this section we study its corollaries. Our main
results in this Section are Prop.3.1 and Th.3.1 \- which, together, with
Th.2.1, prove the Rössler system at trefoil parameters can be reduced to a
hyperbolic flow, unique up to orbital equivalence. Therefore, this Section is
organized as follows - we begin by proving Prop.3.1, where we show the vector
field $G$ given by Th.2.1 is unique (up to orbital equivalence). Following
that, we further analyze its dynamics and prove the periodic dynamics for $G$
can be described by the $L(0,1)$ Template (see Cor.3.0.2), from which Th.3.1
would follow.

We begin with some general remarks. First, by the hyperbolicity of $G$ on its
invariant set, the Birman-Williams Theorem implies we can associate a Template
$T$ with $G$ (see Th.6) - in particular, given any knot type on $T$ (save
perhaps for one), Th.2.1 implies that knot type is also realized as a periodic
trajectory for $F_{p}$, i.e., for the Rössler system at trefoil parameters.
The second conclusion is that provided we can identify $T$ explicitly, we
could characterize the topology of infinitely many periodic trajectories
generated by the Rössler system. However, in order to these heuristic
conclusions rigorous facts, we must first answer the question of how unique is
the vector field $G$. In other words, we need now answer the following
question: does the algorithm described in Th.2.1 always deform $F_{p}$ to the
same vector field - at least up to orbital equivalence? Or, put simply, is the
template $T$ independent of how we construct $G$?

As we will now see, the answer to this question is positive. To prove this
truly is the case, let us begin by introducing the following definition:

######  Definition 3.1.

Let $\phi$ be a flow on a $3$-manifold $M$, with a hyperbolic non-wandering
set, $K\subseteq M$. 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, we have the following theorem, proven in [16]:

######  Theorem 7.

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

Therefore, by applying Th.7, we can now answer the question about the
uniqueness of $G$. We now prove the following Corollary:

######  Proposition 3.1.

Let $p\in P$ be a trefoil parameter for the Rössler system, and let $G$ be the
vector field given by Th.2.1. Then, we have:

  * •

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

  * •

There exists a Template $T$, independent of $G$, s.t. any knot type on $T$ is
generated by $G$ \- and vice versa.

  * •

Every knot type on $T$ (save perhaps for one) is realized as a periodic
trajectory for $F_{p}$ \- that is, one can associate a template with the
Rössler system at trefoil parameters.

###### Proof.

By Prop.2.1, $G$ generates a smooth flow in $S^{3}$, hyperbolic on its non-
wandering set. Also recall that by construction the vector field $G$ has
precisely two fixed points in $S^{3}$, $P^{\prime}_{In}$, a sink, and
$P^{\prime}_{Out}$, a source. Let $B_{In},B_{Out}$ denote open balls centered
at $P^{\prime}_{In},P^{\prime}_{Out}$ respectively s.t. $G$ is transversal to
the respective boundaries of both, $S_{In},S_{Out}$ \- in particular, choose
$B{In}$ and $B_{Out}$ s.t. on $S_{In}$ $G$ points into $B_{In}$ while on
$S_{Out}$ $G$ points outside $B_{Out}$. Additionally, by the construction of
$G$ in Prop.2.1, we can choose $B_{In},B_{Out}$ s.t. both balls lie in some
positive distance from the non-wandering set for $G$. This implies
$S^{3}\setminus(B_{In}\cup B_{Out})=M$ has a total genus $0$ \- therefore,
since by Th.2.1 $G$ is hyperbolic on its non-wandering set in $M$, by Def.3.1
it follows the flow generated by $G$ on $M$ is a model. From Th.7, it follows
the flow generated by $G$ on $M$ is unique up to orbital equivalence.

To conclude the proof, let $T$ denote the template corresponding to $G$, given
by The Birman-Williams Theorem (see Th.6). Additionally, let $p\in P$ denote a
trefoil parameter for the Rössler system, and recall we denote the
corresponding vector field by $F_{p}$ (see Eq.2). By the discussion above, it
follows that when we deform the vector field $F_{p}$ to $G$ as described in
Th.2.1, no matter how we generate $G$, it would always be a model - and these
models are orbitally equivalent to one another by Th.7, it immediately follows
the Template $T$ is also unique. Finally, recall that by Th.2.1, every
periodic trajectory for $G$ (save perhaps for one) can be deformed to a
periodic trajectory for $F_{p}$ by an isotopy of $\mathbf{R}^{3}$ \-
therefore, it follows that given any knot type $J$ on $T$ (save perhaps for
one), there exists a periodic trajectory $T_{0}$ for $F_{p}$ whose type is
$J$. Hence, all in all, the proof Prop.3.1 is now complete. ∎

\begin{overpic}[width=173.44534pt]{images/temp11.png} \put(0.0,260.0){$B$}
\put(350.0,270.0){$B_{2}$} \put(860.0,570.0){$B_{1}$} \end{overpic} Figure 16.
A horseshoe template, $V$ suspended around the trefoil.

Having proven Prop.3.1, we now apply Th.7 yet again - this time, to explicitly
find characterize the template $T$ given by Prop.3.1. To do so, let us first
consider a smooth vector field $H$ on $S^{3}$ which is generated by suspending
a horseshoe map given in Fig.17 as follows. First, assume $H$ has two fixed
points in $S^{3}$ \- first, $p_{1}$, a source, and second, $p_{2}$, a sink.
Now, let $S_{1}$ denotes a sphere, enclosing a ball $B_{1}$ s.t. $p_{1}\in
B_{1}$, on which $H$ points outside of $B_{1}$. Conversely, let $S_{2}$ denote
another sphere, enclosing a ball $B_{2}$ s.t. $p_{2}\in B_{2}$, on which $H$
points into $B_{2}$. Now, suspend the horseshoe in accordance with the
Template $V$ described in Fig.16. By Th.7, the vector field $H$ generates a
model flow on $S^{3}\setminus(B_{1}\cup B_{2})$.

\begin{overpic}[width=195.12767pt]{images/tempret11.jpg}
\put(900.0,300.0){$h(b)$} \put(900.0,240.0){$h(a)$} \put(900.0,160.0){$h(c)$}
\put(900.0,100.0){$h(d)$} \put(480.0,0.0){$F$} \put(220.0,30.0){$c$}
\put(0.0,200.0){$h(F)$} \put(220.0,360.0){$d$} \put(780.0,30.0){$a$}
\put(780.0,360.0){$b$} \put(190.0,200.0){$B$} \put(590.0,360.0){$h(S)$}
\end{overpic} Figure 17. $h:S\to S$, a Smale Horseshoe map. $F$ in Fig.17
denotes the fold-line of $h$ in $S$, and $B$ the branch line on the template
$V$.

Now, recall the first-return map generated by $G$, $g_{p,1}:R\rightarrow R$,
with $R$ denoting the rectangle $ABCD$ (see the proof of Th.2.1), depicted in
Fig.18. Comparing $h:S\rightarrow S$ with $g:R\rightarrow R$, we see the
vector fields $H,G$ suspend a horseshoe in the exact same way, and more
importantly - both are models for the same flow (see the discussion before
Th.7). Hence by Th.7 they are orbitally equivalent. In particular, it follows
$T$, the template given by Prop.3.1 is the same as the template $V$ \-
therefore we immediately conclude:

######  Corollary 3.0.1.

Let $G$ be as in Th.2.1, let $T$ be the template from Prop.3.1, and let $V$
denote the template defined above. Then, $V=T$, and in particular, the
template $V$ describes all the knot-types generated by the vector field $G$.

\begin{overpic}[width=195.12767pt]{images/tempret22.png}
\put(490.0,-15.0){$B$} \put(300.0,130.0){$D_{In}$} \put(200.0,190.0){$g(D)$}
\put(200.0,240.0){$g(C)$} \put(390.0,200.0){$g(A)$} \put(50.0,210.0){$A$}
\put(35.0,690.0){$C$} \put(95.0,780.0){$D$} \put(450.0,110.0){$g(B)$}
\put(750.0,750.0){$D_{Out}$} \put(590.0,340.0){$g(R)$} \end{overpic} Figure
18. The horseshoe map $g_{p,1}:R\to R$ \- for simplicity, we write
$g_{p,1}=g$.

However, we can say more. Consider the template $T=V$ sketched on the left in
Fig.19 below. By deforming that template isotopically (in $\mathbf{R}^{3}$),
$V$ is isotoped to the $L(0,1)$ template, sketched on the right of Fig.19 (see
[7] for more details on $L(0,1)$). In other words, every knot type on $T$ also
lies in $L(0,1)$ and vice versa - hence, by Cor.3.0.1 we immediately conclude:

######  Corollary 3.0.2.

Let $G$ be the vector field given by Th.2.1. Then, every knot type on $L(0,1)$
is generated by $G$ as a periodic trajectory - conversely, every periodic
trajectory for $G$ corresponds to precisely one knot type on the template
$L(0,1)$.

\begin{overpic}[width=195.12767pt]{images/template.jpg}
\put(690.0,330.0){$L(0,1)$} \put(150.0,330.0){$T$} \end{overpic} Figure 19.
The two templates, $T$ and $L(0,1)$. $T$ can be deformed to $L(0,1)$ by an
isotopy of $\mathbf{R}^{3}$.

Having proven the $L(0,1)$ template describes the dynamical complexity
generated by $G$, we are now ready to prove Th.3.1 \- with which we conclude
Section $2$. To do so, recall that given a parameter $p\in P$, we denote by
$F_{p}$ the corresponding vector field (see Eq.2) - that is,
$\dot{x}=F_{p}(x)$ is the Rössler system corresponding to the parameter values
$p=(a,b,c)$. We now prove:

######  Theorem 3.1.

Let $p\in P$ be trefoil parameter for the Rössler system. 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.

###### Proof.

By Th.2.1, every periodic trajectory for $G$ (save perhaps for one) can be
smoothly deformed to a periodic trajectory for $F_{p}$ by some isotopy of
$\mathbf{R}^{3}$. As a consequence, since by Cor.3.0.2 the knots types
corresponding to the periodic trajectories of $G$ are precisely those on the
$L(0,1)$ template, it follows $F_{p}$ also 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 [7]), Th.3.1 follows. ∎

##  4\. Discussion

Before we conclude this paper, let us briefly discuss the meaning of both
Th.3.1 and Th.2.1 \- as well as their possible generalizations. As stated at
the Introduction, the results of this paper are motivated by the Thurston-
Nielsen Classification Theorem and, in more generality, the theory of
topological dynamics on surfaces. Therefore, in the same spirit we are
motivated to propose the following generalizations of Th.2.1 and Th.3.1:

######  Conjecture 1.

Let $p\in P$ be a trefoil parameter for the Rössler system, and recall that
for $v\in P$ we denote the corresponding vector field by $F_{v}$. Then, there
exists a neighborhood of $p$ in $P$, $O\subseteq P$, s.t. for every $v\in O$
the following is satisfied:

  * •

There exists a smooth vector field $G_{v}$, hyperbolic on its non-wandering
set, s.t. $F_{v}$ can be smoothly deformed on $\mathbf{R}^{3}$to $G_{v}$.

  * •

Every periodic trajectory $T_{1}$ of $G_{v}$ (save perhaps for one) can be
deformed by an isotopy of $\mathbf{R}^{3}$ to a periodic trajectory $T_{0}$ of
$F_{v}$. In particular, if $T_{1},T_{2}$ are two distinct periodic
trajectories for $G_{v}$, they cannot be deformed to the same $T_{0}$.

  * •

The Template generated by $G_{v}$, $T_{v}$, is a sub-template of the $L(0,1)$
Template. Consequentially, $F_{v}$ generates every knot type on $T_{v}$ (save
perhaps for one) as a non-constant periodic trajectory.

If correct, Conjecture 1 implies, essentially, that Th.3.1 and Th.2.1 both can
be generalized beyond the trefoil parameters in $P$. In particular, it would
imply that around trefoil parameters in $P$ the Rössler system is essentially
hyperbolic: that is, even though the actual dynamics of the Rössler system may
not necessarily be hyperbolic, it is at least qualitatively so. In other
words, if true, Conjecture $1$ implies the chaotic dynamics generated by
$F_{v}$ are distorted hyperbolic dynamics - which, in itself, implies the
Chaotic Hypotheses (see [18]) holds in some open region of the parameter space
for the Rössler system. Consequentially, Conjecture $1$ implies chaotic
dynamics are an essential part of the dynamics generated by the Rössler
system. As such, it would follow these complex dynamics cannot be easily
removed by an arbitrary smooth deformation of the vector field - hence the
vector field $G_{v}$ is an analogue the Pseudo-Anosov maps known from the
Thurston-Nielsen Classification Theorem.

\begin{overpic}[width=173.44534pt]{images/morecomp.png}
\put(380.0,245.0){$P_{In}$} \put(780.0,460.0){$P_{Out}$} \end{overpic} Figure
20. A heteroclinic knot more complex than a trefoil.

Another remark which must be made is that the $L(0,1)$ Template is not a
universal template (see [7]) - that is, it does not contain every knot type.
Motivated by this fact, we conjecture Th.2.1 and Th.3.1 can be generalized to
other heteroclinic parameters in $P$ (see Def.1.1) - provided they generate
heteroclinic knots complex at least like the trefoil. For example, consider
the heteroclinic knot in Fig.20. If Th.3.1 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 propose the
following conjecture:

######  Conjecture 2.

Let $p\in P$ be a heteroclinic parameter which generates a heteroclinic knot,
$\Lambda\subseteq S^{3}$ (see Def.1.1 and the discussion following it). Then,
the more $\Lambda$ is knotted with itself, the more knot types are in the
template $T_{p}$ associated with $F_{p}$ \- and in particular, every knot type
on $T_{p}$ (save perhaps for one) is realized as a periodic trajectory for
$F_{p}$.

In other words, Conjecture $2$ can be heuristically stated as follows - the
more complex the heteroclinic knot $\Lambda$ is, the more complex are the
dynamics of the corresponding Rössler system. If true, both Conjecture 1 and
Conjecture 2 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 (see, for example, [21],[27],[10]). The second observation is that the same
is true for the associated templates - whose topology also changes with the
variation of $(a,b,c)$, thus forcing the creation of more symbols in the first
return map of the attractor (see [25],[10]).

Finally, before we conclude this paper, it must be remarked that both in this
paper and in [29] the question of the existence (or non-existence) of the
numerically observed Rössler attractor was left unaddressed. In a future paper
it will be proven that given a parameter $p\in P$, provided the invariant
manifolds of the fixed point $P_{Out}$ satisfy a certain condition (which was
observed numerically), there exists an attractor for the flow. Moreover, the
topology of that attractor and the dynamics around it will be studied -
namely, we will prove the dynamics around (and beyond) the said attractor can
be reduced to a one-dimensional model: the Logistic Family. As such, in
addition to giving an analytic explanation to several well-observed numerical
phenomena (see, for example, [21]), these results would showcase the
importance of one-dimensional dynamical systems to the study of the Rössler
system. Additionally, these results can also be considered as a step towards
generalizing the Betsvina-Handel Algorithm (see [9]) to three-dimensional
flows - thus in themselves being a step towards the proof of Conjecture $1$.

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