# Universal bounds on the entropy of toroidal attractors

P. Montealegre Macías  Facultad de Ciencias Matemáticas, Universidad
Complutense de Madrid, 28040 Madrid, España<EMAIL_ADDRESS>and  J.J.
Sánchez-Gabites  Facultad de Ciencias Matemáticas, Universidad Complutense de
Madrid, 28040 Madrid, España<EMAIL_ADDRESS>
###### Abstract.

A toroidal set is a compactum $K\subseteq\mathbb{R}^{3}$ which has a
neighbourhood basis of solid tori. We study the topological entropy of
toroidal attractors $K$, bounding it from below in terms of purely topological
properties of $K$. In particular, we show that for a toroidal set $K$, either
any smooth attracting dynamics on $K$ has an entropy at least $\log 2$, or (up
to continuation) $K$ admits smooth attracting dynamics which are stationary
(hence with a zero entropy).

###### Key words and phrases:

Attractor, toroidal set, entropy

###### 2020 Mathematics Subject Classification:

Primary 57K99, 37B35, 37E99; Secondary 55N99

The second author is funded by the Spanish Ministerio de Ciencia e Innovación
through grant PGC2018-098321-B-I00 and the Ramón y Cajal programme
(RYC2018-025843-I)

##  1\. Introduction

It is well known that attractors can be very complicated both topologically
and dynamically. One wonders to what extent these two sorts of complexity are
related, and in particular whether “topological strangeness” of a compactum
$K$ alone may already force a certain degree of complexity on _any_ possible
attracting dynamics on $K$. With the usual interpretation that a strictly
positive entropy $h$ is indicative of complicated dynamics, we are therefore
interested in bounds of the form

(1.1) $h(f|_{K})\geq\log p(K)>0$

where $f$ is any $\mathcal{C}^{\infty}$ diffeomorphism having the compactum
$K$ as an attractor and, crucially, $p(K)$ depends _only_ on $K$ and _not_ on
$f$. In other words, the bound $\log p(K)$ is universal across all
$\mathcal{C}^{\infty}$ attracting dynamics on a given set $K$.

We shall argue (Subsection 2.4) that, in a certain sense, the simplest class
of compacta $K$ for which universal bounds of the form (1.1) are possible is
that of _toroidal_ sets. These are compacta $K\subseteq\mathbb{R}^{3}$ which
have a neighbourhood basis comprised of solid tori ([1]). These tori can wind
inside each other and be knotted, and so $K$ will usually be a very
complicated continuum. Among toroidal sets one finds every smooth or polygonal
knot, many wild knots, the usual embeddings of $n$–adic solenoids and
generalized solenoids (where $n$ is replaced with a sequence $\\{n_{k}\\}$),
knotted solenoids, every inverse limit of a self map of $\mathbb{S}^{1}$ (see
[20, §3, pp. 375ff.]), classical continua such as the Whitehead continuum,
etc.

To each toroidal set $K$ (no dynamics yet) one can assign a collection of
prime numbers $p_{i}\geq 2$ called its _prime divisors_ ([3]). These capture
purely topological properties of $K$; roughly speaking, they are related to
the amount of “self-winding” of $K$. A toroidal set may have any number of
prime divisors, possibly none or possibly infinite.

In order to analyze dynamics on toroidal sets we shall introduce the notion of
the _geometric degree_ $d_{\mathcal{N}}(f;K)$ of a local homeomorphism $f$
which leaves a toroidal set $K$ invariant. We shall concentrate almost
exclusively on the case when $K$ is an attractor for $f$. Then the geometric
degree $d_{\mathcal{N}}(f;K)$ is a positive integer which, very roughly,
counts the minimum number of “angular preimages” of a point. Unlike the
ordinary (homological) degree, where preimages are counted as $\pm 1$
depending on the local behaviour of the map, in the geometric degree each
preimage contributes $+1$ to the count. This makes it more sensitive than the
ordinary degree while still retaining the usual properties of the latter. In
particular it is invariant under homotopies or, more precisely, under
continuations in the sense of Conley: if $(f_{\lambda})_{\lambda\in[0,1]}$ is
a continuous family of local homeomorphisms, each having a toroidal attractor
$K_{\lambda}$ (also varying continuously with $\lambda$), the prime divisors
of the $K_{\lambda}$ and the geometric degrees
$d_{\mathcal{N}}(f_{\lambda};K_{\lambda})$ are independent of $\lambda$.

The geometric degree provides a link between entropy and prime divisors. On
the one hand, if $K$ is an attractor for a local homeomorphism $f$ then the
prime divisors of the integer $d_{\mathcal{N}}(f;K)$, in the ordinary sense of
arithmetics, coincide with the prime divisors $\\{p_{i}\\}$ of the toroidal
set $K$ (Theorem 3.9) and so in particular
$d_{\mathcal{N}}(f;K)\geq\prod_{i}p_{i}$. On the other, when $f$ is
$\mathcal{C}^{\infty}$ the geometric degree provides the lower bound
$h(f|_{K})\geq\log d_{\mathcal{N}}(f;K)$ for the entropy (Theorem 5.1).
Therefore

(1.2) $h(f|_{K})\geq\log d_{\mathcal{N}}(f;K)\geq\log\prod_{i}p_{i}.$

Since the prime divisors $p_{i}$ depend only on $K$ but not on the dynamics,
this bound is of the desired form (1.1) with $p(K)=\prod_{i}p_{i}$. In
particular as soon as $K$ has at least one prime divisor we get
$h(f|_{K})\geq\log 2>0$. Notice that the bound $\log 2$ is universal not only
across all $\mathcal{C}^{\infty}$ attracting dynamics on a given $K$ but in
fact across all toroidal attractors with at least one prime divisor. In this
sense it is sharp, since it is attained by the standard embedding of the
dyadic solenoid in $\mathbb{R}^{3}$ with its usual dynamics.

If a toroidal attractor $K$ has no prime divisors then (1.2) reduces to the
trivial bound $h(f|_{K})\geq\log 1=0$ and, given the universal nature of the
bound, the (somewhat wild) question arises of whether $K$ actually supports
some attracting dynamics with a zero entropy. Since the geometric degree and
the prime divisors are invariant under continuation, we may also ask whether
$K$ can be continued to an attractor with a zero entropy. It turns out that
the answer is in the affirmative: we shall prove that if $K$ is a toroidal
attractor with no prime divisors, then it can be continued to a smooth knot
$\gamma$ which is an attractor with stationary dynamics (Theorem 4.1). For
this result the use of the geometric degree instead of the ordinary one is
crucial.

Summing up, up to continuation and for smooth dynamics, we obtain a neat
alternative:

######  Theorem 1.1.

Let $K$ be a toroidal attractor for a $\mathcal{C}^{\infty}$ diffeomorphism of
$\mathbb{R}^{3}$.

  * (i)

If $K$ has some prime divisor, then $K$ and all of its $\mathcal{C}^{\infty}$
toroidal continuations have an entropy at least $\log 2$.

  * (ii)

If $K$ has no prime divisors, then $K$ has a $\mathcal{C}^{\infty}$
continuation to a smooth knot which is an attractor with stationary dynamics
(hence with a zero entropy).

We finally discuss briefly the outline of the paper. In Section 2 we recall
some definitions and in particular that of the geometric index of a solid
torus inside another. This was introduced by Schubert ([30]) to study knots
and is basic for our constructions. In Section 3 we define the prime divisors
of a toroidal set $K$ and the geometric degree $d_{\mathcal{N}}(f;K)$, and
relate the two when $K$ is an attractor. We also prove that they are invariant
under continuation. In Section 4 we show that a toroidal attractor $K$ with no
prime divisors can be continued to a smooth knot. Section 5 relates the
entropy of a toroidal attractor to the geometric degree and hence also to the
prime divisors of the attractor. It is here where smoothness assumptions on
the dynamics are needed for the first time in order to apply a bound of Yomdin
on the entropy. Section 6 contains a comparison between the geometric and the
ordinary degree as well as an open question related to the smoothness
assumption on the dynamics.

We have included two appendices. Appendix A contains some technicalities about
the definition of the geometric index. Appendix B is a brief outline of
Yomdin’s bound on entropy, tailored to the very particular case we use in this
paper.

##  2\. Some background

This section gathers some definitions and results that are needed for the rest
of the paper. The most important are the definition and properties of the
geometric index in Subsection 2.2.

###  2.1. Tame solid tori

A solid torus $T$ is a topological space homeomorphic to
$\mathbb{D}^{2}\times\mathbb{S}^{1}$; any homeomorphism
$h:\mathbb{D}^{2}\times\mathbb{S}^{1}\longrightarrow T$ is called a framing of
$T$. The image under $h$ of $\mathbb{D}^{2}\times\\{*\\}$ is called a
meridional disk of $T$ and its boundary (which is a simple closed curve in
$\partial T$) a meridian of $T$. The image under $h$ of the curve
$\\{0\\}\times\mathbb{S}^{1}$ is a called a core of $T$. One should bear in
mind that meridional disks can lie in a very crooked way inside $T$; picturing
them as being “radial” is misleading.

In dimensions three (and higher) there exist wild tori, and these are both
inconvenient and unnecessary for our purposes. Henceforth we will almost
always confine ourselves to working with tame tori only. Recall that a compact
subset $L\subseteq\mathbb{R}^{3}$ is tame if there exists a homeomorphism
$g:\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}$ which sends $L$ onto a
polyhedron, and semilocally tame if there exist an open neighbourhood $U$ of
$L$ and a homeomorphism $g:U\longrightarrow g(U)\subseteq\mathbb{R}^{3}$ which
sends $L$ onto a polyhedron. Evidently a tame set $L$ is also semilocally
tame, and a deep theorem of Moise states that the converse is also true ([24,
Theorem 3, p. 254]).

In the sequel we will make use of the following facts:

(i) Any solid torus $T\subseteq\mathbb{R}^{3}$ can be perturbed an arbitrarily
small quantity to make it tame. To prove this one picks a framing
$h:\mathbb{D}^{2}\times\mathbb{S}^{1}\longrightarrow T\subseteq\mathbb{R}^{3}$
and resorts to a classical approximation theorem (see for example [24, Theorem
2, p. 251]) which ensures that for any $\epsilon>0$ there is another embedding
$h^{\prime}:\mathbb{D}^{2}\times\mathbb{S}^{1}\longrightarrow\mathbb{R}^{3}$
that is $\epsilon$–close to $h$ and is piecewise linear, so in particular its
image $T^{\prime}$ is a polyhedral solid torus $\epsilon$–close to $T$.

(ii) Suppose $f$ is a homeomorphism of $\mathbb{R}^{3}$ or, more generally, a
local homeomorphism of $\mathbb{R}^{3}$, i.e. a homeomorphism
$f:U\longrightarrow f(U)\subseteq\mathbb{R}^{3}$ where $U$ is open in
$\mathbb{R}^{3}$. If $T\subseteq U$ is a tame solid torus, then $f(T)$ is also
tame. This follows because $f(T)$ is evidently semilocally tame and, as
mentioned above, it is therefore tame.

###  2.2. The geometric index

We recall the notion of the geometric index of a solid torus inside another
one. This was introduced by Schubert [30, §9] (who called it “order”) and will
be essential in the paper. The original definition and results by Schubert are
set up for polyhedral tori, but we need a purely topological version. This is
not entirely straightforward because of the existence of wild objects in $3$
dimensions, and sorting this out requires some nontrivial results from
geometric topology. We have deferred the technical details to Appendix A since
the geometric content of the definition is very intuitive.

We first motivate the definition informally. Let $T_{0}$ be a solid torus
containing a simple closed curve $\alpha$ in its interior. Let $\gamma$ be a
core of $T_{0}$, so that $\alpha$ is homologous to a multiple of $\gamma$; say
$\alpha=m\gamma$ with $m\in\mathbb{Z}$. The integer $m$ can be computed (up to
a sign) by counting the number of times that $\alpha$ intersects any
meridional disk $D$ of $T_{0}$. This has to be performed algebraically: each
point in $D\cap\alpha$ contributes $\pm 1$ depending on the sense in which
$\alpha$ crosses $D$ at that point. This algebraic count is independent of the
disk $D$ used to perform it. We shall call it the homological winding number
of $\alpha$ in $T_{0}$ and denote it by $m(\alpha\subseteq T_{0})$.

The geometric index of $\alpha$ inside $T_{0}$ is defined in a similar manner
but counting the points of intersection in $D\cap\alpha$ geometrically; i.e.
each contributes a $+1$ so the count is just the cardinality $|D\cap\alpha|$.
This depends on the disk $D$, and one defines the geometric index
$N(\alpha\subseteq T_{0})$ by minimizing $|D\cap\alpha|$ over all meridional
disks $D$. It is clear that $N(\alpha\subseteq T_{0})\geq|m(\alpha\subseteq
T_{0})|$, and the inequality can be strict. For example the Whitehead curve
shown in the left panel of Figure 1 has $m=0$ but geometric index $2$.

We now proceed to the formal definitions. For our purposes it is more
convenient to define the geometric index of a solid torus (rather than a
simple closed curve) inside another one, but the geometric motivation for the
definition is the one just outlined.

Let $T_{0}\subseteq\mathbb{R}^{3}$ be a solid torus and $T_{1}$ another solid
torus contained in the interior of $T_{0}$. We assume these to be tame. Let
$D$ be a meridional disk of $T_{0}$. We say that $D$ is transverse to $T_{1}$
if there exist a neighbourhood $N$ of $D$ in $T_{0}$ and a homeomorphism
$h:(N,N\cap T_{1})\longrightarrow(\mathbb{D}^{2},D_{1}\cup\ldots\cup
D_{r})\times[-1,1]$ where:

  * (i)

the $D_{i}$ are pairwise disjoint closed disks.

  * (ii)

$h$ carries $D$ onto $\mathbb{D}^{2}\times\\{0\\}$.

This definition is intended to provide a local model for the intersection
$D\cap T_{1}$. Observe that (ii) implies that $D\cap T_{1}$ consists of $r$
disjoint closed disks (the preimages of the $D_{i}\times\\{0\\}$ under $h$).
These are _not_ assumed to be meridional disks of $T_{1}$, in contrast to the
next definition.

######  Definition 2.1.

The geometric index of $T_{1}$ in $T_{0}$ is the minimum $r$ such that there
exists a meridional disk $D$ of $T_{0}$ which is transverse to $T_{1}$ and
such that $D\cap T_{1}$ consists of $r$ meridional disks of $T_{1}$. We denote
this number by $N(T_{1}\subseteq T_{0})$.

At this stage it is not even clear that $N(T_{1}\subseteq T_{0})$ be well
defined in general, since perhaps no meridional disk transverse to $T_{1}$
exists at all. However, for tame tori as we are considering the geometric
index is indeed well defined. Moreover, it has the following three fundamental
properties:

(P1) If $(T_{0},T_{1})$ and $(T^{\prime}_{0},T^{\prime}_{1})$ are homeomorphic
pairs, then $N(T_{1}\subseteq T_{0})=N(T^{\prime}_{1}\subseteq
T^{\prime}_{0})$.

(P2) $N(T_{1}\subseteq T_{0})=0$ if and only if there exists a tame ball $B$
such that $T_{1}\subseteq B\subseteq T_{0}$.

(P3) The geometric index is multiplicative: if $T_{2}\subseteq T_{1}\subseteq
T_{0}$, then

$N(T_{2}\subseteq T_{0})=N(T_{2}\subseteq T_{1})\cdot N(T_{1}\subseteq
T_{0}).$

The first property is direct from the definition, since the notions of
meridional disk and transversality are preserved by homeomorphisms. Proving
the other two involves some woodworking which is best done when the tori are
polyhedral. They are proved, under this assumption, by Schubert ([30,
Hilfssatz 3 and Satz 3, pp. 171 and 175]). We shall show in Appendix A how to
translate those results to our topological setting. Here we discuss briefly
(P2) because it is quite plausible and also provides a good illustration of
why the geometric index is more powerful than the winding number $m$. If the
geometric index $N(T_{1}\subseteq T_{0})$ is zero, there exists a meridional
disk of $T_{0}$ which does not meet $T_{1}$. Then cutting $T_{0}$ along this
meridional disk produces a ball $B$ between $T_{1}$ and $T_{0}$. Conversely,
should such a ball $B$ exist, one can shrink it inwards by an ambient isotopy
$G_{t}$ of $T_{0}$ to a tiny size and find a meridional disk $D$ disjoint from
$G_{1}(B)$. Running the isotopy in the reverse produces a meridional disk
disjoint from $B$ and hence from $T_{1}$. Notice that (P2) is certainly not
true for the winding number $m$, and again the Whitehead curve of Figure 1 is
a classical example.

We finish by returning to the index $N(\alpha\subseteq T_{0})$ of a simple
closed curve $\alpha$ inside a solid torus $T_{0}$. This can be defined just
as $N(T_{1}\subseteq T_{0})$ with the obvious adaptations: in the local model
of transversality one replaces the disks $D_{i}$ with points $p_{i}$ and in
Definition 2.1 one removes the condition that the $D_{i}$ be meridional disks
of $T_{0}$. The resulting definition agrees with the informal one given at the
beginning of this section. It can be shown that $N(\gamma\subseteq
T_{1})=N(T_{0}\subseteq T_{1})$ whenever $\gamma$ is a core of $T_{0}$. (In
fact in Schubert’s original work the geometric index is defined first for
curves, then for solid tori through this relation).

###  2.3. Dynamics

The following definitions are standard and can be found, for example, in [12]
or [21].

Attractors. Suppose $f$ is a homeomorphism. A compact set $N$ such that
$f(N)\subseteq{\rm int}\ N$ is called a trapping region for $f$. The attractor
defined by that trapping region is the set $K:=\bigcap_{n\geq 0}f^{n}(N)$.

A set $P$ is positively invariant if $f(P)\subseteq P$ and invariant if
$f(P)=P$. An attractor $K$ has a neighbourhood basis of compact positively
invariant sets; namely the $\\{f^{n}(N)\\}_{n\geq 0}$. The attractor $K$
itself is invariant.

If the forward orbit of a point $x$ enters $N$ then it converges to $K$
asymptotically, this meaning that for every neighbourhood $V$ of $K$ there
exists $n_{0}$ such that $f^{n}(x)\in V$ for $n\geq n_{0}$. The set of points
with this property is called the basin of attraction of $K$ and is denoted by
$\mathcal{A}(K)$. It is an open, invariant neighbourhood of $K$. One can
easily check that in fact not only points in $\mathcal{A}(K)$ are attracted by
$K$, but also compact sets as well. Explicitly: for every neighbourhood $V$ of
$K$ and every compact set $C\subseteq\mathcal{A}(K)$ there exists $n_{0}$ such
that $f^{n}(C)\subseteq V$ for all $n\geq n_{0}$.

An attractor can also be defined intrinsically (i.e. without reference to a
trapping region) as a compact invariant set which attracts nearby points and
has a neighbourhood basis of positively invariant sets. In turn, the latter
condition can be replaced by requiring that $K$ be stable in the sense of
Lyapunov. Thus the type of attractors we are considering are sometimes called
stable attractors.

In this paper we will consider dynamics generated by a local homeomorphism of
$\mathbb{R}^{3}$. By this we mean a map $f$ defined on an open subset
$U\subseteq\mathbb{R}^{3}$ and which is a homeomorphism onto its image
$f(U)\subseteq\mathbb{R}^{3}$. By the invariance of domain theorem it suffices
to require that $f$ be continuous and injective. The definition of an
attractor given above generalizes immediately to this situation.

Continuations. We now recall the notion of a “continuation” introduced by
Conley [6], tayloring the definition to our particularly simple case of
attracting dynamics.

Suppose we have a parametrized family of local homeomorphisms $f_{\lambda}$
all defined on some open set $U\subseteq\mathbb{R}^{3}$. Here $\lambda$ ranges
in some set of parameters which for definiteness we shall take to be the
interval $[0,1]\subseteq\mathbb{R}$. Formally, we consider a continuous map
$f:[0,1]\times U\longrightarrow\mathbb{R}^{3}$ such that every partial mapping
$f_{\lambda}:U\longrightarrow\mathbb{R}^{3}$ defined by
$f_{\lambda}(x):=f(\lambda,x)$ is injective (hence a local homeomorphism).

Now let $[0,1]\ni\lambda\longmapsto K_{\lambda}$ be a map where each
$K_{\lambda}$ is an attractor for $f_{\lambda}$. We want to provide a
reasonable definition of “continuity” of this map. Fix some
$\lambda_{0}\in[0,1]$ and let $N$ be a trapping region for $K_{\lambda_{0}}$,
so that $f_{\lambda_{0}}(N)\subseteq{\rm int}\ N$. Evidently the latter
inclusion still holds for $\lambda$ close enough to $\lambda_{0}$, and so $N$
contains an attractor for $f_{\lambda}$. We say that $\lambda\longmapsto
K_{\lambda}$ is continuous at $\lambda_{0}$ if there is a neighbourhood
$I\subseteq[0,1]$ of $\lambda_{0}$ such that for every $\lambda\in I$ the
attractor $K_{\lambda}$ is precisely the one determined by the trapping region
$N$ under the dynamics $f_{\lambda}$; that is, $K_{\lambda}:=\bigcap_{n\geq
0}f_{\lambda}^{n}(N)$. It is easy to check that this condition is independent
of $N$, although $I$ will usually depend on it. Of course, we say that
$\lambda\longmapsto K_{\lambda}$ is continuous if it is continuous at every
$\lambda_{0}\in[0,1]$. This _ad hoc_ definition should be reasonable on
intuitive grounds and will suffice for our purposes, but in fact one can set
up a topology in the collection of attractors of the $f_{\lambda}$ so that
this map is continuous in the ordinary sense of Topology.

A continuous mapping as just described is called a continuation, or a
continuation from $K_{\lambda=0}$ to $K_{\lambda=1}$. Each intermediate
$K_{\lambda}$ is also called a continuation of $K_{\lambda=0}$.

Entropy. We also recall the definition of “topological entropy”. For this
purpose, we consider a compact metric space, $(X,\operatorname{d})$, and a
continuous map $g:X\longrightarrow X$.

For each $n\geq 0$ we define the metric

$\operatorname{d}_{n}^{g}(x,y):=\max_{0\leq i\leq
n}\\{\operatorname{d}(g^{i}(x),g^{i}(y))\\}$

and, for a given $x\in X$ and the open ball $B(x,\epsilon):=\\{y\in X\ |\
\operatorname{d}(x,y)<\varepsilon\\}$, the $(n,\epsilon)$-dynamical ball

$B_{g}(x,\epsilon,n)=\bigcap_{i=0}^{n}g^{-i}(B(x,\epsilon)).$

The continuity of $g$ ensures that $B_{g}(x,n,\epsilon)$ is open. Since $X$ is
compact, for any $\epsilon>0$ there is a minimum number of
$(n,\epsilon)$-dynamical balls needed to cover $X$. We denote this number by
$S_{\operatorname{d}}(g,n,\epsilon)$. It can be interpreted as the minimum
amount of initial conditions which, at a scale $\epsilon$ and up to time $n$,
are representative of all possible trajectories of the system.

Now, the topological entropy of $g$ is defined as the following double limit:

$h_{\operatorname{d}}(g):=\lim_{\varepsilon\rightarrow
0}\limsup_{n}\frac{1}{n}\log{S_{\operatorname{d}}(g,n,\varepsilon)}.$

Although the definition of $h_{\operatorname{d}}$ involves the distance
$\operatorname{d}$, it is actually invariant among equivalent distances which
define the same topology. This can be proved explicitly or by means of a
purely topological definition of the entropy that does not involve distances
(as it was defined originally). Henceforth we will omit the distance
$\operatorname{d}$ from the notation.

###  2.4. Why toroidal sets?

It was mentioned in the Introduction that toroidal sets are the simplest for
which universal bounds of the form (1.1) are possible. The following example
justifies this. For part (iv), recall that a compactum
$K\subseteq\mathbb{R}^{n}$ is called cellular if it has a neighbourhood basis
comprised of cells; that is, of sets homeomorphic to the standard closed
$n$–ball in $\mathbb{R}^{n}$.

######  Example 2.2.

Let $K\subseteq\mathbb{R}^{n}$. If

  * (i)

$K$ is an attractor for a flow, or

  * (ii)

$K$ is a global attractor, or

  * (iii)

$K$ is an attractor in dimension $n\leq 2$, or

  * (iv)

$K$ is a cellular set,

then $K$ can be realized as an attractor with stationary dynamics and so with
a zero entropy.

###### Proof.

(i) One just needs to stop the flow on $K$. This is straightforward to do when
the flow is $\mathcal{C}^{1}$ (by multiplying its vectorfield by a nonnegative
function which vanishes precisely on $K$) but also true when the flow is
merely continuous; see [2] and references therein.

(ii) and (iv) Every global attractor is cellular and, conversely, every
cellular set can be realized as a (global) attractor of a flow which is
stationary on the set (these are results of Garay [13]).

(iii) An attractor in $\mathbb{R}^{n}$ has a finitely generated Čech
cohomology ([27, Theorem 1, p. 2827]) and then in dimensions $n\leq 2$ a
result of Günther and Segal ([16, Corollary 3, p. 326]) reduces the situation
to that of flows. ∎

######  Remark 2.3.

Regarding smoothness of the dynamics, in case (i) the slowed down flow can
evidently be made as smooth as the original flow, and in the remaining cases
in dimension $n\neq 4$ the dynamics can be made $\mathcal{C}^{\infty}$. This
follows from results by Grayson, Norton and Pugh ([14] and [26]).

Notice that the example shows that several well known strange attractors (for
instance, the Lorenz or Hénon attractors) can also be realized as attractors
with stationary dynamics. Thus, “dynamical strangeness” is not always
necessary for “topological strangeness”.

Bearing in mind our goal of obtaining positive bounds on the entropy which are
universal across all attracting dynamics on a given compactum $K$, the
preceding example justifies that we focus on discrete dynamical systems and on
dimension $n$ at least $3$. Also, we need to go beyond cellular sets. If one
regards cells as handlebodies of genus zero, a natural next step in complexity
consists in considering compacta $K\subseteq\mathbb{R}^{3}$ that have a
neighbourhood basis comprised of handlebodies of genus one; i.e. solid tori.
These are precisely toroidal sets.

###  2.5. Toroidal sets and toroidal attractors

A compactum $K\subseteq\mathbb{R}^{3}$ is toroidal if it is not cellular and
has a neighbourhood basis comprised of (not necessarily tame) solid tori
$\\{T_{k}\\}$. One can always choose the $\\{T_{k}\\}$ to satisfy the
following conditions:

  * (i)

Each $T_{k}$ is a tame solid torus.

  * (ii)

Each $T_{k+1}$ is contained in the interior of $T_{k}$.

  * (iii)

The geometric indices $N(T_{k+1}\subseteq T_{k})$ are all nonzero.

To show that these bases exist, first start with any neighbourhood basis
$\\{T_{k}\\}$ of $K$ satisfying (ii). This exists by the definition of a
toroidal set. Then one perturbs each $T_{k}$ to a tame torus $T^{\prime}_{k}$
by a perturbation of size $\epsilon_{k}\longrightarrow 0$ chosen inductively
to ensure that the tame $T^{\prime}_{k}$ still contains $T_{k+1}$ in its
interior and is contained in the interior of $T^{\prime}_{k-1}$. The new tame
tori $\\{T^{\prime}_{k}\\}$ for a neighbourhood basis of $K$ which satisfy (i)
and (ii). Finally, condition (iii) can be achieved by discarding finitely many
of the $T^{\prime}_{k}$. Indeed, for each $k$ such that
$N(T^{\prime}_{k+1}\subseteq T^{\prime}_{k})=0$ we have (by property (P2) of
the geometric index) a ball $T^{\prime}_{k+1}\subseteq B_{k}\subseteq
T^{\prime}_{k}$. Thus if $N(T^{\prime}_{k+1}\subseteq T^{\prime}_{k})=0$ for
infinitely many $k$ then we have $\\{B_{k}\\}$ a neighbourhood basis for $K$
comprised of balls, so $K$ would be cellular. This contradicts the definition
of a toroidal set. Hence $N(T^{\prime}_{k+1}\subseteq T^{\prime}_{k})=0$ for
finitely many $k$, and so by discarding these we may achieve the three
conditions enumerated above. From now on whenever we speak of “a basis” of a
toroidal set we will always mean a neighbourhood basis that satisfies the
conditions enumerated above.

When a toroidal set $K$ is an attractor for a local homeomorphism $f$ there is
a natural way of constructing bases $\\{T_{k}\\}$. Since $K$ is toroidal and
its basin of attraction is an open neighbourhood of $K$, it contains some
solid torus which is a neighbourhood of $K$. This compact set is attracted by
$K$, and so there exists $n_{0}$ such that $f^{n}(T)\subseteq{\rm int}\ T$ for
every $n\geq n_{0}$. Thus
$\\{T_{k}\\}:=\\{T,f^{n_{0}}(T),f^{2n_{0}}(T),\ldots\\}$ is a neighbourhood
basis of $K$ comprised of nested solid tori. If $T$ is taken to be tame (this
can always be done by perturbing it to a polyhedral torus) then all the
iterates $f^{kn_{0}}(T)$ are semilocally tame and hence tame. Thus
$\\{T_{k}\\}$ satisfies properties (i) to (iii) listed above. It has the
additional crucial property that each pair $(f^{(k+1)n_{0}}(T),f^{kn_{0}}(T))$
is homeomorphic to $(T,f^{n_{0}}(T))$ and so the geometric indices
$N(T_{k+1}\subseteq T_{k})$ are all equal. We will call a basis constructed in
this manner a dynamically generated basis.

######  Remark 2.4.

A toroidal attractor has a neighbourhood basis of solid tori (because it is
toroidal) and a neighbourhood basis of positively invariant sets (because it
is an attractor), but there is in principle no guarantee that it has a basis
of neighbourhoods which satisfy simultaneously both conditions; i.e. which are
positively invariant solid tori. We do not know if this is generally true.

We conclude by observing that not every toroidal set can be realized as an
attractor, and characterizing topologically which can is an open problem. For
example, (i) any smooth knot in $\mathbb{R}^{3}$ can be realized as a
(toroidal) attractor with stationary dynamics, but (ii) there are toroidal
knots which are smooth everywhere except at a single point and cannot be
realized as attractors whatsoever; however (iii) there are toroidal knots
which are nowhere smooth (in fact, they are wild everywhere in the sense of
Geometric Topology) and they can again be realized as attractors with
stationary dynamics. Thus a rather natural geometric gradation of complexity
(smooth everywhere, smooth but at a single point, smooth nowhere) does not
have a consistent dynamical counterpart.

##  3\. Prime divisors and the geometric degree

In this section we associate to each toroidal set a collection of prime
numbers (possibly empty or infinite) called its prime divisors. They capture
purely topological information about the self-winding of $K$. We also
associate to each local homeomorphism between two toroidal sets a rational
number called its geometric degree. This is then particularized to the case of
a toroidal attractor of a local homeomorphism. Some motivation for the
definitions to come can be found in Section 6.

###  3.1. Prime divisors

Let $K$ be a toroidal set and $\\{T_{k}\\}$ a basis for $K$ as described in
Subsection 2.5.

######  Definition 3.1.

A prime divisor of $K$ is a prime number $p\geq 2$ which divides
$N(T_{k+1}\subseteq T_{k})$ for infinitely many $k$.

By multiplicativity of the geometric index and the primality of $p$ this is
equivalent to requiring that the number $N(T_{k}\subseteq T_{1})$ contains
arbitrarily large powers of $p$ as $k\rightarrow+\infty$. A yet equivalent
condition is that for any fixed $k_{1}$, the numbers $N(T_{k}\subseteq
T_{k_{1}})$ should contain arbitrarily large powers of $p$ as
$k\rightarrow+\infty$.

######  Proposition 3.2.

Being a prime divisor is independent of the basis $\\{T_{k}\\}$.

###### Proof.

Let $\\{T_{k}\\}$ and $\\{T^{\prime}_{\ell}\\}$ be two bases for $K$ and
suppose that $p$ satisfies Definition 3.1 for $\\{T_{k}\\}$. We show that it
also satisfies the definition for $\\{T^{\prime}_{\ell}\\}$. Pick $k_{1}$ so
that $T_{k_{1}}\subseteq T^{\prime}_{1}$. Now, for a given power $p^{n}$ of
$p$ let $k\geq k_{1}$ be big enough so that $p^{n}|N(T_{k}\subseteq
T_{k_{1}})$. Finally, pick $\ell$ so that $T^{\prime}_{\ell}\subseteq T_{k}$.
Then by multiplicativity of the geometric index applied to
$T^{\prime}_{1}\supseteq T_{k_{1}}\supseteq T_{k}\supseteq T^{\prime}_{\ell}$
we have $N(T_{k}\subseteq T_{k_{1}})|N(T^{\prime}_{\ell}\subseteq
T^{\prime}_{1})$. Thus $p^{n}|N(T^{\prime}_{\ell}\subseteq T^{\prime}_{1})$
and so $N(T_{\ell}\subseteq T_{1})$ contains arbitrarily large powers of $p$
as $\ell\rightarrow+\infty$, as was to be shown. ∎

The prime divisors of a toroidal set were first defined in [3] in a less
elementary fashion. The definition given here is equivalent.

######  Example 3.3.

(1) Suppose $K\subseteq\mathbb{R}^{3}$ is a smooth knot. Picking a closed
tubular neighbourhood $T$ of $K$ and a diffeomorphism
$(T,K)\cong(\mathbb{D}^{2}\times\mathbb{S}^{1},0\times\mathbb{S}^{1})$ it is
straightforward to construct a neighbourhood basis of $K$ which consists of
nested, solid tori such that the geometric index of each consecutive pair is
$1$. Thus any smooth knot is a toroidal set with no prime divisors. The same
holds for a polygonal knot taking pl regular neighbourhoods instead of tubular
neighbourhoods.

(2) Suppose $K$ is the intersection of a nested family of solid tori
$\\{T_{i}\\}$ such that each $T_{i+1}$ winds monotonically (i.e. without
doubling back) $n_{i}\geq 1$ times inside $T_{i}$. The monotonicity condition
ensures that there exists a meridional disk of $T_{i}$ which intersects
$T_{i+1}$ along $n_{i}$ disks (so $N(T_{i+1}\subseteq T_{i})\leq n_{i}$) and
at each of these intersections $T_{i+1}$ crosses the meridional disk in the
same sense (so $m(T_{i+1}\subseteq T_{i})=n_{i}$). Thus $N(T_{i+1}\subseteq
T_{i})=n_{i}$, and so the prime divisors of $K$ are exactly those prime
numbers that divide infinitely many of the $n_{i}$. In particular when
$n_{i}=n$ (as in the standard embedding of an $n$–adic solenoid) the prime
divisors of $K$ are the prime divisors of $n$ without multiplicity.

A more interesting example is provided by Whitehead continua.

######  Example 3.4.

Start with an unknotted solid torus $T_{1}$ and place a thinner one $T_{2}$ in
its interior along the black curve depicted in the left panel of Figure 1.
Then place an even thinner torus $T_{3}$ inside $T_{2}$ following the same
pattern (by this we mean that $(T_{2},T_{3})$ is homeomorphic to
$(T_{1},T_{2})$) and so on. This produces a nested sequence of solid tori
$\\{T_{k}\\}$ whose intersection $K$ is a toroidal set called a Whitehead
continuum, an example of which is shown in the right panel of Figure 1. This
prescription only determines the isotopy class of the core of $T_{k+1}$ inside
$T_{k}$, so $K$ is far from being uniquely determined.

To compute the prime divisors of $K$ notice that $N(T_{k+1}\subseteq
T_{k})=N(T_{2}\subseteq T_{1})$ by construction (and invariance of the
geometric index under homeomorphisms). Thus the prime divisors of $K$ are
exactly the prime numbers that divide $N(T_{2}\subseteq T_{1})$. It is clear
from the drawing that $N(T_{2}\subseteq T_{1})\leq 2$ since there are obvious
meridional disks of $T_{1}$ which intersect $T_{2}$ in exactly two points. In
fact one has $N(T_{2}\subseteq T_{1})=2$; i.e. no meridional disk of $T_{1}$
transverse to $T_{2}$ intersects it in less than two disks. This seems
intuitively reasonable but is not completely trivial to prove. A quick
argument goes as follows. Since the geometric index $N(T_{2}\subseteq T_{1})$
and the homological winding number $m(T_{2}\subseteq T_{1})$ are both given by
counting intersections with a meridional disk, one with a sign and the other
without, they must differ in an even number. But $m(T_{2}\subseteq T_{1})$ is
clearly zero, so $N(T_{2}\subseteq T_{1})$ must be even and hence either $0$
or $2$. The core of $T_{2}$ is linked with a meridian of $T$ (in the sense of
knot theory) and therefore it cannot be contained in a ball in $T_{1}$, so
$N(T_{2}\subseteq T_{1})$ must be nonzero, showing that $N(T_{2}\subseteq
T_{1})=2$ indeed. Thus $K$ has exactly the prime divisor $p=2$.

$\text{core of }T_{2}$$K$$T_{1}$ Figure 1. Whitehead’s continuum

###  3.2. The geometric degree

Let $K$ and $K^{\prime}$ be two toroidal sets and $f$ a continuous map such
that $f(K)=K^{\prime}$ and $f$ is injective on some open neighbourhood $U$ of
$K$. By invariance of domain this implies that $f$ is a homeomorphism onto the
open set $U^{\prime}=f(U)$.

Let $\\{T_{k}\\}$ and $\\{T^{\prime}_{\ell}\\}$ be bases for $K$ and
$K^{\prime}$ respectively. Notice that for large enough $k$ the tori $T_{k}$
are contained in the domain of $f$ and in fact $\\{f(T_{k})\\}$ is also a
basis of $K^{\prime}$: tameness of $f(T_{k})$ is ensured by the discussion at
the end of Subsection 2.1 while the invariance of the geometric index under
homeomorphisms ensures that $N(f(T_{k+1})\subseteq
f(T_{k}))=N(T_{k+1}\subseteq T_{k})\neq 0$.

Let $k$ be big enough so that $T_{k}$ is contained in the domain of $f$ and
$f(T_{k})\subseteq T^{\prime}_{1}$. We define a rational number by

(3.1) $d:=\frac{N(f(T_{k})\subseteq T^{\prime}_{1})}{N(T_{k}\subseteq
T_{1})}\in\mathbb{Q}.$

The denominator is nonzero because $N(T_{k}\subseteq T_{1})$ is the product of
the geometric indices of the consecutive pairs $T_{2}\subseteq T_{1}$,
$T_{3}\subseteq T_{2}$, etc. and these are nonzero by definition of a basis.

######  Proposition 3.5.

$d$ does not depend on $k$.

###### Proof.

We provisionally denote $d$ by $d_{k}$ to reflect its dependence on $k$.
Suppose that the condition $f(T_{k})\subseteq T^{\prime}_{1}$ holds. Then it
also holds with $k+1$ instead of $k$, and to prove the proposition it suffices
to show that $d_{k}=d_{k+1}$. We have

$d_{k+1}=\frac{N(f(T_{k+1})\subseteq T^{\prime}_{1})}{N(T_{k+1}\subseteq
T_{1})}.$

Inserting $f(T_{k})$ in the numerator and $T_{k}$ in the denominator using
multiplicativity gives

$d_{k+1}=\frac{N(f(T_{k+1})\subseteq T^{\prime}_{1})}{N(T_{k+1}\subseteq
T_{1})}=\frac{N(f(T_{k+1})\subseteq f(T_{k}))N(f(T_{k})\subseteq
T^{\prime}_{1})}{N(T_{k+1}\subseteq T_{k})N(T_{k}\subseteq T_{1})}.$

By the invariance of the geometric index under homeomorphisms
$N(f(T_{k+1})\subseteq f(T_{k}))=N(T_{k+1}\subseteq T_{k})$ and so a
cancellation occurs in the expression above, yielding $d_{k+1}=d_{k}$. ∎

We call $d$ the _geometric degree_ of $f$ with respect to the bases
$\\{T_{k}\\}$ and $\\{T^{\prime}_{\ell}\\}$ and denote it by
$d_{\mathcal{N}}(f;\\{T_{k}\\},\\{T^{\prime}_{\ell}\\})$. This degree is
nonzero. This follows readily from Proposition 3.6 below but can also seen
directly. If $d_{\mathcal{N}}(f;\\{T_{k}\\},\\{T^{\prime}_{\ell}\\})=0$ then
$N(f(T_{k})\subseteq T^{\prime}_{1})=0$ for every $k$. For every $\ell$ we may
choose $k$ large enough so that $f(T_{k})\subseteq T^{\prime}_{\ell}$, and
then $0=N(f(T_{k})\subseteq T^{\prime}_{1})=N(f(T_{k})\subseteq
T^{\prime}_{\ell})N(T^{\prime}_{\ell}\subseteq T^{\prime}_{1})$ implies
$N(f(T_{k})\subseteq T^{\prime}_{\ell})=0$ since the second factor is nonzero
by definition of a basis. But this implies that there is a ball $B_{k}$
between $f(T_{k})$ and $T^{\prime}_{\ell}$, and this being true for every
$\ell$ would imply that $K^{\prime}$ is cellular, a contradiction.

As with the usual degree, the geometric degree is multiplicative under
composition. Suppose we have three toroidal sets $K$, $K^{\prime}$,
$K^{\prime\prime}$ with bases $\\{T_{k}\\}$, $\\{T^{\prime}_{\ell}\\}$,
$\\{T^{\prime\prime}_{m}\\}$ and local homeomorphisms $f,g$ that carry $K$ to
$K^{\prime}$ and $K^{\prime}$ to $K^{\prime\prime}$ respectively.

######  Proposition 3.6.

The geometric degree is multiplicative:

$d_{\mathcal{N}}(g\circ
f;\\{T_{k}\\},\\{T^{\prime\prime}_{m}\\})=d_{\mathcal{N}}(f;\\{T_{k}\\},\\{T^{\prime}_{\ell}\\})d_{\mathcal{N}}(g;\\{T^{\prime}_{\ell}\\},\\{T^{\prime\prime}_{m}\\}).$

###### Proof.

Choose $\ell$ so that $g(T^{\prime}_{\ell})\subseteq T^{\prime\prime}_{1}$ and
then $k$ so that $f(T_{k})\subseteq T^{\prime}_{\ell}$. Then $(g\circ
f)(T_{k})\subseteq g(T^{\prime}_{\ell})$, so again using the multiplicativity
and invariance under homeomorphisms of the geometric index we have

$\displaystyle N((g\circ f)(T_{k})\subseteq T^{\prime\prime}_{1})$
$\displaystyle=N((g\circ f)(T_{k})\subseteq
g(T^{\prime}_{\ell}))N(g(T^{\prime}_{\ell})\subseteq T^{\prime\prime}_{1})$
$\displaystyle=N(f(T_{k})\subseteq
T^{\prime}_{\ell})N(g(T^{\prime}_{\ell})\subseteq T^{\prime\prime}_{1})$
$\displaystyle=\frac{N(f(T_{k})\subseteq
T^{\prime}_{1})}{N(T^{\prime}_{\ell}\subseteq
T^{\prime}_{1})}N(g(T^{\prime}_{\ell})\subseteq T^{\prime\prime}_{1})$
$\displaystyle=N(f(T_{k})\subseteq
T^{\prime}_{1})d_{\mathcal{N}}(g;\\{T^{\prime}_{\ell}\\},\\{T^{\prime\prime}_{m}\\}).$

Dividing through by $N(T_{k}\subseteq T_{1})$ gives

$\frac{N((g\circ f)(T_{k})\subseteq T^{\prime\prime}_{1})}{N(T_{k}\subseteq
T_{1})}=d_{\mathcal{N}}(f;\\{T_{k}\\},\\{T^{\prime}_{\ell}\\})d_{\mathcal{N}}(g;\\{T^{\prime}_{\ell}\\},\\{T^{\prime\prime}_{m}\\}).$

The left hand side is, by definition, $d_{\mathcal{N}}(g\circ
f;\\{T_{k}\\},\\{T^{\prime\prime}_{m}\\})$, proving the desired equality. ∎

Now we particularize to the case when $K^{\prime}=K$; i.e. $f$ is a local
homeomorphism which leaves a toroidal set $K$ invariant. Then there is a
natural choice of bases for the geometric degree of $f$; namely the same basis
$\\{T_{k}\\}$ of $K$ for its role both as a source and target space.

######  Proposition 3.7.

Let $f$ be a local homeomorphism which leaves a toroidal set $K$ invariant.
Then $d_{\mathcal{N}}(f;\\{T_{k}\\},\\{T_{k}\\})$ is independent of the basis
$\\{T_{k}\\}$.

###### Proof.

Let $\\{T_{k}\\}$ and $\\{T^{\prime}_{\ell}\\}$ be two bases for $K$. Observe
that by multiplicativity of the geometric degree

$d_{\mathcal{N}}({\rm
Id};\\{T^{\prime}_{\ell}\\},\\{T_{k}\\})d_{\mathcal{N}}({\rm
Id};\\{T_{k}\\},\\{T^{\prime}_{\ell}\\})=d_{\mathcal{N}}({\rm
Id};\\{T^{\prime}_{\ell}\\},\\{T^{\prime}_{\ell}\\})=1$

where the last equality is trivial from the definition. Writing $f={\rm
Id}\circ f\circ{\rm Id}$, again by multiplicativity

$d_{\mathcal{N}}(f;\\{T^{\prime}_{\ell}\\},\\{T^{\prime}_{\ell}\\})=d_{\mathcal{N}}({\rm
Id};\\{T^{\prime}_{\ell}\\},\\{T_{k}\\})d_{\mathcal{N}}(f;\\{T_{k}\\},\\{T_{k}\\})d_{\mathcal{N}}({\rm
Id};\\{T_{k}\\},\\{T^{\prime}_{\ell}\\})$

and the first and third terms on the right hand side cancel each other by the
previous paragraph. ∎

The previous proposition justifies the following definition:

######  Definition 3.8.

If $f$ is a local homeomorphism which leaves a toroidal set $K$ invariant and
$\\{T_{k}\\}$ is any basis for $K$, we call
$d_{\mathcal{N}}(f;\\{T_{k}\\},\\{T_{k}\\})\in\mathbb{Q}$ the geometric degree
of $f$ (on $K$) and denote it by $d_{\mathcal{N}}(f;K)$.

###  3.3. The case of attractors

We now discuss the case when $K$ is an attractor for $f$ (and not merely an
invariant set as in Proposition 3.7). The following theorem is the main result
in this section. It relates a purely topological property of $K$ (its prime
divisors) to the geometric degree of any attracting dynamics on it.

######  Theorem 3.9.

Let $K$ be a toroidal attractor for a local homeomorphism $f$. Then:

  * (i)

$K$ has only finitely many prime divisors $p_{i}$ (possibly none).

  * (ii)

$d_{\mathcal{N}}(f;K)$ is an integer whose prime divisors in the ordinary
sense of arithmetic are exactly those of $K$ (perhaps with multiplicities).

As a consequence $d_{\mathcal{N}}(f;K)\geq\prod_{i}p_{i}$, and
$d_{\mathcal{N}}(f;K)=1$ if and only if $K$ has no prime divisors.

###### Proof.

Let $T\subseteq\mathcal{A}(K)$ be a solid torus neighbourhood of $K$. Pick $r$
big enough so that $f^{r}(T)\subseteq T$ and set $n:=N(f^{r}(T)\subseteq T)$.
We are free to perform our computations in any convenient basis, and we choose
the basis $\\{T_{k}\\}$ generated from $T$ by using the dynamics; namely
$\\{T\supseteq f^{r}(T)\supseteq f^{2r}(T)\supseteq\ldots\\}$. Thus by
definition $T_{k+1}=f^{r}(T_{k})$, starting with $T_{1}:=T$.

(i) By construction each pair $T_{k+1}\subseteq T_{k}$ is homeomorphic to
$f^{r}(T)\subseteq T$, and so $N(T_{k+1}\subseteq T_{k})=N(f^{r}(T)\subseteq
T)=n$ for every $k$. Thus a prime $p$ divides infinitely many of the
$N(T_{k+1}\subseteq T_{k})$ if and only if it divides $n$; in other words, the
prime divisors of $K$ are precisely those of $n$. In particular, if $n=1$ then
$K$ has no prime divisors.

(ii) We first compute the geometric degree of $f^{r}$ instead of $f$. By
definition

$d_{\mathcal{N}}(f^{r};K)=\frac{N(f^{r}(T_{k})\subseteq
T_{1})}{N(T_{k}\subseteq T_{1})}=\frac{N(T_{k+1}\subseteq
T_{1})}{N(T_{k}\subseteq T_{1})}$

and, since $T_{k+1}\subseteq T_{k}$ by construction of the basis
$\\{T_{k}\\}$, we may interpolate a $T_{k}$ in the numerator and cancel with
the denominator to get $d_{\mathcal{N}}(f^{r};K)=N(T_{k+1}\subseteq T_{k})=n$.

By multiplicativity of the geometric degree we must have

$n=d_{\mathcal{N}}(f^{r};K)=(d_{\mathcal{N}}(f;K))^{r}.$

An elementary argument shows that if an integer ($n$, which is an integer
because it is a geometric index) has an $r$th root in $\mathbb{Q}$ (namely
$d_{\mathcal{N}}(f;K)$) then in fact the root is an integer, so we deduce
$d_{\mathcal{N}}(f;K)\in\mathbb{Z}$. Again using the equality
$n=(d_{\mathcal{N}}(f;K))^{r}$ we see that the prime factors of $n$ must be
exactly those of $d_{\mathcal{N}}(f;K)$. ∎

For later use we record here the following fact established in the proof of
Theorem 3.9:

######  Remark 3.10.

If $T\subseteq\mathcal{A}(K)$ is any torus neighbourhood of $K$ and $r$ is
such that $f^{r}(T)\subseteq T$, then
$d_{\mathcal{N}}(f;K)^{r}=N(f^{r}(T)\subseteq T)$ and the prime divisors of
$K$ are the prime divisors of $N(f^{r}(T)\subseteq T)$. When $T$ is positively
invariant this is trivial from the definitions, but as mentioned in Remark 2.4
we do not know if a toroidal attractor always has a positively invariant
neighbourhood which is a solid torus.

######  Example 3.11.

(1) A variation on the construction of Example 3.4 produces Whitehead continua
which are automatically attractors. First fix a pair $(T_{1},T_{2})$ with the
usual pattern. Now observe that there is a homeomorphism
$f:\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}$ that carries $T_{1}$ onto
$T_{2}$. This must be the case since both are unknotted solid tori, but it is
also easy to picture such an $f$. One starts with $T_{1}$, stretches it along
some direction to obtain an tube shaped like an ellipse, then twists it one
whole turn along its long axis, and finally folds it over. This makes the tube
resemble the pattern shown in Figure 1. It is then just a matter of shrinking
the diameter of the tube and moving it around to make it fit inside $T_{1}$.
This whole procedure defines an isotopy of $\mathbb{R}^{3}$ whose final stage
is the required homeomorphism $f$. Then setting $T_{k+1}:=f^{k}(T_{1})$ one
obtains a Whitehead continuum $K:=\bigcap_{k\geq 1}T_{k}$ which is by
construction an attractor for $f$. We emphasize that the choice of $f$ is far
from unique and can very well lead to different Whitehead continua or even to
the same continuum but with essentially different dynamics. For example
Garity, Jubran and Schori ([7], [8], [19]) show how to construct two
homeomorphisms $f$ and $f^{\prime}$ of $\mathbb{R}^{3}$ which both fit the
framework just described and even have the same attractor $K$ but have the
property that $f|_{K}$ is not transitive whereas $f^{\prime}|_{K}$ is
transitive (and in fact chaotic in the sense of Devaney).

(2) Whenever a local homeomorphism $f$ has a Whitehead continuum $K$ as an
attractor, its geometric degree must be $d_{\mathcal{N}}(f;K)=2,4,8,\ldots$
This is a consequence of Theorem 3.9 and the fact that $K$ has $2$ as its only
prime divisor. All possible powers of $2$ can appear: for the specific
construction of part (1) one has $d_{\mathcal{N}}(f;K)=2$ (because $T_{1}$ is
positively invariant and Remark 3.10.(1) applies with $r=1$), and then the
homeomorphisms $f^{2},f^{3},\ldots$ have the same continuum $K$ as an
attractor and have geometric degrees $4,8,\ldots$

###  3.4. Invariance under continuation

Recall the setup for continuations: $U\subseteq\mathbb{R}^{3}$ is some open
set and $f_{\lambda}:U\longrightarrow\mathbb{R}^{3}$ is a family of
continuous, injective maps (hence homeomorphisms onto their image) which
depends continuously on a parameter $\lambda\in[0,1]$. We assume that
$\lambda\longmapsto K_{\lambda}$ is a continuation of attractors for the
$f_{\lambda}$. We begin by analyzing the local situation:

######  Proposition 3.12.

Let $\lambda\longmapsto K_{\lambda}$ be a continuation of attractors and
suppose that $K_{\lambda_{0}}$ is toroidal. Then for $\lambda$ close enough to
$\lambda_{0}$, the attractor $K_{\lambda}$ is also toroidal. Moreover, its
prime divisors and the geometric degree of $f_{\lambda}$ at $K_{\lambda}$ are
the same as those at $\lambda_{0}$.

###### Proof.

Let $N$ be a trapping region for $f_{\lambda_{0}}$. Since $K_{\lambda_{0}}$ is
toroidal, there exists a (tame) solid torus $T\subseteq N$ which is a
neighbourhood of $K_{\lambda_{0}}$. This $T$ is contained in the region of
attraction of $K_{\lambda_{0}}$, so there exists an iterate $r\geq 1$ such
that $f^{r}_{\lambda_{0}}(T)\subseteq{\rm int}\ T$. Finally, pick another
trapping region $N^{\prime}$ for $K_{\lambda_{0}}$ contained in $T$. There is
an interval of parameters $I\subseteq[0,1]$ which is a neighbourhood of
$\lambda_{0}$ and such that for every $\lambda\in I$ we still have the
conditions (i) $f_{\lambda}(N)\subseteq{\rm int}\ N$,
$f_{\lambda}(N^{\prime})\subseteq{\rm int}\ N^{\prime}$ (that is, $N$ and
$N^{\prime}$ are still trapping regions), (ii) the continuation $K_{\lambda}$
derived from $N$ and $N^{\prime}$ is the same, and (iii)
$f_{\lambda}^{r}(T)\subseteq{\rm int}\ T$.

Since $N\supseteq T\supseteq N^{\prime}$, evidently

$\underbrace{\bigcap_{k\geq
0}f_{\lambda}^{k}(N)}_{=K_{\lambda}}\supseteq\bigcap_{k\geq
0}f_{\lambda}^{k}(T)\supseteq\underbrace{\bigcap_{k\geq
0}f_{\lambda}^{k}(N^{\prime})}_{=K_{\lambda}}.$

and so $K_{\lambda}=\bigcap_{k\geq 0}f^{k}_{\lambda}(T)$. The same computation
holds with $(f_{\lambda}^{r})^{k}$ instead of $f_{\lambda}^{k}$. Thus we see
that $K_{\lambda}$ is a toroidal set; in fact, it has the dynamical basis
$\\{(f_{\lambda}^{r})^{k}(T)\\}$. If we now show that
$N(f_{\lambda}^{r}(T)\subseteq T)$ is independent of $\lambda$ the theorem
will follow from Remark 3.10.(1).

Let $C:=f_{\lambda_{0}}^{r}(T)$. Pick any $\lambda\in I$ and define an isotopy
of $C$ inside $T$ by $h_{t}:C\longrightarrow T$ with
$h_{t}:=f^{r}_{\lambda_{0}+t(\lambda-\lambda_{0})}\circ f_{\lambda_{0}}^{-r}$
and $t\in[0,1]$. This isotopy actually takes place in the interior of $T$,
since $h_{t}(C)=f^{r}_{\lambda_{0}+t(\lambda-\lambda_{0})}(T)\subseteq{\rm
int}\ T$ because $\lambda_{0}+t(\lambda-\lambda_{0})\in I$ for every
$t\in[0,1]$ and condition (iii) holds on $I$. Trivially $h_{t}$ can be
extended to an open neighbourhood of $C$ in ${\rm int}\ T$; the same
definition given above provides the extension. Now a deep result of Edwards
and Kirby ([10, Corollary 1.2, p. 63]) ensures that the isotopy $h_{t}$
extends to an ambient isotopy, that is, there exists an isotopy
$H_{t}:T\longrightarrow T$ such that $h_{t}=H_{t}|_{C}$. Then $H_{1}$ is a
homeomorphism of $T$ which sends $C=f_{\lambda_{0}}^{r}(T)$ onto
$h_{1}(C)=f_{\lambda}^{r}(T)$, and by the invariance of the geometric index
under homeomorphisms we have $N(f^{r}_{\lambda_{0}}(T)\subseteq
T)=N(f^{r}_{\lambda}(T)\subseteq T)$ as desired. ∎

The previous proposition shows in particular that the set
$\\{\lambda\in[0,1]:K_{\lambda}\text{ is toroidal}\\}$ is open. In general it
need not be closed, so to obtain a global continuation theorem one needs to
place some extra assumption:

######  Theorem 3.13.

Let $\lambda\longmapsto K_{\lambda}$ be a continuation through toroidal sets
(i.e. each $K_{\lambda}$ is toroidal). Then all the $K_{\lambda}$ have the
same prime divisors and all the $f_{\lambda}$ have the same geometric degree
at $K_{\lambda}$.

###### Proof.

This is completely straightforward. Since each $K_{\lambda}$ is toroidal by
assumption, the previous proposition shows that the prime divisors of
$K_{\lambda}$ and the degree of $f_{\lambda}$ at $K_{\lambda}$ depend on
$\lambda$ in a locally constant manner. Since $[0,1]$ is connected, they must
be constant. ∎

##  4\. Toroidal attractors with no prime divisors

Suppose that $\gamma\subseteq\mathbb{R}^{3}$ is a smooth knot. We saw earlier
(Example 3.3.(1)) that $\gamma$ is a toroidal set with no prime divisors, and
it can be easily realized as an attractor for homeomorphism (even a
$\mathcal{C}^{\infty}$ diffeomorphism) of $\mathbb{R}^{3}$. Thus smooth knots
provide a particularly simple example of toroidal attractors with no prime
divisors. In this section we show that, up to continuation, this is the only
model for such attractors:

######  Theorem 4.1.

Let $K$ be a toroidal attractor for a homeomorphism $f$ of all
$\mathbb{R}^{3}$. Then $K$ has no prime divisors if and only if it can be
continued through toroidal attractors to a smooth knot $\gamma$. Moreover, the
dynamics on $\gamma$ can be taken to be stationary.

If $f$ is $\mathcal{C}^{\infty}$ then the continuation can also be made
$\mathcal{C}^{\infty}$.

######  Remark 4.2.

(1) Notice that the dynamical system $f$ is assumed to be a homeomorphism of
all of $\mathbb{R}^{3}$ and not only a local homeomorphism as in the previous
sections. Without this assumption it is easy to see that the theorem is false.

(2) The use of the geometric index (through the prime divisors of $K$) instead
of the homological winding number is crucial. To see this consider again
Example 6.2. The toroidal attractor $K$ has no homological prime divisors but
it cannot be continued to a smooth knot since it has $p=3$ as a geometric
prime divisor.

Implication $(\Leftarrow)$ of Theorem 4.1 is a direct consequence of Example
3.3.(1) and the invariance of prime divisors under continuation (Theorem
3.13). The remaining of this section is devoted prove the converse
implication. We will prove it in the $\mathcal{C}^{\infty}$ case. The
(slightly simpler) argument in the topological case follows essentially the
same steps and we will just make a couple of comments where appropriate.

Recall that a diffeotopy of $\mathbb{R}^{3}$ is a $\mathcal{C}^{\infty}$ map
$G:[0,1]\times\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}$ such that each
partial map $G_{t}$ is a diffeomorphism of $\mathbb{R}^{3}$ and $G_{0}={\rm
Id}$. A diffeotopy is supported on a set $U$ if each $G_{t}$ is the identity
outside $U$. This implies in particular that $G_{t}(U)=U$ for each $t$.
Diffeotopies can be concatenated; i.e. given two diffeotopies $G^{(1)}$ and
$G^{(2)}$ one can first perform $G^{(1)}_{2t}$ for $0\leq
t\leq\nicefrac{{1}}{{2}}$ and then apply $G^{(2)}$ to the end result of the
first diffeotopy, namely $G_{2t-1}^{(2)}\circ G_{1}^{(1)}$ for
$\nicefrac{{1}}{{2}}\leq t\leq 1$. This is generally not differentiable at
$t=\nicefrac{{1}}{{2}}$ but can be easily smoothed out (see [18, p. 111]).

We will use diffeotopies to produce $\mathcal{C}^{\infty}$ continuations as
follows. Suppose that $g=g_{0}$ is a diffeomorphism of $\mathbb{R}^{3}$ which
has an attractor $K$ with a trapping region $N$. If $G_{t}$ is a diffeotopy
supported on $N$ then $[0,1]\ni\lambda\longmapsto G_{\lambda}\circ g$ produces
a continuation of $g_{0}$, and the condition that $G$ be supported on $N$
implies that $g_{\lambda}(N)=G_{\lambda}\circ g(N)\subseteq G_{\lambda}({\rm
int}\ N)={\rm int}\ N$ for every $\lambda$. Thus $N$ is a trapping region
throughout the whole continuation, and in particular its maximal invariant
subset $K_{\lambda}$ is a continuation of $K$.

The proof of ($\Rightarrow$) of Theorem 4.1 requires two auxiliary lemmas. We
state them, then prove the theorem, and finally prove the lemmas. Recall that
two solid tori $T_{1}\subseteq{\rm int}\ T_{0}$ are concentric if
$\overline{T_{0}\setminus T_{1}}$ is homeomorphic (or equivalently
diffeomorphic, if the tori are smooth) to $\mathbb{T}^{2}\times[0,1]$. The
first auxiliary lemma states that given two concentric solid tori in
$\mathbb{R}^{3}$ we can drag the smaller one via an ambient an ambient
deformation until it matches the bigger one.

######  Lemma 4.3.

Let $T_{1}\subseteq{\rm int}\ T_{0}\subseteq\mathbb{R}^{3}$ be two concentric
smooth solid tori. There exists a diffeotopy $G$ of all $\mathbb{R}^{3}$ such
that:

  * (i)

$G_{t}(T_{1})\subseteq T_{0}$ for all $t\in[0,1]$.

  * (ii)

$G_{1}(T_{1})=T_{0}$.

Moreover, $G$ can be taken to be supported on any prescribed neighbourhood $U$
of $\overline{T_{0}\setminus T_{1}}$.

The second lemma essentially states that if a diffeomorphism of
$\mathbb{R}^{3}$ leaves a solid torus $T$ invariant and is homologically the
identity on its boundary, then it can be deformed to be the identity on $T$.

######  Lemma 4.4.

Let $T\subseteq\mathbb{R}^{3}$ be a differentiable solid torus. Let
$g:\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}$ be a diffeomorphism such that
$g(T)=T$ and $g|_{\partial T}$ induces the identity in $H_{1}(\partial
T;\mathbb{Z})$. Then there exists a diffeotopy $G$ of $\mathbb{R}^{3}$ such
that:

  * (i)

$G_{t}(T)=T$ for every $t\in[0,1]$.

  * (ii)

$G_{1}\circ g|_{T}={\rm Id}_{T}$.

Moreover, $G$ can be taken to be supported on any prescribed neighbourhood $U$
of $T$.

###### Proof of ($\Rightarrow$) in Theorem 4.1.

We assume that $K$ is a toroidal attractor for some diffeomorphism
$f:\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}$ and that $K$ has no prime
divisors. We are going to show that some power of $f$ can be continued to a
smooth curve with trivial dynamics.

Let $T$ be a smooth solid torus in $\mathcal{A}(K)$ which is a neighbourhood
of $K$. Choose a power $r$ such that $f^{r}(T)\subseteq{\rm int}\ T$. By
Remark 3.10 the prime divisors of $K$ are the prime divisors of
$N(f^{r}(T)\subseteq T)$, and so the latter must be $1$. Moreover, since $f$
is defined on all of $\mathbb{R}^{3}$, the tori $f^{r}(T)$ and $T$ are
equivalently knotted. A result of Edwards ([9, Theorem 3, p. 4]) then implies
that $f^{r}(T)$ and $T$ are concentric.

Set $g_{0}:=f^{2r}$, where the role of the extra factor $2$ in the exponent
will be clear shortly. This diffeomorphism will be the starting point of our
continuation of the dynamics. Notice that $g_{0}(T)\subseteq{\rm int}\ T$. Let
$T^{\prime}$ be a solid differentiable torus which is a thickening of $T$ thin
enough so that $g_{0}(T^{\prime})\subseteq{\rm int}\ T$ still.

Let $G$ be the diffeotopy given by Lemma 4.3 applied to $T_{0}=T$,
$T_{1}=g_{0}(T)$, and $U=T^{\prime}$. Then $g_{\lambda}:=G_{\lambda}\circ
g_{0}$ defines a continuation of $g_{0}$ such that $g_{1}(T)=T$ and
$T^{\prime}$ is a trapping region for each $g_{\lambda}$ because $G$ is
supported in $T^{\prime}$.

Claim. The map $g_{1}|_{\partial T}$ induces the identity in $H_{1}(\partial
T;\mathbb{Z})$.

Proof of claim. The homology group $H_{1}(\partial T;\mathbb{Z})$ is generated
by the homology classes $m$ and $\ell$ of a meridian and a longitude of
$\partial T$. The element $m$ is uniquely determined up to sign by the
condition that when included in $T$ it becomes zero. The element $\ell$
depends on the framing of $T$, but one can fix it up to sign by requiring it
to be nullhomologous in $\overline{\mathbb{R}^{3}\setminus T}$ (this is called
a preferred longitude in knot theory). Now, since $g_{1}$ is defined not only
on $\partial T$ but in all of $\mathbb{R}^{3}$ and leaves both $T$ and
$\overline{\mathbb{R}^{3}\setminus T}$ invariant, $(g_{1}|_{\partial
T})_{*}(m)$ and $(g_{1}|_{\partial T})_{*}(\ell)$ are again nullhomologous
when included in $T$ and $\overline{\mathbb{R}^{3}\setminus T}$ respectively,
so $(g_{1}|_{\partial T})_{*}(m)=\pm m$ and $(g_{1}|_{\partial
T})_{*}(\ell)=\pm\ell$ in $H_{1}(\partial T;\mathbb{Z})$.

Observe that both $g_{0}$ and $G_{1}$ are orientation preserving; the first
because it is an even power of a homeomorphism and the second because it is
isotopic to the identity. Thus $g_{1}$ is also orientation preserving, and so
are $g_{1}|_{T}$ and consequently $g_{1}|_{\partial T}$. This implies that the
determinant of the endomorphism $(g_{1}|_{\partial T})_{*}$ of $H_{1}(\partial
T;\mathbb{Z})$ must be positive. The computation from the previous paragraph
shows that in the basis $\\{m,\ell\\}$ the matrix of $(g_{1}|_{\partial
T})_{*}$ is diagonal with $\pm 1$ entries; so we conclude that these must
either both be positive or both be negative. Thus to prove that
$(g_{1}|_{\partial T})_{*}={\rm Id}$ we only need to show that the sign in
$(g_{1}|_{\partial T})_{*}(\ell)=\pm\ell$ is actually a $+$.

Consider the map $f^{r}|_{T}:T\longrightarrow T$. It can be regarded as the
composition of $f^{r}$ and the inclusion $f^{r}(T)\subseteq T$, both of which
induce isomorphisms in homology: the first because it is a homeomorphism; the
second because $f^{r}(T)\subseteq T$ are concentric. Hence $(f^{r}|_{T})_{*}$
is an isomorphism of $H_{1}(T;\mathbb{Z})\cong\mathbb{Z}$, i.e. multiplication
by $\pm 1$. Since by definition $g_{0}=f^{2r}$, we have that
$(g_{0}|_{T})_{*}$ is the square of $(f^{r}|_{T})_{*}$ so it follows that
$(g_{0}|_{T})_{*}={\rm Id}$ in $H_{1}(T;\mathbb{Z})$. The map $G_{t}\circ
g_{0}|_{T}$ is a homotopy between $g_{0}|_{T}$ and $g_{1}|_{T}$, and this
homotopy takes place in $T$ by condition (i) of Lemma 4.3 so
$(g_{1}|_{T})_{*}=(g_{0}|_{T})_{*}={\rm Id}$ in $H_{1}(T;\mathbb{Z})$.

We can now show that the sign in $(g_{1}|_{\partial T})_{*}(\ell)=\pm\ell$ is
in fact a $+$. Regarding this as an equality in $H_{1}(T;\mathbb{Z})$ via the
inclusion $\partial T\subseteq T$ we have $(g_{1}|_{T})_{*}(\ell)=\pm\ell$,
and since $(g_{1}|_{T})_{*}={\rm Id}$ we get $\ell=\pm\ell$. Since
$\ell\neq-\ell$ in $H_{1}(T;\mathbb{Z})\cong\mathbb{Z}$ we conclude that the
sign on the right hand side must be a $+$. ■

Now we apply Lemma 4.4 to $g_{1}$ and $U=T^{\prime}$ to obtain a second
diffeotopy $G_{t}$. Letting $\lambda$ run from $1$ to $2$ and setting
$g_{\lambda}:=G_{\lambda-1}\circ g_{1}$ we obtain a further continuation of
$g_{1}$ to some $g_{2}$ such that $g_{2}|_{T}$ is the identity and again
$T^{\prime}$ is a trapping region for each $g_{\lambda}$.

Since $T^{\prime}$ is a trapping region for $g_{\lambda}$ for every
$\lambda\in[0,2]$ and its maximal invariant subset for $g_{0}$ is precisely
$K$, the map $\lambda\longmapsto K_{\lambda}=\bigcap_{k\geq
0}g_{\lambda}^{k}(T^{\prime})$ defines a continuation of $K$ through toroidal
attractors. The attractor $K_{2}$ is contained in $T^{\prime}$ and contains
$T$ since the latter is compact and invariant under $g_{2}$. The last part of
the proof consists in perturbing the dynamics further by gradually adding a
“radial” component on $T^{\prime}$ towards a core $\gamma$ of $T$. We first
describe this idea in the abstract.

Consider the nested triple of solid tori
$\mathbb{D}^{2}\times\mathbb{S}^{1}\subseteq(2\mathbb{D}^{2})\times\mathbb{S}^{1}\subseteq(3\mathbb{D}^{2})\times\mathbb{S}^{1}$,
where $r\mathbb{D}^{2}$ denotes the closed unit disk of radius $r$. Let $\rho$
be a diffeotopy of the interval $[0,3]$ which is supported on $[0,5/2]$ and
such that $\rho_{t}|_{[0,2]}(r)=(1-t/2)r$. We use this to define a diffeotopy
$f_{t}$ of the solid torus $(3\mathbb{D}^{2})\times\mathbb{S}^{1}$ by
$f_{t}(x,s):=(\rho_{t}(\|x\|)x/\|x\|,s)$. The action of $f_{t}$ on any
meridional disk $(2\mathbb{D}^{2})\times\\{s\\}$ is just given by
$x\longmapsto(1-t/2)x$ so for $t\in(0,1]$ it is a radial contraction towards
the origin. Hence $f_{t}$ sends $(2\mathbb{D}^{2})\times\mathbb{S}^{1}$ into
its interior and has the core $\\{0\\}\times\mathbb{S}^{1}$ as an attractor
with stationary dynamics. The same is true of the restriction
$f_{t}|_{\mathbb{D}^{2}\times\mathbb{S}^{1}}$. At the final stage $t=1$ we
have
$f_{1}((2\mathbb{D}^{2})\times\mathbb{S}^{1})=\mathbb{D}^{2}\times\mathbb{S}^{1}$.

Now we copy this abstract construction to our setting. Recall that
$T^{\prime}$ was obtained by thickening $T$ very slightly. Let
$T^{\prime\prime}$ be obtained from $T^{\prime}$ in the same way and consider
a diffeomorphism
$h:(T^{\prime\prime},T^{\prime},T)\longrightarrow(3\mathbb{D}^{2},2\mathbb{D}^{2},\mathbb{D}^{2})\times\mathbb{S}^{1}$.
Then we set $G$ to be the diffeotopy of $T^{\prime\prime}$ obtained by copying
$f_{t}$ through $h$; i.e. $G_{t}(x):=h^{-1}\circ f_{t}\circ h(x)$. By
construction each $G_{t}$ is the identity on a neighbourhood of $\partial
T^{\prime\prime}$, and so we can extend $G$ to a diffeotopy of all
$\mathbb{R}^{3}$. This has the following properties:

  * (i)

$G_{t}(T^{\prime})\subseteq T^{\prime}$ for each $t\in[0,1]$ and
$G_{1}(T^{\prime})\subseteq T$.

  * (ii)

$G_{1}(T)\subseteq T$ and the restriction $G_{1}|_{T}$ is conjugate (via $h$)
to a radial contraction of $\mathbb{D}^{2}\times\mathbb{S}^{1}$ onto its core
$\gamma:=h^{-1}(\\{0\\}\times\mathbb{S}^{1})$.

Letting $\lambda$ run from $2$ to $3$, consider the continuation
$g_{\lambda}:=G_{\lambda-2}\circ g_{2}$ of $g_{2}$. We have
$g_{\lambda}(T^{\prime})\subseteq{\rm int}\ T^{\prime}$ by (i) and the
corresponding property for $g_{2}$, so again the map $\lambda\longmapsto{\rm
Inv}(T^{\prime})$ is a continuation of $K_{2}$ through toroidal attractors to
the attractor $K_{3}$ of $g_{3}$. We claim that $K_{3}=\gamma$. To check this
first notice that for $x\in T$ we have $g_{3}(x)=G_{1}\circ g_{2}(x)=G_{1}(x)$
which belongs to $G_{1}(T)\subseteq T$ again, so it follows inductively that
$g_{3}^{k}|_{T}=G_{1}^{k}|_{T}=(G_{1}|_{T})^{k}$ for every $k\geq 0$. Thus by
(ii) the dynamics of $g_{3}$ on $T$ is conjugate (via $h$) to the dynamics of
a radial contraction on $\mathbb{D}^{2}\times\mathbb{S}^{1}$. The latter
clearly has $\\{0\\}\times\mathbb{S}^{1}$ as an attractor, and so $\gamma$ is
an attractor for $g_{3}|_{T}$. Since $T^{\prime}$ is positively invariant
under $g_{2}$ and $G_{1}(T^{\prime})\subseteq T$, we have
$g_{3}(T^{\prime})\subseteq T$ and so $\gamma$ is an attractor also for
$g_{3}|_{T^{\prime}}$, so in particular $K_{3}=\gamma$. Moreover,
$g_{3}|_{\gamma}=G_{1}|_{\gamma}={\rm Id}|_{\gamma}$.

The proof is complete. The full continuation from $K$ to $K_{3}$ is given by
the (smoothed out) concatenation of the $G$s obtained in the succesive steps
of the proof. ∎

Now we prove the auxiliary Lemmas 4.3 and 4.4. The first is an easy exercise
in differential topology and we will only sketch the proof:

###### Proof of Lemma 4.3.

Let $C_{1}\subseteq U$ be a closed collar of $\partial T_{1}$ inside $T_{1}$;
$C$ the set $\overline{T_{0}\setminus T_{1}}$, and $C_{0}\subseteq U$ a closed
collar of $\partial T_{0}$ in $\overline{\mathbb{R}^{3}\setminus T_{0}}$. (In
the topological category one needs to require that $T_{0}$ be tame to ensure
that this last collar exists). Each of these sets is diffeomorphic to a
$2$–torus $S$ times an interval: the first and third are collars; the second
is a product because of the hypothesis that $T_{0}$ and $T_{1}$ are
concentric. It is then a standard exercise to construct an ambient diffeotopy
that is the identity outside $C_{1}\cup C\cup C_{0}$ and stretches $C_{1}$ so
much that it becomes $C_{1}\cup C$, while shrinking $C\cup C_{1}$
appropriately so that they fit in $C_{0}$. (Pick a diffeomorphism $b:C_{1}\cup
C\cup C_{0}\longrightarrow\mathbb{T}^{2}\times[-1,2]$ such that
$b(C_{1})=\mathbb{T}^{2}\times[-1,0]$, $b(C)=\mathbb{T}^{2}\times[0,1]$ and
$b(C_{0})=\mathbb{T}^{2}\times[1,2]$, so that $\partial T_{1}$ and $\partial
T_{0}$ correspond to $\mathbb{T}^{2}\times 0$ and $\mathbb{T}^{2}\times 1$
respectively. Let $\rho:[-1,2]\longrightarrow[-1,2]$ be a diffeotopy which is
stationary near $-1$ and $2$ and whose final stage sends $0$ to $1$ and $1$ to
$\nicefrac{{3}}{{2}}$. Define a diffeotopy $a_{t}$ of
$\mathbb{T}^{2}\times[-1,2]$ by $a_{t}(x,s):=(x,\rho_{t}(s))$ and use this to
construct a diffeotopy $G_{t}$ of $\mathbb{R}^{3}$ given by
$G_{t}(x)=b^{-1}\circ a_{t}\circ b(x)$ if $x\in C_{1}\cup C\cup C_{0}$ and the
identity outside). ∎

For the second lemma we need the following result: any diffeomorphism
$g:T\longrightarrow T$ of a solid torus $T$ such that $g|_{\partial T}={\rm
Id}_{\partial T}$ is diffeotopic to the identity ${\rm Id}_{T}$; i.e. there
exists a diffeotopy $G$ of $T$ such that $G_{1}\circ g={\rm Id}_{T}$. This is
relatively easy to prove in the topological case, but in the differentiable
category the proof is much more involved and in fact the result is equivalent
to a conjecture of Smale which was settled by Hatcher in [17] (for the
formulation used here see [17, (9), p. 606]).

###### Proof of Lemma 4.4.

Since diffeomorphisms of the $2$–torus are classified up to diffeotopy by the
homomorphism they induce in homology ([11, Theorem 2.5, p. 55]), it follows
from the hypothesis that there exists a diffeotopy $G^{(1)}$ defined on
$\partial T$ such that $G^{(1)}_{1}\circ g|_{\partial T}={\rm Id}_{\partial
T}$. That diffeotopy can be extended, by using a collar of $\partial T$ in
$T$, to a diffeotopy of all of $T$ which we still denote by $G^{(1)}$.
Applying the result recalled before the proof to $G^{(1)}_{1}\circ g$, there
is a diffeotopy $G^{(2)}$ of $T$ such that $G^{(2)}_{1}\circ G^{(1)}_{1}\circ
g={\rm Id}_{T}$. Then the concatenation of $G^{(1)}$ and $G^{(2)}$ gives a
diffeotopy $G$ of $T$ which carries $g|_{T}$ onto the identity. Using a
sufficiently thin collar of $\partial T$ in $\overline{\mathbb{R}^{3}\setminus
T}$ one can extend the diffeotopy to all of $\mathbb{R}^{3}$ having it be the
identity outside any prescribed neighbourhood of $T$. (The proof of the lemma
in the topological case requires that one assumes $T$ to be tame so that
$\partial T$ indeed has a collar in $\overline{\mathbb{R}^{3}\setminus T}$). ∎

##  5\. The entropy of toroidal attractors

The goal of this section is to prove that the geometric degree provides a
lower bound on the entropy of a toroidal attractor. It is here where for the
first time we need the dynamics to be smooth.

######  Theorem 5.1.

Let $K$ be a toroidal attractor for a $\mathcal{C}^{\infty}$ local
diffeomorphism $f$. Then $h(f|_{K})\geq\log d_{\mathcal{N}}(f;K)$.

Coupling the above with the inequality
$d_{\mathcal{N}}(f;K)\geq\prod_{i}p_{i}$ from Theorem 3.9 yields
$h(f|_{K})\geq\log d_{\mathcal{N}}(f;K)\geq\log\prod_{i}p_{i}$ which is bound
(1.2) from the Introduction. The rest of the argument leading to Theorem 1.1
was already detailed there.

The proof of Theorem 5.1 depends, in turn, on the following result concerning
the entropy of dynamics on a solid torus. For definiteness we denote by
$V\subset\mathbb{R}^{3}$ the solid torus obtained by rotating around the
$z$–axis the disk of radius $\nicefrac{{1}}{{2}}$ and center
$(0,\nicefrac{{3}}{{2}},0)$ contained in the $yz$–plane.

######  Theorem 5.2.

Let $f:V\longrightarrow V$ a $\mathcal{C}^{\infty}$ embedding. Then the
entropy of $f$ is bounded below by $h(f)\geq\log N(f(V)\subseteq V)$.

###### Proof of Theorem 5.1 from Theorem 5.2.

Let $T\subseteq\mathcal{A}(K)$ be a smooth solid torus neighbourhood of $K$
and $r$ big enough so that $f^{r}(T)\subseteq T$. Consider the restriction
$f^{r}|_{T}:T\longrightarrow T$, which is a dynamical system on $T$ that still
has $K$ as its (global in $T$) attractor. This implies that $K$ contains the
non-wandering set of $f^{r}|_{T}$ and therefore by a result of Bowen (see [5]
or [28]) it concentrates all the entropy: $h(f^{r}|_{K})=h(f^{r}|_{T})$. Now
observe that Theorem 5.2 is valid not only for dynamics on $V$ but on any
smooth solid torus. This is a direct consequence of the invariance of both
entropy and the geometric index under conjugation. Thus for $f^{r}$ on the
smooth solid torus $T$ we may write

$h(f^{r}|_{T})\geq\log N(f^{r}(T)\subseteq T)=r\log d_{\mathcal{N}}(f;K)$

where in the last step we have used Remark 3.10.(1). Finally, a standard
property of entropy ensures that $h(f^{r}|_{K})=rh(f|_{K})$ and putting all
this together yields $h(f|_{K})\geq\log d_{\mathcal{N}}(f;K)$. ∎

It remains to prove Theorem 5.2. The argument proceeds by showing that the
length of a curve in a solid torus is bounded below by its geometric index
(Lemma 5.3) and then applying an inequality of Yomdin ([31]) which relates
volume growth rate and topological entropy for smooth dynamics. We only need a
very particular case of the inequality, which we describe now.

Given a $\mathcal{C}^{\infty}$ path in $V$, $\sigma:[0,1]\longrightarrow V$,
its length is given by the usual formula

$\ell(\sigma):=\int_{[0,1]}\|\sigma^{\prime}(t)\|dt.$

Suppose $f:V\longrightarrow V$ is a $\mathcal{C}^{\infty}$ map (not
necessarily injective). We consider the iterates of $\sigma$ under the
dynamics generated by $f$; i.e. the paths $f^{n}\circ\sigma$, and the
exponential growth rate of their lengths

$\overline{\lim}_{n\rightarrow\infty}\frac{1}{n}\log\ell(f^{n}\circ\sigma).$

The inequality that we need is the following:

(5.1)
$\overline{\lim}_{n\rightarrow\infty}\frac{1}{n}\log\ell(f^{n}\circ\sigma)\leq
h(f).$

We have included a proof of (5.1) in Appendix B. This is done both for
completeness and because, while in the general case the proof is very
involved, in our particular setting it becomes fairly accessible while still
retaining the essential ideas.

An extremely crude intuition of why (5.1) might be reasonable is as follows.
Suppose $[0,1]$ is partitioned into intervals $I_{i}$ so that each portion
$\sigma(I_{i})$ of the curve is contained in an $(n,\epsilon)$-dynamical ball.
This implies that the endpoints of $f^{k}\circ\sigma|_{I_{i}}$ lie at a
distance less than $\epsilon$ for $k=0,\ldots,n$ and, if the curve
$f^{k}\circ\sigma|_{I_{i}}$ does not oscillate too much, its length will then
be of order $\epsilon$. Thus the total length $\ell(f^{k}\circ\sigma)$ will be
of the order of $\epsilon$ times the number of intervals $I_{i}$, which in
turn is related to the minimal number of dynamical balls $S(n,\epsilon)$
needed to cover $V$ and therefore to the entropy $h(f)$. This heuristic idea
breaks down if the curve $f^{k}\circ\sigma$ oscillates a lot. If it does so
confined within a small ball the oscillations will have a large contribution
to the length but not to the entropy. If it oscillates entering and exiting a
dynamical ball a large number of times, then the number of intervals $I_{i}$
might grossly overestimate $S(n,\epsilon)$. The assumption on the smoothness
of $f$ provides control over this phenomenon.

In order to apply (5.1) we need to bound from below the length of paths in the
solid torus $V$. This is the content of the following lemma:

######  Lemma 5.3.

For every regular $\mathcal{C}^{\infty}$ parametrization
$\gamma:[0,1]\longrightarrow V$ of a simple closed curve, its length
$\ell(\gamma)$ is bounded as follows:

$\ell(\gamma)\geq 2\pi N(\gamma\subseteq V).$

###### Proof.

We shall regard $\gamma$ as a diffeomorphic embedding
$\gamma:\mathbb{S}^{1}\longrightarrow V$ and denote its image also by
$\gamma$.

Let $S\subseteq V$ be the shortest longitude of the solid torus $V$; namely
the circumference in the $\\{z=0\\}$ plane, of radius $1$ and centered at the
origin. Fix some orientation on $S$. We define a mapping
$h:\mathbb{S}^{1}\longrightarrow S$ which captures just the angular
information in $\gamma$ as follows: $h$ is the composition of (i) the
parameterization $\gamma$, followed by (ii) the orthogonal projection of
$\mathbb{R}^{3}\setminus z\text{-axis}$ onto the punctured plane
$\\{z=0\\}\setminus\\{(0,0,0)\\}$, followed finally by (iii) the radial
retraction of the latter onto $S$. Each of these maps is differentiable so $h$
is differentiable as well.

Fix $0<\epsilon<2\pi$. By Sard’s theorem, the set of critical values of $h$
can be covered by a family of open arcs $c_{i}$ whose lengths add up to less
than $\epsilon$. The $c_{i}$ can be taken to be mutually disjoint and finite
in number. It is possible that $h$ has no critical values: this happens
precisely when $\gamma$ winds monotonically inside $V$, without doubling back.
To avoid having to discuss that somewhat trivial case separately, we then take
the family of arcs $\\{c_{i}\\}$ to consist of a single arc of length less
than $\epsilon$.

Write $c_{1},\ldots,c_{n}$ for the arcs in the covering with the convention
that indices are taken cyclically (modulo $n$) as we move along $S$ in the
positive orientation and denote by $e_{i}^{1}$ and $e_{i}^{2}$ the endpoints
of $c_{i}$. Let $D_{i}^{1}$ and $D_{i}^{2}$ be the radial meridional disks of
$V$ that go through the points $e_{i}^{1}$ and $e_{i}^{2}$ (that is,
$D_{i}^{j}$ is the intersection of $V$ with the plane that contains the
$z$-axis and the point $e_{i}^{j}$). Figure 2 illustrates the definitions
showing $V$ viewed from the top.

$e_{1}^{2}$$e_{1}^{1}$$e_{4}^{1}$$e_{3}^{1}$$e_{4}^{2}$$e_{3}^{2}$$D_{2}^{1}$$D_{2}^{2}$$D_{3}^{1}$$D_{3}^{2}$$D_{1}^{1}$$D_{1}^{2}$$\gamma$$e_{2}^{1}$$e_{2}^{2}$$p$$D_{4}^{2}$$D_{4}^{1}$$S$

Figure 2. Setup for Lemma 5.3

Fix any one of the indices $i$ and denote by $V_{i}$ the closed sector of $V$
comprised between the disks $D_{i}^{2}$ and $D_{i+1}^{1}$. Suppose
$t\in\mathbb{S}^{1}$ is such that $p:=\gamma(t)$ belongs to the interior of
the sector $V_{i}$. Then:

Claim. $p$ lies in an arc of $\gamma$ which is entirely contained in the
sector $V_{i}$ and joins the disks $D_{i}^{2}$ and $D_{i+1}^{1}$.

Proof of claim. Define $J\subseteq\mathbb{S}^{1}$ to be the closure of the
connected component of
$\mathbb{S}^{1}\backslash\gamma^{-1}(\bigcup_{l,m}D_{l}^{m})$ which contains
$t$. Notice that $\gamma^{-1}(\bigcup_{l,m}D_{l}^{m})$ is closed in
$\mathbb{S}^{1}$, so it is disjoint from ${\rm int}\ J$ and contains $\partial
J$. Hence $\gamma(J)$ is an arc in $V$ whose interior is disjoint from
$\bigcup_{l,m}D_{l}^{m}$ and whose endpoints are contained in
$\bigcup_{l,m}D_{l}^{m}$. Since the arc is connected and contains $p$, it must
be completely contained in the sector $V_{i}$ and its endpoints must lie in
$D_{i}^{2}\cup D_{i+1}^{1}$. Thus $h(J)$ is an arc in $S$ whose endpoints are
contained in $\\{e_{i}^{2},e_{i+1}^{1}\\}$ and which is disjoint from all the
$c_{j}$. The latter implies that $h|_{J}$ has no critical points, and so in
particular it cannot map both endpoints of $J$ onto the same point $e_{i}^{2}$
or $e_{i+1}^{1}$ (otherwise it would reach some local extremum on $J$). Hence
the endpoints of $\gamma(J)$ indeed lie one in $D_{i}^{2}$ and the other in
$D_{i+1}^{1}$. ${\blacksquare}$

Now pick a point $p_{0}\in D_{i}^{2}\cap\gamma$. The following holds:

Claim. $p_{0}$ is an endpoint of an arc of $\gamma$ which is entirely
contained in the sector $V_{i}$ and whose other endpoint is contained in
$D_{i+1}^{1}$.

Proof of claim. Write $p_{0}=\gamma(t_{0})$ for some $t_{0}\in\mathbb{S}^{1}$.
Since $\gamma$ is transverse to the disk $D_{i}^{2}$, there exists a small arc
$I\subseteq\mathbb{S}^{1}$ centered at $t_{0}$ such that $\gamma(I)$
intersects $D_{i}^{2}$ precisely at $p_{0}$ and both components of
$\gamma(I\backslash\\{t_{0}\\})$ lie on different sides of $D_{i}^{2}$. Since
the latter is the common boundary of $V_{i}$ and $V_{i-1}$, one of the
components $I_{0}$ of $I\backslash\\{t_{0}\\}$ satisfies that $\gamma(I_{0})$
is contained in the interior of $V_{i}$. It only remains to pick any $t\in
I_{0}$ and apply the previous claim to $p:=\gamma(t)$. ${\blacksquare}$

The two claims above imply that the number of arcs of $\gamma$ in the sector
$V_{i}$ is precisely $|D_{i}^{2}\cap\gamma|$, and each of them has a length at
least $d_{i}^{i+1}$ where $d_{i}^{i+1}$ is the angle between $D_{i}^{2}$ and
$D_{i+1}^{1}$. Since $|D_{i}^{2}\cap\gamma|\geq N(\gamma\subseteq V)$ by the
definition of the geometric index, the total contribution of these arcs to the
length of $\gamma$ is bounded below by $d_{i}^{i+1}N(\gamma\subseteq V)$.
Hence

$\ell(\gamma)\geq N(\gamma\subseteq V)\sum_{i}d_{i}^{i+1}\geq
N(\gamma\subseteq V)(2\pi-\varepsilon)$

where the last inequality makes use of the fact that $\sum_{i}d_{i}^{i+1}\geq
2\pi-\varepsilon$ by the choice of the covering $\\{c_{i}\\}$. Since this is
true for every $\varepsilon>0$ the lemma follows. ∎

The proof of Theorem 5.2 is now straightforward. Recall that
$f:V\longrightarrow V$ is a $\mathcal{C}^{\infty}$ embedding. Let $\sigma$ be
a smooth core of $V$ (for example its centerline). By the preceding lemma
applied to $\gamma=f^{n}\circ\sigma$

$\ell(f^{n}\circ\sigma)\geq 2\pi N(f^{n}\circ\sigma\subseteq V)=2\pi
N(f^{n}(V)\subseteq V)$

where the last equality owes to the definition of the geometric index of a
curve and the fact that $f^{n}\circ\sigma$ is a core of $f^{n}(V)$. Observe
that for every $n\geq 1$ we have $f^{n}(V)\subseteq
f^{n-1}(V)\subseteq\ldots\subseteq f(V)\subseteq V$ and so by multiplicativity
and invariance under homeomorphisms of the geometric index
$N(f^{n}(V)\subseteq V)=\prod_{k=0}^{n-1}N(f^{k+1}(V)\subseteq
f^{k}(V))=\prod_{k=0}^{n-1}N(f(V)\subseteq V)=N(f(V)\subseteq V)^{n}$.
Therefore $\ell(f^{n}\circ\sigma)\geq 2\pi N(f(V)\subseteq V)^{n}$ and now by
inequality (5.1)

$h(f)\geq\overline{\lim}_{n\rightarrow\infty}\frac{1}{n}\log\ell(f^{n}\circ\sigma)\geq\log
N(f(V)\subseteq V).$

as was to be shown.

##  6\. Concluding remarks

### 6.1.

The formal similarities between the homological winding number $m$ and the
geometric index $N$ imply that one can obtain “homological” counterparts to
the definitions and results above by replacing $N$ with $m$ throughout. Thus
one can define homological prime divisors $q_{i}$ of a toroidal set $K$, a
homological degree $d(f;K)$, etc. These turn out to be describable in terms of
Čech cohomology. One needs to assume that $\check{H}^{1}(K;\mathbb{Z})\neq 0$
for the definitions to make sense. Then:

  * (D1)

The homological prime divisors of $K$ correspond to those prime numbers $q$
which divide every element in $\check{H}^{1}(K;\mathbb{Z})$ in the sense that
for every $x\in\check{H}^{1}(K;\mathbb{Z})$ there exists
$y\in\check{H}^{1}(K;\mathbb{Z})$ such that
$x=y+\stackrel{{\scriptstyle(q)}}{{\ldots}}+y$.

  * (D2)

Suppose $f$ is a local homeomorphism which leaves a toroidal set $K$
invariant. Then the induced homomorphism
$f^{*}:\check{H}^{1}(K;\mathbb{Z})\longrightarrow\check{H}^{1}(K;\mathbb{Z})$
turns out to be multiplication by the homological degree $d(f;K)$.

A straightforward adaptation of the proof of Theorem 3.9 yields:

  * (D3a)

When $K$ is an attractor for $f$, the homological degree $d(f;K)$ is an
integer whose prime divisors are exactly the prime divisors $q_{i}$ of $K$.
Moreover,

  * (D3b)

if $T\subseteq\mathcal{A}(K)$ is a solid torus neighbourhood of $K$ and $r$ is
such that $f^{r}(T)\subseteq T$ then the $q_{i}$ are precisely the prime
divisors of $|m(f^{r}(T)\subseteq T)|$ and the homological degree $d(f;K)$
satisfies $|d(f;K)|^{r}=|m(f^{r}(T)\subseteq T)|$.

Since $|m|\leq N$ in general, (D3b) and Remark 3.10 yield the inequality
$|d(f;K)|\leq d_{\mathcal{N}}(f;K)$.

For a $\mathcal{C}^{0}$ embedding $f:V\longrightarrow V$ the induced map
$f_{*}:H_{1}(V)\longrightarrow H_{1}(V)$ is just multiplication by
$m(f(V)\subseteq V)$, and so a classical theorem of Manning ([22]) implies
that $h(f)\geq|m(f(V)\subseteq V)|$. This is the homological counterpart to
Theorem 5.2, where notably the smoothness assumption is not needed. Replacing
$N$ with $m$ in the derivation of Theorem 5.1 from Theorem 5.2 shows that:

  * (D4)

If $K$ is a toroidal attractor for a local homeomorphism and
$\check{H}^{1}(K;\mathbb{Z})\neq 0$ then $h(f|_{K})\geq\log|d(f;K)|$.

Combining (D3a) and (D4) one has $d(f;K)\geq\prod_{i}q_{i}$ and as a
consequence the universal bound

(6.1) $h(f|_{K})\geq\log|d(f;K)|\geq\log\prod_{i}q_{i}$

where $f$ is now a local homeomorphism having a toroidal set $K$ as an
attractor. This is structurally similar to (1.2) and holds without any
smoothness assumptions.

######  Example 6.1.

Let $K\subseteq\mathbb{R}^{3}$ be the standard embedding of an $n$–adic
solenoid in $\mathbb{R}^{3}$ (see for example [21, §17.1]). This particular
embedding is toroidal and its prime divisors, both geometric and homological,
are just the prime divisors $p_{i}=q_{i}$ of the integer $n$ (counted just
once). Thus if $f$ is any local homeomorphism having $K$ as an attractor then
$h(f|_{K})\geq\log\prod_{i}q_{i}$. There exists a homeomorphism of
$\mathbb{R}^{3}$ (in fact, a $\mathcal{C}^{\infty}$ diffeomorphism) which
realizes the embedded solenoid $K$ as an attractor with its standard dynamics.
These have an entropy $\log\prod_{i}q_{i}$, and so in this case the universal
bound $\log\prod_{i}q_{i}$ for any attracting dynamics on $K$ is in fact
sharp.

In spite of the previous example, the homological approach has several
shortcomings. It is not applicable when $\check{H}^{1}(K;\mathbb{Z})=0$, so
for instance it leaves out examples as simple as the Whitehead continua
described in Example 3.4. Also, Theorem 4.1 fails if one replaces (geometric)
prime divisors with homological prime divisors. The consequence of this is
that the bound $h(f|_{K})\geq\log\prod_{i}q_{i}$ is not sharp even up to
continuation, and so the homological degree is not powerful enough to prove
Theorem 1.1. The following example provides an illustration of these
phenomena.

######  Example 6.2.

Consider a variation of Example 3.11.(1) where $T_{1}$ and $T_{2}=f(T_{1})$
are still unknotted and the core of $f(T_{1})$ looks like the pattern in the
left panel of Figure 1 but with an extra full winding inside $T_{1}$. The
resulting toroidal set $K$ can still be realized as an attractor for a
$\mathcal{C}^{\infty}$ diffeomorphism $f$ of $\mathbb{R}^{3}$. Now
$m(T_{2}\subseteq T_{1})=1$ and $N(T_{2}\subseteq T_{1})=3$. Therefore:

  * •

$K$ has no homological prime divisors and $d(f;K)=1$.

  * •

$K$ has $p=3$ as a (geometric) prime divisor, and $d_{\mathcal{N}}(f;K)=3$.

This illustrates that the inequality $|d(f;K)|\leq d_{\mathcal{N}}(f;K)$ can
be strict. Since $f$ is $\mathcal{C}^{\infty}$, according to Theorem 5.1 we
have $h(f|_{K})\geq\log d_{\mathcal{N}}(f;K)=\log 3$ whereas the homological
bound only says $h(f|_{K})\geq\log d(f;K)=\log 1=0$. Notice that $K$ cannot be
continued to a smooth knot through toroidal attractors (because it has $3$ as
a prime divisor) but the homological bound on the entropy vanishes.

### 6.2.

It is natural to ask whether the bound $h(f|_{K})\geq\log
d_{\mathcal{N}}(f;K)$ is valid when $f$ is just a (local) homeomorphism,
without any smoothness assumptions. The derivation of Theorem 5.1 from Theorem
5.2 works equally well in the $\mathcal{C}^{0}$ case, so this boils down to
the following:

Question. Let $f:V\longrightarrow V$ be continuous and injective. Is it true
that $h(f)\geq\log N(f(V)\subseteq V)$?

A heuristic argument which might suggest an affirmative answer goes as
follows. Let again $\sigma$ be the centerline of $V$. After some technical
work it can be arranged that the angular behaviour of $f^{n}\circ\sigma$ be
piecewise monotone, and then the number of monotonicity intervals is bounded
below by $N(f^{n}\circ\sigma\subseteq V)=N(f(V)\subseteq V)^{n}$. A classical
result of Misiurewicz and Szlenk ([23, Theorem 1, p. 48]) concerning the
entropy of self-maps of $\mathbb{S}^{1}$ then suggests that this angular
behaviour alone should contribute $\log N(f(V)\subseteq V)$ to the entropy of
$f$.

If the answer to the question posed above is affirmative then, since
$d_{\mathcal{N}}(f;K)\geq\prod_{i}p_{i}$ holds generally, one would have the
universal bound $h(f|_{K})\geq\log\prod_{i}p_{i}$ for any local homeomorphism
$f$ having $K$ as an attractor. In turn this leads to the following theorem
which improves Theorem 1.1 from the Introduction in that it gives a conclusion
about $K$ itself, with no continuations involved, and makes no smoothness
assumptions:

Theorem. Let $K$ be a toroidal attractor for a homeomorphism $f$ of
$\mathbb{R}^{3}$. Then either $K$ admits stationary attracting dynamics or
every attracting dynamics on $K$ has an entropy at least $\log 2$.

###### Proof.

If $K$ has a prime divisor then $h(f|_{K})\geq\log 2$. If $K$ has no prime
divisors, as in the proof of Theorem 4.1 this implies that $K$ has a basis of
concentric solid tori. Using these one can construct a $\mathcal{C}^{0}$ flow
which has $K$ as an attractor (see [3, Theorem B, p. 71]) and is stationary on
it. In particular, the time-one map of such a flow provides a homeomorphism of
$\mathbb{R}^{3}$ which has $K$ as an attractor and is stationary on it. ∎

## Appendix A

In this Appendix we relate our topological definition of the geometric index
to the standard piecewise linear (pl) one and show how the properties (P2) and
(P3) from p. 2.1 can be derived from their pl counterparts. Since we will be
working simultaneously with the topological and the piecewise linear geometric
indices, we shall distinguish them notationally with a subscript thus: $N_{\rm
top}$ and $N_{\rm pl}$.

We recall the pl definition. Let $T\subseteq\mathbb{R}^{3}$ be a polyhedral
solid torus and $\gamma\subseteq{\rm int}\ T$ a polygonal simple closed curve.
We consider all possible polyhedral meridional disks $D$ of $T$ (which may be
very contorted) and count the number of points of intersection in
$D\cap\gamma$. The geometric index of $\gamma$ inside $T$ is defined as

$N_{\rm pl}(\gamma\subseteq T):=\min\\{|D\cap\gamma|:D\text{ is a polyhedral
meridional disk of }T\\}.$

It should be intuitively clear that this number is finite: since the objects
involved are polyhedral, a slight perturbation of any meridional disk $D$ will
always make it transverse to $\gamma$ and then $D\cap\gamma$ will consist of
finitely many points.

Now suppose $T_{1}\subseteq T_{0}$ is a pair of polyhedral solid tori with
$T_{1}$ contained in the interior of $T_{0}$. Let $\gamma_{1}$ be a polyhedral
core of $T_{1}$. Then one defines the geometric index of $T_{1}$ inside
$T_{0}$ as $N_{\rm pl}(T_{1}\subseteq T_{0}):=N_{\rm pl}(\gamma_{1}\subseteq
T_{0})$. This definition is correct because cores are unique up to isotopy. An
equivalent definition which does not make use of cores and is more directly
related to our definition of $N_{\rm top}$ is the following. Consider a
meridional disk $D$ of $T_{0}$. By perturbing it slightly if necessary we may
achieve that it intersects $\partial T_{1}$ along disjoint simple closed
curves $\gamma_{i}$ which bound meridional disks $E_{i}$ of $T_{1}$. Then
$N_{\rm pl}(T_{1}\subseteq T_{0})$ is the smallest possible number of these
curves. The equivalence of this definition and the previous one is a
consequence of [30, Hilfssatz 5, p. 174].

For a polyhedral pair of tori $T_{1}\subseteq T_{0}$ we now have two
definitions of a geometric index. The following proposition shows that both
are equivalent.

######  Proposition A.1.

Suppose $T_{1}\subseteq T_{0}$ is a pair of polyhedral solid tori. Then
$N_{\rm top}(T_{1}\subseteq T_{0})=N_{\rm pl}(T_{1}\subseteq T_{0})$.

###### Proof.

Let $p$ and $t$ be the polyhedral and topological geometric indices of $T_{1}$
inside $T_{0}$, respectively. As mentioned above, there is a pl meridional
disk of $T_{0}$ which intersects $T_{1}$ in precisely $p$ meridional disks.
This can clearly be made transverse to $T_{1}$ by slightly perturbing its
vertices if necessary; hence $t\leq p$.

To prove the converse we shall show that any topological meridional disk $D$
of $T_{0}$ which is transverse to $T_{1}$ in the sense of Definition 2.1 can
be used to construct a polyhedral disk which is still transverse to $T_{1}$
and intersects it in the same number of meridional disks, thus showing that
$p\leq t$.

Since $T_{1}$ is polyhedral, the set $M:=T_{0}\backslash{\rm int}\ T_{1}$ is a
compact $3$–manifold whose boundary is the disjoint union of $\partial T_{0}$
and $\partial T_{1}$. Consider $D^{*}:=D\cap M$, which is a disk with holes.
The boundaries of these holes are simple closed curves
$\gamma_{1},\ldots,\gamma_{k}$ in $\partial T_{1}$ which are meridional curves
of $T_{1}$.

By the topological transversality condition in Definition 2.1 the set $D^{*}$
is semi-locally tame and hence tame ([4, Theorem 9, p. 157]). Thus there
exists a homeomorphism $h$ of $M$ onto itself which moves points less than any
prescribed $\epsilon$ and sends $D^{*}$ to a polyhedral disk with holes. Due
to the invariance of the boundary of a manifold under homeomorphisms, $h$
sends $\partial T_{0}$ and $\partial T_{1}$ to themselves. By choosing
$\epsilon$ sufficiently small, each $h(\gamma_{i})$ is homotopic to
$\gamma_{i}$ in $\partial T_{1}$; in particular each is still a meridional
curve (non-nullhomotopic in $\partial T_{1}$ but nullhomotopic in $T_{1}$).
Thus each $h(\gamma_{i})$ bounds a polyhedral meridional disk $E_{i}$ of
$T_{1}$; these can clearly be taken to be disjoint by removing their possible
intersections. Then $h(D^{*})\cup E_{1}\cup\ldots\cup E_{k}$ is a polyhedral
meridional disk of $T_{0}$ which intersects $T_{1}$ in $k$ disks. ∎

The analogues of properties (P1) to (P3) in p. 2.1 were established, in the pl
context, by Schubert. The above proposition then implies that they also hold
in the topological context. We shall argue this for the multiplicativity
property, for example.

Consider three nested solid tori $T_{2}\subseteq T_{1}\subseteq T_{0}$, where
as usual we take the tori to be tame. Pick any homeomorphism $h_{0}$ from
$T_{0}$ onto any polyhedral solid torus $P_{0}$. This sends $h_{0}(T_{1})$
onto a topological solid torus inside $T_{0}$ which is semilocally tame (it
being the image under a local homeomorphism of a tame object). Thus it is
tame; in fact there is a homeomorphism $h_{1}:P_{0}\longrightarrow P_{0}$
which is the identity outside a neighbourhood of $h_{0}(T_{1})$ and sends the
latter onto a polyhedral solid torus $P_{1}$. Thus the homeomorphism
$h_{1}\circ h_{0}$ sends both $T_{0}$ and $T_{1}$ onto polyhedral tori.
Applying the same argument once more we obtain a homeomorphism $h$ which sends
the triple $(T_{0},T_{1},T_{2})$ onto a triple of polyhedral solid tori
$(P_{0},P_{1},P_{2})$. By the invariance of $N_{\rm top}$ under homeomorphisms
and the proposition above $N_{\rm top}(T_{j}\subseteq T_{i})=N_{\rm
top}(P_{j}\subseteq P_{i})=N_{\rm pl}(P_{j}\subseteq P_{i})$. Then the pl
multiplicativity property $N_{\rm pl}(P_{2}\subseteq P_{0})=N_{\rm
pl}(P_{2}\subseteq P_{1})\cdot N_{\rm pl}(P_{1}\subseteq P_{0})$ implies
directly its topological counterpart $N_{\rm top}(T_{2}\subseteq T_{0})=N_{\rm
top}(T_{2}\subseteq T_{1})\cdot N_{\rm top}(T_{1}\subseteq T_{0})$.

##  Appendix B Yomdin’s inequality

In this Appendix we prove Yomdin’s inequality (5.1) for our very specific case
of a map $f:V\longrightarrow V$ and exponential growth rates of lengths of
curves. We have followed the papers by Yomdin [31] and Gromov [15]. Although
we will eventually take $f$ to be $\mathcal{C}^{\infty}$, for the proof we
work with a fixed degree of smoothness $k$. Define $M(f)$ to be the maximum of
$\|df\|_{\infty}$ over $V$, and $R(f)$ as the limit

$R(f):=\lim_{n\rightarrow\infty}\frac{1}{n}\log\max_{x\in
V}\|df^{n}\|_{\infty},$

or in other words the exponential growth rate of $M(f^{n})$. Notice that
$M(f^{n+m})\leq M(f^{n})M(f^{m})$ so the sequence $\log M(f^{n})$ is
subadditive. Thus $R(f)$ exists and satisfies $R(f)\leq M(f)<+\infty$.

For the sake of brevity we shall denote by ${\rm EGR}\ a_{n}$ the exponential
growth rate of any sequence $a_{n}\geq 0$; i.e.
$\overline{\lim}_{n\rightarrow\infty}\frac{1}{n}\log a_{n}$. Now, the
topological entropy of $f$ is bounded as follows:

######  Theorem B.1.

Let $f:V\longrightarrow V$ be of class $\mathcal{C}^{k}$. For any
$\mathcal{C}^{k}$ curve $\sigma:[0,1]\longrightarrow V$ the following
inequality holds:

(B.1) ${\rm EGR}\ \ell(f^{n}\circ\sigma)\leq h(f)+\frac{1}{k}R(f).$

Notice that $R(f)$ is finite and does not depend on $k$. Hence when $f$ and
$\sigma$ are of class $\mathcal{C}^{\infty}$ then $k$ can be taken to be
arbitrarily large and so the $R(f)/k$ term can be removed, leading to
inequality (5.1).

######  Remark B.2.

Inequality (B.1) actually holds for arbitrary compact manifolds and simplices
of arbitrary dimension instead of paths, replacing length with the appropriate
volume. This is the celebrated theorem of Yomdin. Combining this with an
inequality of Newhouse ([25]) shows that when $f$ is $\mathcal{C}^{\infty}$
its entropy $h(f)$ is in fact equal to the supremum of the left hand side of
(5.1) over all smooth simplices of arbitrary dimensions.

Examination of (B.1) under time rescaling (i.e. replacing $f$ with $f^{q}$ for
some fixed $q\geq 1$) leads to the observation that it is enough to prove the
following a priori slightly weaker bound:

(B.2) ${\rm EGR}\ \ell(f^{n}\circ\sigma)\leq h(f)+A(k)+\frac{1}{k}M(f)$

where $A(k)$ is some number which only depends on $k$ but not on $f$ or
$\sigma$. This implies inequality (B.1) as follows. Fix the path
$\sigma:[0,1]\longrightarrow V$ and pick some sequence
$n_{k}\rightarrow+\infty$ such that $1/n_{k}\log\ell(f^{n_{k}}\circ\sigma)$
converges to the exponential growth rate ${\rm EGR}\ \ell(f^{n}\circ\sigma)$.
Fix some $q\geq 1$ and write $n_{k}=m_{k}q+r_{k}$ with $0\leq r_{k}<q$. After
passing to a subsequence we may take all $r_{k}$ to be equal to some fixed
$r$. Trivially

$\frac{1}{n_{k}}\log\ell(f^{n_{k}}\circ\sigma)=\frac{m_{k}}{n_{k}}\frac{1}{m_{k}}\log\ell((f^{q})^{m_{k}}\circ(f^{r}\circ\sigma))$

and taking limits as $k\rightarrow+\infty$ yields

${\rm EGR}\ \ell(f^{n}\circ\sigma)\leq\frac{1}{q}{\rm EGR}\
\ell((f^{q})^{m}\circ(f^{r}\circ\sigma)),$

where the exponential growth rate on the right hand side is taken as
$m\rightarrow+\infty$. Now the weaker inequality (B.2) applied to the map
$f^{q}$ and the path $f^{r}\circ\sigma$ on the right hand side yields

${\rm EGR}\
\ell(f^{n}\circ\sigma)\leq\frac{1}{q}\left(h(f^{q})+A(k)+\frac{1}{k}\log
M(f^{q})\right).$

A standard property of entropy ensures that $h(f^{q})=qh(f)$. Plugging this
above and letting $q\rightarrow+\infty$ removes the constant term $A(k)$ and
leads to (5.1) by the definition of $R(f)$.

###  B.1. Basic strategy of the proof

Ultimately, the entropy is related to the minimum number of dynamical balls
needed to cover $V$. One possible way of estimating this from below is the
following.

Fix some curve $\sigma:[0,1]\longrightarrow V$ and any open covering
$\mathcal{B}$. For each member $B$ of the covering we want to analyze the
intersection of (the image of) $\sigma$ with $B$, which will generally be a
union of curves, and how the length of these grows as we iterate them with
$f$. Thus we consider the set $\sigma^{-1}(B)\subseteq[0,1]$ which is a
disjoint union of compact intervals (its connected components), and the length
of $f^{n}\circ\sigma|_{\sigma^{-1}(B)}$ for each $B\in\mathcal{B}$. Obviously
as $B$ ranges over the covering $\mathcal{B}$ the sets $\sigma^{-1}(B)$
provide a covering of $[0,1]$, and so
$\ell(f^{n}\circ\sigma)\leq\sum_{B\in\mathcal{B}}\ell(f^{n}\circ\sigma|_{\sigma^{-1}(B)})$.
In particular when $\mathcal{B}$ is a subcover of $\mathcal{B}(n,\epsilon)$
with the minimal number of elements, namely $S(n,\epsilon)$, we have

$\ell(f^{n}\circ\sigma)\leq
S(n,\epsilon)\sup_{B\in\mathcal{B}(n,\epsilon)}\ell(f^{n}\circ\sigma|_{\sigma^{-1}(B)})$

because the number of summands is $S(n,\epsilon)$ and every summand can be
overestimated by the supremum of the lengths over the whole cover
$\mathcal{B}(n,\epsilon)$ instead of only the minimal subcover. Taking
exponential growth rates on both sides and then letting $\epsilon\rightarrow
0$ leads to

(B.3) ${\rm EGR}\ \ell(f^{n}\circ\sigma)\leq h(f)+\lim_{\epsilon\rightarrow
0}{\rm EGR}\
\sup_{B\in\mathcal{B}(n,\epsilon)}\ell(f^{n}\circ\sigma|_{\sigma^{-1}(B)}).$

Comparing this with (B.2) we see that one needs to show that the second
summand is bounded above by an expression of the form $A(k)+M(f)/k$. The key
to doing this is an analysis of the set $\sigma^{-1}(B)$. This is provided by
Lemma B.3 below. We introduce a preliminary definition and then state the
lemma.

If $\tau:[0,1]\longrightarrow V$ is a curve such that all the derivatives of
$\tau$ up to order $k$ have a norm $\leq 1$ we shall say that $\tau$ is
normalized. If $\tau$ is defined only on a subinterval $J\subseteq[0,1]$ we
reparameterize it by letting $\psi:[0,1]\longrightarrow J$ be the unique
affine bijection which preserves orientation, and say that $\tau$ is
normalized if $\tau\circ\psi$ is normalized. Clearly the length of a
normalized curve is bounded above by $1$.

######  Lemma B.3.

Let $f:V\longrightarrow V$ be a $\mathcal{C}^{k}$ map. There exist numbers
$\mu(k)$ and $\epsilon_{0}(f,k)$ with the following property. For any
$\mathcal{C}^{k}$ curve $\sigma:[0,1]\longrightarrow V$ and any
$(n,\epsilon)$-dynamical ball $B\subseteq V$ with $\epsilon<\epsilon_{0}$ the
set $\sigma^{-1}(B)$ can be covered by at most
$c(\sigma,\epsilon)(\mu(k)M(f)^{\nicefrac{{1}}{{k}}})^{n}$ intervals $J_{i}$
such that for each of these the curve $f^{n}\circ\sigma|_{J_{i}}$ is
normalized.

The notation $c(\sigma,\epsilon)$ means that $c$ depends only on $\sigma$ and
$\epsilon$ but not on $f$ or $B$. It follows immediately from the lemma that
for $\epsilon<\epsilon_{0}(f,k)$

$\ell(f^{n}\circ\sigma|_{\sigma^{-1}(B)})\leq\sum_{i}\ell(f^{n}\circ\sigma|_{J_{i}})\leq
c(\sigma,\epsilon)(\mu(k)M(f)^{\nicefrac{{1}}{{k}}})^{n}$

since each piece $f^{n}\circ\sigma|_{J_{i}}$ is normalized. The right hand
side of the above inequality is independent of $B$, and so the supremum of
these lengths over $B\in\mathcal{B}(n,\epsilon)$ has the same upper bound.
Therefore extracting the exponential growth rate one has

${\rm EGR}\
\sup_{B\in\mathcal{B}(n,\epsilon)}\ell(f^{n}\circ\sigma|_{\sigma^{-1}(B)})\leq\log\mu(k)+\frac{1}{k}\log
M(f)$

whenever $\epsilon<\epsilon_{0}(f,k)$. In particular the inequality holds in
the limit $\epsilon\rightarrow 0$, so coupled with (B.3) we get

${\rm EGR}\ \ell(f^{n}\circ\sigma)\leq h(f)+\log\mu(k)+\frac{1}{k}\log M(f)$

which is valid for any $\sigma:[0,1]\longrightarrow V$. With $A(k)=\log\mu(k)$
this has the desired form (B.2) and therefore proves (B.1) via time rescaling
as explained above.

###  B.2. The inductive step for Lemma B.3

Suppose $t\in[0,1]$ belongs to $\sigma^{-1}(B)$ where $B$ is an
$(n,\epsilon)$-dynamical ball. By definition, there is some $x$ such that
$\sigma(t)\in B(x,\epsilon)$, $f\circ\sigma(t)\in B(f(x),\epsilon)$, etc. up
to $f^{n}\circ\sigma(t)\in B(f^{n}(x),\epsilon)$. Thus finding the set
$\sigma^{-1}(B)$ can be approached iteratively as follows. For each $r$ let
$S_{r}\subseteq[0,1]$ be the set of parameters $t$ such that $\sigma(t)\in
B(x,\epsilon),f\circ\sigma(t)\in B(f(x),\epsilon)\ldots$, up to iterate $r$.
Clearly the $S_{r}$ form a decreasing sequence of compact sets and
$\sigma^{-1}(B)=S_{n}$. Suppose we have already found some $S_{r}$. To find
$S_{r+1}$ one may proceed as follows. Each $S_{r}$ is a disjoint union of
closed intervals (its connected components; maybe some degenerate). If we
write $S_{r}=\bigcup J$ for that decomposition, one can describe $S_{r+1}$ by
restricting attention to each $(f^{r}\circ\sigma)|_{J}$ at a time and finding
what parameter values $t\in J$ satisfy the additional condition
$f^{r+1}\circ\sigma(t)\in B(f^{r+1}(x),\epsilon)$. In other words, letting
$\tau$ be the curve $f^{r}\circ\sigma|_{J}$, which is contained in the ball
$B(z,\epsilon)$ with $z=f^{r}(x)$, we need to find the set of parameters $t\in
J$ such that $f\circ\tau(t)\in B(f(z),\epsilon)$. This set might be quite
complicated (for instance, it might have infinitely many connected components)
and it will be technically convenient to overestimate it slightly; that is, we
will actually show that it is contained in a union of closed intervals which
we shall be able to bound in number.

The proof of Lemma B.3 thus boils down to a one-step estimate. Much as we did
with the time rescaling earlier on, by taking advantage of a degree of freedom
related to the metric size of the problem we can reduce the one-step estimate
to the convenient case when the derivatives of $f$ of orders $2,\ldots,k$ are
bounded above by $1$ and the radius of the ball $B$ is $\epsilon=1$. We
therefore state the one-step estimate for a solid torus $V_{\lambda}$ which is
just the solid torus $V$ scaled up by a factor $\lambda\geq 1$:

######  Lemma B.4.

Let $f_{\lambda}:V_{\lambda}\longrightarrow V_{\lambda}$ be a
$\mathcal{C}^{k}$ map with $\|d^{s}f\|_{\infty}\leq 1$ for every
$s=2,\ldots,k$. Let $\sigma_{\lambda}:[0,1]\longrightarrow V_{\lambda}$ be a
$\mathcal{C}^{k}$ normalized curve and $B\subseteq V_{\lambda}$ a closed ball
of radius $1$ and an arbitrary center. Finally, let
$S=\\{t\in[0,1]:f_{\lambda}\circ\sigma_{\lambda}(t)\in B\\}$. Then there
exists a family of no more than $\mu(k)M(f_{\lambda})^{\nicefrac{{1}}{{k}}}$
intervals $J_{i}$ such that:

  * (i)

The set $S$ is covered by the $J_{i}$.

  * (ii)

For each of the $J_{i}$ the curve $f_{\lambda}\circ\sigma_{\lambda}|_{J_{i}}$
is normalized.

Applying this one-step case to each of the curves
$f_{\lambda}\circ\sigma_{\lambda}|_{J_{i}}$ (after the standard affine
reparametrization) and repeating this inductively as described earlier we
obtain that for an $(n,1)$-dynamical ball the set $\sigma^{-1}(B)$ is covered
by at most $(\mu(k)M(f_{\lambda})^{\nicefrac{{1}}{{k}}})^{n}$ intervals
$J_{i_{1}i_{2}\ldots i_{n}}$ such that for each of them the curve
$f^{n}_{\lambda}\circ\sigma_{\lambda}|_{J_{i_{1}i_{2}\ldots i_{n}}}$ is
normalized.

###### Proof of Lemma B.3 from Lemma B.4.

First we scale up all the elements in our problem by a factor $\lambda\geq 1$
to be fixed later. The solid torus $V$ becomes the solid torus
$V_{\lambda}:={\lambda}\cdot V$ and the map $f$ is replaced with its conjugate
via the rescaling; i.e. $f_{\lambda}:V_{\lambda}\longrightarrow V_{\lambda}$
given by $f_{\lambda}(x):={\lambda}\cdot f(x/{\lambda})$. Notice that this
rescaling does not change the first derivatives of $f$ (so
$M(f)=M(f_{\lambda})$) but divides its $s$th derivatives, $s\geq 1$, by a
factor ${\lambda}^{s-1}$. Hence by choosing ${\lambda}$ large enough we can
achieve that all the derivatives of $f_{\lambda}$ of orders $2,3,\ldots,k$ be
bounded above by $1$ on $V_{\lambda}$. The cutoff value $\lambda_{0}$ of
$\lambda$ at which this happens depends only on the derivatives of $f$ on $V$.
Let $\epsilon_{0}(f,k):=1/{\lambda_{0}}$.

Now consider an $(n,\epsilon)$-dynamical ball $B\subseteq V$, with
$\epsilon<\epsilon_{0}$. It is straightforward to see that its scaled up
version $\lambda\cdot B$ is just an $(n,\lambda\epsilon)$-dynamical ball for
$f_{\lambda}$. Choose $\lambda:=\nicefrac{{1}}{{\epsilon}}$ as the scaling
factor, so that $\lambda B$ is actually an $(n,1)$-dynamical ball. Since
$\epsilon<\epsilon_{0}$ we have $\lambda>\lambda_{0}$ and so the derivatives
of $f_{\lambda}$ of order $2\leq s\leq k$ are all bounded by $1$. This sets us
under the assumptions of Lemma B.4 save for the normalization of the curve
$\sigma$. This is the origin of the $c(\sigma,\epsilon)$ factor in the
statement of the lemma, as follows. Split $[0,1]$ into $N$ intervals $I_{j}$
of equal length $1/N$. It is then trivial to check that the standard
reparameterization $\sigma_{\lambda}\circ\psi_{j}$ satisfies
$\|d^{s}\sigma_{\lambda}\circ\psi_{j}\|_{\infty}=\lambda\|d^{s}\sigma\|_{\infty}/N^{s}$.
Set $c(\sigma,\epsilon):=\max_{1\leq s\leq
k}(\lambda\|d^{s}\sigma\|_{\infty})^{\nicefrac{{1}}{{s}}}=\max_{1\leq s\leq
k}(\epsilon^{-1}\|d^{s}\sigma\|_{\infty})^{\nicefrac{{1}}{{s}}}$. Clearly by
choosing $N\approx c(\sigma,\epsilon)$, by which we mean $c(\sigma,\epsilon)$
rounded up to the closest integer, we have that
$\sigma_{\lambda}\circ\psi_{j}$ is normalized for all $j$. Then an inductive
application of Lemma B.4 as described before to each piece
$\sigma_{\lambda}\circ\psi_{j}$ in turn shows that each
$S_{j}:=\\{t\in[0,1]:\sigma_{\lambda}\circ\psi_{j}(t)\in\lambda\cdot B\\}$ can
be covered by at most
$(\mu(k)M(f_{\lambda})^{\nicefrac{{1}}{{k}}})^{n}=(\mu(k)M(f)^{\nicefrac{{1}}{{k}}})^{n}$
intervals $J_{ji_{1}i_{2}\ldots i_{n}}$. Scaling back down to $V$ one has
$S_{j}=\\{t\in[0,1]:\sigma\circ\psi_{j}(t)\in B\\}$, and so
$\sigma^{-1}(B)=\bigcup_{j}S_{j}\subseteq\bigcup_{ji_{1}\ldots
i_{n}}J_{ji_{1}\ldots i_{n}}$ is then covered by at most
$c(\sigma,\epsilon)(\mu(k)M(f)^{\nicefrac{{1}}{{k}}})^{n}$ intervals. ∎

###  B.3. The one-step estimate

Here we finally prove Lemma B.4. The basic strategy of the proof consists in
replacing $f\circ\sigma$ with local Taylor approximations $Q$ of degree $k$
and solving the problem for these.

The polynomial case. In all the following statements
$Q:[0,1]\longrightarrow\mathbb{R}^{3}$ is a polynomial curve of degree $k$
which will eventually be a Taylor approximation to $f\circ\sigma$. Any number
expressed in the form $\alpha(k)$ is meant to depend only on $k$ and not on
$Q$, $\lambda$, etc.

(1) Let $B^{\prime}\subseteq V_{\lambda}$ be a ball of radius
$\nicefrac{{3}}{{2}}$ and set

$S^{\prime}=\\{t\in[0,1]:Q(t)\in B^{\prime}\\}.$

Then $S^{\prime}$ can be covered by at most $\alpha_{1}(k)$ intervals $J_{i}$
with the property that $Q(J_{i})$ is contained in a ball of radius $2$.

Proof. $B^{\prime}$ is the intersection of an Euclidean ball with the solid
torus $V_{\lambda}$. These can be described by polynomial inequations of
degrees $2$ and $4$ respectively, and so $S^{\prime}$ is described by two
polynomial inequalities of degrees $2k$ and $4k$. The set $S^{\prime}$ is
therefore a union of intervals (possibly degenerated) whose endpoints are
roots of those polynomials or $0$ or $1$. Thus $S^{\prime}$ consists of at
most $\alpha_{1}(k):=6k+2$ connected components (if all of them are
singletons; if all of them are nondegenerate intervals there can be at most
$3k+1$ of these). If a component of $S^{\prime}$ is a singleton, we just
enlarge it ever so slightly that its image under $Q$ is contained in the ball
of radius $2$ concentric with $B^{\prime}$.

(2) Assume that the image of $Q$ is contained in a ball of radius $2$. Then
$[0,1]$ can be partitioned into $\alpha_{2}(k)$ intervals $J_{i}$ such that
the reparameterized curves $Q\circ\psi_{i}$ satisfy the
$\nicefrac{{1}}{{2}}$–normalization condition
$\|d^{s}(Q\circ\psi_{i})\|_{\infty}\leq\nicefrac{{1}}{{2}}$ for
$s=1,\ldots,k$.

Proof. A result of Markov (see for example [29]) implies that there exists a
constant $c(k)$ such that for every polynomial
$Q:[0,1]\longrightarrow\mathbb{R}^{3}$ of degree $k$ one has
$\|d^{s}Q\|_{\infty}\leq c(k)\|Q\|_{\infty}$ for $s=1,\ldots,k$. Under our
assumptions this is bounded above by $2c(k)$. Thus splitting $[0,1]$ into
$N\approx 4c(k)$ intervals $J_{i}$ of equal length and reparameterizing
affinely via $\psi_{i}$, each curve $Q_{i}:=Q\circ\psi_{i}$ will satisfy
$\|d^{s}(Q\circ\psi_{i})\|_{\infty}=N^{-s}\|d^{s}Q\|_{\infty}\leq
N^{-1}\|d^{s}Q\|_{\infty}\leq N^{-1}2c(k)\leq\nicefrac{{1}}{{2}}$ for
$s=1,\ldots,k$.

Combining (1) and (2) we get the following:

######  Lemma B.5.

Let $Q:[0,1]\longrightarrow\mathbb{R}^{3}$ be a polynomial curve of degree
$k$. Let $B^{\prime}$ be a ball of radius $\nicefrac{{3}}{{2}}$ in
$V_{\lambda}$. Then the set

$S^{\prime}=\\{t\in[0,1]:Q(t)\in B^{\prime}\\}$

can be covered by at most $\alpha(k)$ intervals $J_{i}$ such that $Q|_{J_{i}}$
is $\nicefrac{{1}}{{2}}$–normalized for each $i$.

###### Proof.

By (1) the set $S^{\prime}$ can be covered by $\alpha_{1}(k)$ intervals
$J_{i}$. For each of these we consider the reparameterized curve
$Q\circ\psi_{i}$. Its image is contained in a ball of radius $2$ as stated in
(1), and so applying (2) to $Q\circ\psi_{i}$ the interval $[0,1]$ can be split
into no more than $\alpha_{2}(k)$ intervals such that $(Q\circ\psi_{i})$
restricted to each of them is $\nicefrac{{1}}{{2}}$-normalized. ∎

Taylor approximations. The basic strategy of the proof of Lemma B.4 will
consist in partitioning the interval $[0,1]$ into subintervals and replacing
$f\circ\sigma$ with its Taylor polynomial of degree $k$ on each of those
subintervals. We now make some comments about these approximations. Recall
that we have a $\mathcal{C}^{k}$ map $f_{\lambda}:V_{\lambda}\longrightarrow
V_{\lambda}$ with $\|d^{s}f_{\lambda}\|_{\infty}\leq 1$ for $s=2,\ldots,k$ and
also a $\mathcal{C}^{k}$ normalized curve
$\sigma_{\lambda}:[0,1]\longrightarrow V_{\lambda}$. For notational ease we
will henceforth drop the subindex $\lambda$ from all objects except for
$V_{\lambda}$ itself.

(1) The derivatives of order $s$ of $f\circ\sigma$ can be expressed via the
chain rule and, using the fact that $f$ has derivatives $\leq 1$ for orders
$s\geq 2$ and $\sigma$ has derivatives $\leq 1$ for all orders, we see that
for every $s=1,\ldots,k$ one has $\|d^{s}(f\circ\sigma)\|\leq A(k)+B(k)M(f)$
for some constants $A(k)$ and $B(k)$ that depend only on $k$. Assuming that
$M(f)\geq 1$ this can be written as $\|d^{s}(f\circ\sigma)\|_{\infty}\leq
c_{1}(k)M(f)$ for some constant $c_{1}(k)=A(k)+B(k)$ which depends only on
$k$. When $M(f)<1$ we can just write it as
$\|d^{s}(f\circ\sigma)\|_{\infty}\leq c_{1}(k)$, but this case will be
somewhat trivial in what follows.

(2) Let $J\subseteq[0,1]$ be a closed interval of length $\delta$ (which one
thinks of as being short). Let $t_{0}$ be the midpoint of $J$ and denote by
$P$ the Taylor polynomial of degree $k$ for $f\circ\sigma$ at $t_{0}$.
Starting with the very crude trivial bound
$\|(f\circ\sigma)^{(k)}(t)-(f\circ\sigma)^{(k)}(t_{0})\|\leq
2\|(f\circ\sigma)^{(k)}\|_{\infty}$ and integrating it one obtains

$\|(f\circ\sigma)^{(s)}|_{J}-P^{(s)}|_{J}\|_{\infty}\leq\frac{2\|(f\circ\sigma)^{(k)}\|_{\infty}}{(k-s)!}\delta^{k-s}\leq
2c_{1}(k)M(f)\delta^{k-s}$

for $s=0,1,\ldots,k$. Suppose that $\psi:[0,1]\longrightarrow J$ is the
standard affine reparameterization, so that $\psi^{\prime}=\delta$. Clearly
$(f\circ\sigma\circ\psi)^{(s)}=\delta^{s}(f\circ\sigma)^{(s)}\circ\psi$, and
so

(B.4)
$\|(f\circ\sigma\circ\psi)^{(s)}-(P\circ\psi)^{(s)}\|_{\infty}=\delta^{s}\|(f\circ\sigma)^{(s)}|_{J}-P^{(s)}_{J}\|_{\infty}\leq
2c_{1}(k)M(f)\delta^{k}.$

Proof of Lemma B.4. We are finally ready to prove the one-step estimate. Let
$B\subseteq V_{\lambda}$ be a ball of radius $1$ and denote by $B^{\prime}$
the closed ball with the same center as $B$ but radius $\nicefrac{{3}}{{2}}$.
As in the statement of the lemma, set $S=\\{t\in[0,1]:f\circ\sigma(t)\in
B\\}$.

Partition $[0,1]$ as the union of
$N\approx(4c_{1}(k)M(f))^{\nicefrac{{1}}{{k}}}$ closed intervals $J_{i}$ of
equal length $\delta=1/N$, so that
$2c_{1}(k)M(f)\delta^{k}\leq\nicefrac{{1}}{{2}}$. Denote by $P_{i}$ the Taylor
approximation to $(f\circ\sigma)|_{J_{i}}$ as described above and by
$\psi_{i}:[0,1]\longrightarrow J_{i}$ the standard affine reparametrization.
Let us focus on one of the intervals $J_{i}$. By (B.4) we have that
$\|(f\circ\sigma\circ\psi_{i})-(P_{i}\circ\psi_{i})\|_{\infty}\leq\nicefrac{{1}}{{2}}$.
This ensures that if $t\in S\cap J_{i}$ then $P_{i}(t)$ belongs to the ball
$B^{\prime}$. Therefore $S\cap J_{i}\subseteq\\{t\in J_{i}:P_{i}(t)\in
B^{\prime}\\}=\psi_{i}^{-1}(\\{t\in[0,1]:P_{i}\circ\psi_{i}(t)\in
B^{\prime}\\})$. By Lemma B.5 applied to $Q_{i}=P_{i}\circ\psi_{i}$ the set
$\\{t\in[0,1]:P_{i}\circ\psi_{i}(t)\in B^{\prime}\\}$ can be covered by at
most $\alpha(k)$ intervals $J^{\prime}_{ij}$ such that
$(P_{i}\circ\psi_{i})|_{J^{\prime}_{ij}}$ is $1/2$–normalized. Taking the
preimage of these via $\psi_{i}$ produces a family of at most $\alpha(k)$
intervals $J_{ij}$ which cover $S\cap J_{i}$. Therefore $S$ itself is covered
by no more than $\alpha(k)N\approx\mu(k)M(f)^{\nicefrac{{1}}{{k}}}$ intervals
$J_{ij}$, where $\mu(k)$ simply absorbs all factors other than $M(f)$.

To conclude the proof it only remains to show that each
$f\circ\sigma|_{J_{ij}}$ is normalized. Let $\psi_{ij}:[0,1]\longrightarrow
J_{ij}$ and $\psi^{\prime}_{ij}:[0,1]\longrightarrow J^{\prime}_{ij}$ be the
standard affine reparameterizations. Observe that the definition of $J_{ij}$
as $\psi_{i}^{-1}(J^{\prime}_{ij})$ ensures that
$\psi_{ij}=\psi_{i}\circ\psi^{\prime}_{ij}$. The condition that
$(P_{i}\circ\psi_{i})|_{J^{\prime}_{ij}}$ be $1/2$–normalized thus reads
$\|d^{s}(P_{i}\circ\psi_{ij})\|\leq\nicefrac{{1}}{{2}}$ for $s=1,\ldots,k$. We
also have

$\|d^{s}(f\circ\sigma\circ\psi_{ij})-d^{s}(P_{i}\circ\psi_{ij})\|\leq\|d^{s}(f\circ\sigma\circ\psi_{i})-d^{s}(P_{i}\circ\psi_{i})\|\leq\nicefrac{{1}}{{2}}$

where the first inequality follows because $\psi^{\prime}_{ij}$ has a
derivative less than $1$ and the second from (B.4) and the choice of $N$.
Therefore $\|d^{s}(f\circ\sigma\circ\psi_{ij})\|\leq 1$ as required.

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