An example on movable approximations of a
minimal set in a continuous flow
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
Petra ?Sindel?a?rov?a
Certificate of Approval:
Jack B. Brown
Professor
Department of Mathematics
Krystyna Kuperberg, Chair
Professor
Department of Mathematics
Michel Smith
Professor
Department of Mathematics
Stewart Baldwin
Professor
Department of Mathematics
Stephen L. McFarland
Acting Dean
Graduate School
An example on movable approximations of a
minimal set in a continuous flow
Petra ?Sindel?a?rov?a
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 11, 2006
An example on movable approximations of a
minimal set in a continuous flow
Petra ?Sindel?a?rov?a
Permission is granted to Auburn University to make copies of this dissertation at
its discretion, upon the request of individuals or institutions and at their expense.
The author reserves all publication rights.
Signature of Author
May 11, 2006
Date of Graduation
iii
Vita
Petra ?Sindel?a?rov?a, daughter of Petr ?Sindel?a?r and Eva (Vyslou?zilov?a) ?Sindel?a-
?rov?a, was born September 5, 1976, in Krnov, Czech Republic. She graduated cum
laude from bilingual Czech?French section of ?Slovansk?e gymn?azium? at Olomouc,
Czech Republic, in 1996. She attended Silesian University at Opava, Czech Republic,
and graduated with a Bachelor of Informatics and Computer Sciences in 1999, and
a Master of Mathematics in 2001. In 2000 she spent one semester at University of
W?urzburg, Germany. In 2003 she obtained a one year Marie Curie Postgraduate
Studentship to study at University of Warwick, United Kingdom. She entered the
graduate program in mathematics at Auburn University in 2004. She is not married
and has no children.
iv
Dissertation Abstract
An example on movable approximations of a
minimal set in a continuous flow
Petra ?Sindel?a?rov?a
Doctor of Philosophy, May 11, 2006
(M.A., Silesian University at Opava, Czech Republic, 2001)
(B.S., Silesian University at Opava, Czech Republic, 1999)
38 Typed Pages
Directed by Krystyna Kuperberg
In the present dissertation the study of flows on n-manifolds in particular in
dimension three, e.g., R3, is motivated by the following question. Let A be a compact
invariant set in a flow on X. Does every neighbourhood of A contain a movable
invariant set M containing A? Here, a dynamical system (a flow) is the pair (X,pi),
where X, in general, is a manifold, pi : X ?R ? X is continuous, pi(x,0) = x and
pi(pi(x,t1),t2) = pi(x,t1 + t2), for each x ? X and each t1,t2 ? R. A nonempty set
A ? X is invariant if pi(A,t) = A for each t ? R. A compact invariant set A ? X is
stable if for every neighbourhood U of A there exists a neighbourhood V of A with
V ? U, such that pi(V,t) ? U for all t ? 0. The topological notion of movability
(also called the UV-property) is in the sense of K. Borsuk and is closely related to
the notion of stability in dynamics. A continuum M in X is said to be movable if for
every neighbourhood U of M there exists a neighbourhood U0 ? U of M such that
v
for every neighbourhood W of M there is a continuous map ? : U0?I ? U satisfying
the conditions ?(x,0) = x and ?(x,1) ? W for every point x ? U0. It is known that
a stable solenoid (an intersection of a nested sequence of solid tori positioned one
inside another in some regular way) in a flow on a 3-manifold has approximating
periodic orbits in each of its neighbourhoods. The solenoid with the approximating
orbits form a movable set, although the solenoid is not movable. Not many such
examples are known. The main part of the dissertation consists of constructing
an example in R3 which uses Denjoy?like invariant approximating sets instead of
periodic orbits. This gives a partial answer to the above question. The construction
involves both, the adding machines and Denjoy maps, and the suspension of specially
defined Cantor set homeomorphisms.
vi
Acknowledgments
The author would like to thank Professor Krystyna Kuperberg for directing the
research leading up to this dissertation, and for all the help she has given her during
the graduate studies in Auburn. She also wishes to express her gratitude to Professor
Jack B. Brown and his wife Jane Brown. Additional thanks go to all members of
her advisory committee for many suggestions and corrections that improved this
dissertation. Special thanks are due to author?s parents for their support, patience
and love.
vii
Style manual or journal used Transactions of the American Mathematical
Society (together with the style known as ?auphd?). Bibliography follows van
Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically
LATEX) together with the departmental style-file auphd.sty.
viii
Table of Contents
1 Introduction 1
2 Minimal sets 4
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Solenoidal and Denjoy minimal sets . . . . . . . . . . . . . . . . . . . 6
2.2.1 Adding machines and solenoids . . . . . . . . . . . . . . . . . 6
2.2.2 Irrational rotation, blowing up orbits, and Denjoy continuum . 11
3 An example of a suspension on a mapping torus ? 13
3.1 The set ? is not a solenoid . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 The set ? is not movable . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Embedding of ? in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Bibliography 28
ix
Chapter 1
Introduction
The main subject of this dissertation is the study of continuous dynamical sys-
tems. The work is inspired by an open problem stated for invariant sets: Let A
be a compact invariant set in a flow on an n?dimensional manifold. Does every
neighbourhood of A contain a movable compact invariant set containing A?
It is known that the answer is positive for a stable set called a solenoid in
dimension three. Such an example appeared in a paper by H. Bell and K. R. Meyer
[1]. In their constructions the resulting stable solenoid has periodic orbits in every of
its neighbourhoods. By a modification of this example they also proved that analogue
result for a stable solenoid in higher dimension does not hold. Later M. Kulczycki
showed in his dissertation [10], that it is possible to drop the stability assumption but
only under some extra requirements on the flow. Another result by E. S. Thomas,
Jr. in [18] guarantees that a minimal solenoid in dimension three is never an isolated
invariant set, i.e., in every neighbourhood of the solenoid there are other invariant
sets.
The author of this dissertation gives a partial answer to the above question by
constructing an example in dimension three and by considering a set that is not
stable and is not a solenoid. To the knowledge of the author, such a case has not
been published yet.
1
In Chapter 2, first the definition of a dynamical system or what is also called a
continuous flow is introduced. Definitions of a minimal set and almost periodicity are
reviewed next. Later we recall the key notions of our study, in particular definitions
of special minimal sets, solenoids, and Denjoy continua, and we summarize their
basic properties. For the construction of these sets, we first need to discuss a map of
a Cantor set that is known as the adding machine, and describe a process of blowing
up orbits, that was first published in a paper by A. Denjoy in [7]. Then the notion
of suspension is established. It is a continuous dynamical system obtained from a
discrete dynamical system. All these objects and maps constitute a significant part
of the example constructed in the last chapter of this dissertation. They have been
a popular field of study of many authors.
Chapter 3 starts with introducing a dynamical system (a suspension) on a set ?.
The set ? is minimal under the considered flow. The main original results provided
in this chapter are the following.
The first two theorems describe the set ?.
Theorem 1.1 The set ? is not a solenoid.
Theorem 1.2 The curve ? is not movable.
The next two theorems show that the set ? is an invariant set in a flow in
dimension three. Moreover, in every neighbourhood of ? there is a compact invariant
set that we call Denjoy?like. These Denjoy?like sets are proved to be movable. The
set ? cannot have approximating periodic orbits in each of its neighbourhoods as in
the case of a stable solenoid in [1]. This is due to the fact that the flow defined on
? is not almost periodic.
2
Theorem 1.3 There exists an embedding of ? in a mapping torus in R3 with the
property that ? is approximated by invariant Denjoy?like sets Dn, n ? N.
Theorem 1.4 Every Denjoy?like set Dn, n ? N, is movable.
Finally we prove that ? together with any sequence of its approximating movable
sets is a movable set.
Theorem 1.5 Let Dprime =uniontext?n=k Dn. For any k ? N, the union of ? and Dprime is movable.
To complete the description of the properties of ? and the approximating sets
Dn, n ? N, we show that none of those sets is stable. It is a corollary of a result by
J. Buescu and I. Stewart [6].
Theorem 1.6 The set ? and the sets Dn, n ? N, are not stable.
Although the sets ? and Dn, n ? N, are not stable as the sets in the example
by H. Bell and K. R. Meyer, and moreover ? is not movable, the union of ? and Dprime
is movable. Therefore, this case still yields a kind of ?stability?.
3
Chapter 2
Minimal sets
2.1 Preliminaries
In this section, we introduce the definition of a dynamical system that is some-
times also called a continuous flow, the definition of a minimal set, and we establish
the notation.
Throughout the paper we usually consider metric spaces unless stated otherwise.
The symbol R is the the real line, Z and N stand for all integer and all natural
numbers, respectively. We denote by I the compact unit interval [0,1]. Let A
denotes the closure of a set A. By a neighbourhood of a set A we understand an
open set containing A.
A dynamical system on X is the triplet (X,R,pi) where pi is a continuous map
(also called a continuous flow) from the product space X ? R into the space X
satisfying pi(x,0) = x and pi(pi(x,t1),t2) = pi(x,t1+t2) for every x ? X and t1,t2 ? R.
The phase map pi determines two other maps when one of the variables x or t
is fixed. For a fixed t ? R, the map pit : X ? X is defined by pit(x) = pi(x,t) and is
called a motion through x. For each t ? R, pit is a homeomorphism of X onto itself
(see [2]). For a fixed x ? X, the map pix : R ? X is given by pix(t) = pi(x,t).
A discrete dynamical system on X is the triplet (X,Z,f) where f is a continuous
map of X into itself. The dynamics is defined through iterations of f. The n?th
4
iterate of f is the map fn = f ? fn?1, n ? N. The negative iterates are given by
f?n = (fn)?1, n ? N. We use the notation f0 = f.
The following definitions concern dynamical systems (X,R,pi). The reader can
easily reformulate all the notions for the discrete case. The orbit of a point x ? X is
the set {pit(x) | t ? R} and the positive half orbit is the set {pit(x) | t ? 0}. A point
x ? X is said to be a fixed point (or a critical point) if pi(x,t) = x for all t ? R. A
point x ? X is periodic if there is a T negationslash= 0 such that pi(x,t) = pi(x,t+T) for all t ? R.
In this case the smallest such number T ? R will be called a period of x. A nonempty
set A ? X is called invariant whenever pi(x,t) ? A for all x ? A and t ? R. A closed
invariant set is minimal if it contains no proper closed invariant subset. It is easy to
see that if A is compact, then A is minimal if and only if the positive half orbit of
every point in A is dense in A. The simplest example of minimal sets are the orbits
of fixed or periodic points.
Minimal sets can also arise in the following way. Suppose (X,d) is a metric space.
A point x ? X is said to be almost periodic (as defined in [15] on page 384) if, given
? > 0, there is a set E ? R which is relatively dense such that d(pit(x),pit+?(x)) < ?
for all ? ? E and t ? R. A set E ? R is relatively dense means that for some number
L > 0 every interval in R of length L contains a point of E. If x is almost periodic
and the closure ? of the orbit of x is compact and metrizable, then ? is a minimal
set (see [15], page 385). One?dimensional minimal sets of this type are described in
the next section.
A compact invariant set A ? X is stable if for every neighbourhood U of A there
exists a neighbourhood V of A with V ? U, such that pi(V,t) ? U for all t ? 0.
5
2.2 Solenoidal and Denjoy minimal sets
The main construction of this paper involves solenoids and Denjoy continua.
They are defined in this section. We also introduce some other well known objects
and recount their basic properties. We use similar background as it can be found in
[1], [6] and [17].
2.2.1 Adding machines and solenoids
First we recall the abstract definition, via symbolic dynamics, of the class of
maps of the Cantor set called adding machines. Let k = {kn}n?1 be a sequence of
integers with kn > 1 for all n ? N. Let ?k =producttext?n=1{0,1,2,...,kn?1} be the space of
all one?sided infinite sequences i = {in}n?1 such that 0 ? in < kn with the product
topology. One can see that ?k is metrizable and the metric
d(i,j) =
?summationdisplay
n=1
|in ?jn|
knn
is compatible with this topology.
The adding machine with base k = (k1,k2,...) is the map
?k : ?k ? ?k
defined by ?k(...,iq,...) = (...,jq,...) in the following way
? if iq = kq ?1 for all q then jq = 0 for all q, i.e. ?k(...,iq,...) = (0,0,...);
or
6
? if the first index q with iq < kq ?1 is r then jq = 0 for 1 ? q < r, jr = ir + 1,
and iq = jq for q > r, i.e. ?k(...,iq,...) = (0,0,...,ir + 1,ir+1,ir+2,...).
A familiar description of this operation is ?add one and carry? because roughly
speaking we add one to the first term of the sequence, and if the result is zero we
add one to the next term, and so on. It is also well?known that ?k is a minimal
homeomorphism of ?k (cf., e.g., [6], page 277, [1], page 411?2, or [12], pages 242?3).
Let us now construct a Cantor set by the following common algorithm. It is
especially known for the ternary (or so called middle?third) Cantor set which can be
seen as all members in the compact unit interval I = [0,1] with ternary expansion
using only digits 0 and 2.
Take the interval I and let k = (k1,k2,...) be as previously. In the first step
remove from I a collection of k1 ?1 nonempty, open intervals with pairwise disjoint
closure and not containing 0 or 1 as an endpoint. Moreover, the intervals that are
removed and that remain must all have the same length. Inductively, at the n?th
step remove from each of the remaining intervals kn ? 1 intervals in the same way
and denote the remaining collection of closed intervals by In. At each step we obtain
a compact set that is a subset of the compact set resulting from the previous step.
As a limit of this process we take the intersection of this nested sequence of compact
sets and denote it by C, i.e. C =intersectiontext?n=1 In. It is well known that C is a non?empty,
perfect, totally disconnected compact metric space called the Cantor set.
We can easily see that the space ?k is homeomorphic to such a Cantor set. In-
deed, any point c ? C is ?coded? as follows to obtain a point i ? ?k. If c lies in
7
the (i1 +1)?th interval from the left of the collection of intervals I1 (let?s denote this
interval by Ii11 ) then the first coordinate of i is i1. Inductively, in the n?th step, if c
lies in the (in + 1)?th interval from the left of the collection of intervals Iin?1n?1 (let?s
denote this interval by Iinn ) then the n?th coordinate of i is in.
Adding machines occur in a natural way in the study of solenoids. To see it, we
need to introduce some auxiliary definitions.
An inverse sequence {Xi,fji } of topological groups is a sequence of topological
groups {Xi}i?N together with a collection of continuous homomorphisms {fji : Xj ?
Xi}i?j satisfying
? fii : Xi ? Xi is the identity for all i ? N; and
? fki = fji ?fkj for all i ? j ? k, i,j,k ? N.
Notice, that it is sufficient to define fi+1i (called bonding maps) for each i ? N
to determine all fji by the second part above.
The inverse limit of an inverse sequence {Xi,fji } is the topological group
X = lim? {Xi,fji } = {(x1,x2,...) ?
productdisplay
i?N
Xi | xi = fi+1i (xi+1) for all i ? N}
with the topology inherited from the product producttexti?N Xi with the product topology.
A solenoid can be defined in several ways. The presented definitions disclose
homeomorphic objects, we omit the technical proof.
8
For example, by a solenoid we mean a space that is homeomorphic to the inverse
limit of a sequence of bonding maps fi+1i : S1 ? S1 given by fi+1i (z) = zni, where
S1 is the unit circle in the complex plane and ni ? {2,3,...}.
Geometrically, a solenoid is the intersection of a nested sequence of solid tori in
R3 such that each torus is positioned in a specific way inside the previous one as on
the picture below.
Before we discuss another way to construct a solenoid, we need the definition of
a suspension on a mapping torus.
Let A be a set and h : A ? A a homeomorphism. The mapping torus TA of the
homeomorphism h is the set obtained by the following identification. Consider the
set A?I. For each x ? A we identify the point (x,1) with the point (h(x),0). We
9
define a dynamical system on TA by piTA((x,0),t) = (x,t) for each x ? A and each
t ? [0,1] and extend piTA in a unique way to a dynamical system on the whole of TA
by the equivalence relation ?
(x,t) ? (y,s) if and only if
(x = y and t = s) or
(t = 1,s = 0 and h(x) = y) or
(t = 0,s = 1 and h(y) = x).
A dynamical system defined as above for any homeomorphism h of an arbitrary
set is called a suspension of h on the mapping torus TA (see also [17], Appendix).
Now we are ready to construct a solenoid ?. Consider the space S?[0,1], where
S is a Cantor set, and a homeomorphism h?k : S ? S that is the adding machine
as defined above. Denote by ? the mapping torus of the homeomorphism involved
and by pi? the dynamical system on ? that is given by the suspension of h on the
mapping torus ?.
Because the orbit of every point in any adding machine is dense, the whole ? is
minimal under pi?.
To define a dynamical system onR3 with a subspace homeomorphic to a solenoid
as a minimal set see Section 2 in [1].
10
2.2.2 Irrational rotation, blowing up orbits, and Denjoy continuum
This section is devoted to a construction of another useful minimal set. We start
with a rotation through the angle 2pi? of the unit circle r? : S1 ? S1, where ? is
an irrational number. We will change this map and obtain a new homeomorphism
hr? with a minimal set which is neither a single closed orbit, nor the whole space.
Let us consider the circle S1 to be obtained from the interval [0,1] by identifying its
endpoints. We choose a point x0 ? S1, and at each point xn = rn?(x0) of its orbit
we insert a small closed interval In into the circle. To fit again into a new circle
of circumference 1 + a denoted by S1a, the intervals In have to satisfy the condition
a = summationtextn?Z length(In) < ?. There is a continuous onto map g : S1a ? S1 which
collapses each interval In ? S1a to the corresponding point xn ? S1 and is one?to?one
otherwise. We can now define the new map hr? : S1a ? S1a, which is topologically
semi?conjugate to r? under a topological semi?conjugacy g, i.e.
g ?hr? = r? ?g (2.1)
and g is continuous and onto by definition. This semi?conjugacy determines hr? at
all points at which g is one?to?one. We can define g at the remaining points such
that hr? is a homeomorphism. Moreover, it is possible to obtain a C1 diffeomorphism
hr?, for details see [17]. It is an easy exercise to show that the orbits of r? are mapped
onto orbits of hr? by means of a topological semi?conjugacy g, thus ?the dynamics
is preserved?.
The irrationality of ? implies that r? and, by 2.1, also hr? have no periodic points.
Hence, the compact invariant set S1a \Intuniontextn?ZIn contains a minimal set (under hr?)
11
D which is clearly a Cantor set and is neither a single closed orbit, nor the whole
space S1a.
Remark 2.1 Note that to be a topological conjugacy the map g has to be a home-
omorphism.
Take again the suspension pi? of hr? (restricted to D) on the mapping torus ?
obtained from D. The whole ? is minimal under pi?. The set ? is referred to as a
Denjoy continuum. The process of inserting intervals is called ?blowing up orbits?.
The construction of pi? was first described by A. Denjoy in [7], page 352?5. For
details of this construction see [17], Appendix or [14].
12
Chapter 3
An example of a suspension on a mapping torus ?
We construct the following example of a suspension. Suppose h?k : S ? S,
hr? : D ? D, pi? and pi? are as in the previous Section 2.2.
Take the product h?k ?hr? and denote it by F : S ?D ? S ?D. Let ? be the
mapping torus of F and consider the suspension pi? of F on ?.
In this chapter we will show that ? is not a solenoid (and that F is not an
adding machine) and that ? is not a movable set. Then we will embed ? in R3 and
we will discuss the properties of this embedding. We will also state that ? and its
approximating sets are not stable.
3.1 The set ? is not a solenoid
Using the fact that pi? is not almost periodic for any point we will show that ?
is not a solenoid.
Lemma 3.1 Every point is almost periodic for pi?.
Proof. The proof can be found in [15]. It also follows from [6], page 277. a50
The proof of the next lemma uses Theorem 1 by E. S. Thomas, Jr. [18].
13
Theorem 3.2 (Thomas) If ? is a compact 1?dimensional metric space which is
minimal under some flow and if some point of ? is almost periodic, then ? is a
solenoid or a circle.
Lemma 3.3 There are no almost periodic points for pi?.
Proof. Suppose there exits an almost periodic of pi?. Then by Theorem 3.2 the set
? is a solenoid or a circle. But it is clearly not a circle and, by [6] Remark 7.9,
hr? : D ? D is not an adding machine (and not topologically conjugate to one).
Contradiction. a50
Proposition 3.4 There are no almost periodic points for pi?. Hence, F is not (topo-
logically conjugate to) an adding machine and ? is not a solenoid.
Proof. Let ((x1,y1),t1),((x2,y2),t2) ? ?. We denote and define a metric on ? by
d?(((x1,y1),t1),((x2,y2),t2)) = d?((x1,t1),(x2,t2)) +d?((y1,t1),(y2,t2)), (3.1)
where d? and d? is a metric on ?, and on ?, respectively.
An easy check verifies that d? is a well defined metric on ?. Indeed, let
d?(((x1,y1),t1),((x2,y2),t2)) = 0. Then by (3.1) and the fact that both d? andd? are
metrics, we have d? = d? = 0. It means that (x1,t1) = (x2,t2) and (y1,t1) = (y2,t2).
Consequently, x1 = x2, y1 = y2 and t1 = t2, i.e. ((x1,y1),t1) = ((x2,y2),t2). The
converse is trivial. This completes the proof of positivity of the metric d?. Symme-
try and triangular inequality are immediate using (3.1) and symmetry and triangular
inequality of d? and d?.
14
A more natural way to define a metric on ? would be to establish a general
metric for any suspension. Roughly, such a metric would reflect naturally the length
of the orbit of a point in the direction of the flow. But since we want to avoid
technicalities, the presented metric is more convenient for our purpose.
We need to introduce projections p1 and p2 of ? on ? and on ?, respectively.
These projections p1 : ? ? ? and p2 : ? ? ? are defined by p1(?) = ? and
p2(?) = ? where ? = ((x,y),t), ? = (x,t) and ? = (y,t). The maps p1 and p2
are well defined continuous, surjective maps preserving the suspension. Indeed, let
((x1,y1),t1),((x2,y2),t2) ? ? and pi?(((x1,y1),t1),t) = ((x2,y2),t2) for some t ? R.
By definition of suspension, it means that F(t1+t) div 1(x1,y1) = (x2,y2) and t2 = (t1+
t) mod 1, where t1+t = (t1+t) div 1+(t1+t) mod 1. Recall that F = h?k ?hr?. To
prove that the projections are well defined we must prove that pi?(p1((x1,y1),t1),t) =
p1((x2,y2),t2), and similarly for p2. We have pi?(p1((x1,y1),t1),t) = pi?((x1,t1),t) =
(x2,t2) = p1((x2,y2),t2), where again, by the defition of suspension, h(t1+t) div 1?k (x1) =
(x2) and t2 = (t1 + t) mod 1. The proof for p2 is analogous. Surjectivity and
continuity are obvious.
Suppose ? ? ? is almost periodic with respect to pi?. Let ? > 0. Then by
definition, there is a relatively dense set E ? R such that d?(pit?(?),pit+?? (?)) < ? for
every ? ? E and every t ? R. Since pi?,pi? and pi? are suspensions and by (3.1) we
have
d?(pit?(?),pit+?? (?)) = d?(pit?(?),pit+?? (?)) +d?(pit?(?),pit+?? (?)) < ?.
15
Hence, d?(pit?(?),pit+?? (?)) < ?. But it is not possible by Lemma 3.3. a50
The fact that the flow pi? on ? is not almost periodic implies that ? cannot
have approximating orbits in each of its neighbourhoods as in the case of a stable
solenoid in [1].
Remark 3.5 Let f be a homeomorphism defined on a Cantor set that is minimal
underf. We have proved that if the product of an adding machine with the functionf
is (topologically conjugate to) an adding machine, then f must also be (topologically
conjugate to) an adding machine. The reader can also convince himself, that a
product of two adding machines is (topologically conjugate to) an adding machine.
But we will not need it in this dissertation.
3.2 The set ? is not movable
As a corollary of results by K. Borsuk, J. Krasinkiewicz and A. Trybulec, we
will state that ? is not movable.
The notion of movability and n?movability was introduced by K. Borsuk (see
[4] and [5]) and is closely related to stability in dynamical systems.
Definition 3.6 A set which is both compact and connected is called a continuum.
Definition 3.7 A continuous map r : X ? A is said to be a retraction of X to A if
A ? X and r(A) = A. In this case, A is said to be a retract of X. A space Y is said
to be an absolute retract (abbreviated AR), provided that for each homeomorphism
h mapping Y onto a closed subset h(Y) of a space X the set h(Y) is a retract of
16
X. A space Y is called an absolute neighbourhood retract (abbreviated ANR), if for
every homeomorphism h mapping Y onto a closed subset of a space X there is a
neighbourhood U of the set h(Y) in the space X such that h(Y) is a retract of U.
Definition 3.8 Let X be an ANR. A continuum M ? X is said to be movable in X
if for every neighbourhood U of M there exists a neighbourhood U0 ? U of M such
that for every neighbourhood W of M there is a continuous map ? : U0 ? I ? U
satisfying the condition ?(x,0) = x and ?(x,1) ? W for every point x ? U0.
In several places we will need a result by Borsuk (see [4], page 142) about
independence of movability on the embedding.
Theorem 3.9 (Borsuk) Movability is a topological property. Thus, a continuum is
movable if it is homeomorphic to a continuum movable in the previous sense.
Definition 3.10 By a curve we understand any 1?dimensional continuum.
The following theorem combines Theorem 4.1 in [9] (see also [11]) with a theorem
in [19].
Theorem 3.11 (Krasinkiewicz, Trybulec) If f is a continuous map from a mov-
able curve X onto a curve Y, then Y is movable.
The proof of the next theorem appears in [4].
Theorem 3.12 (Borsuk) If ? is a solenoid then ? is not movable.
Corollary 3.13 The curve ? is not movable.
17
Proof. Let ? is a solenoid given by a mapping torus obtained from a Cantor set S as
presented in Section 2.2.1. Suppose that the Cantor set S here is the same one that
is used in construction of ?. Notice that both, ? and ?, are curves. Let p1 : ? ? ?
be a function defined by p1((x,y),t) = (x,t) (see the proof of Proposition 3.4). It is
a continuous well?defined map of ? onto ?, therefore ? is not movable by Theorems
3.12 and 3.11. a50
3.3 Embedding of ? in R3
In this section, we will first show that ? can be embedded in a flow in R3 in such
a way that it is approximated by Denjoy?like sets that are movable. We construct
them as a mapping torus of the product of the Denjoy map hr? on D and a map that
constitutes just of one periodic orbit O of a point in a discrete dynamical system.
These Denjoy?like sets (orbits) are ?stretched along? the orbits of the points from
?, i.e. for every point in ? we can find a point of the same Denjoy?like set that is as
close to the selected point in ? as we like if the Denjoy?like set is chosen sufficiently
long (in the sense that the periodic orbit O is sufficiently long) and sufficiently close
to ? in the sense of Hausdorff metric.
Then we will prove that although ? is not movable, its union with the approxi-
mating Denjoy?like sets is movable. We will complete the description by a corollary
giving that none of the sets ? and its approximating Denjoy?like sets are stable.
For the formulation of the theorems of this section we need some auxiliary
definitions.
18
Definition 3.14 Let O be a periodic orbit in a discrete dynamical system. Consider
the product D ?O with the product of the corresponding maps. We say that D is
a Denjoy?like set if it is the mapping torus of this product.
Definition 3.15 Let M be a complete metric space with a metric d, and CM be the
collection of all compact subsets of M. The Hausdorff metric dH on CM is defined
as follows. For A,B ? CM,
dH = sup{d(a,B),d(b,A) : a ? A,b ? B},
where
d(b,A) = inf{d(b,a) : a ? A}
and similarly for d(a,B).
Definition 3.16 We say that ? is approximated by Denjoy?like sets Dn, n ? N,
if every for every ? > 0 there is a Denjoy?like set Dj, for some j ? N, such that
dH(?,Dj) < ?.
The sets S and D can be embedded in R and therefore ? and the sets Dn, n ? N,
can be embedded in R3. Let the metric d needed in the previous definition be the
Euclidean metric of R3.
Before we state the main Theorem 3.18 of this section, we need the following
theorem that is proved, e.g., in [13] in Chapter 12 and in more general settings also
in Chapter 13.
19
Theorem 3.17 Let C1 and C2 be Cantor sets in R2 and h : C1 ? C2 a homeomor-
phism. Then there exists an orientation preserving homeomorphism H : R2 ? R2
such that H|C1 = h.
Theorem 3.18 There exists an embedding of ? in a mapping torus in R3 with the
property that ? is approximated by invariant Denjoy?like sets Dn, n ? N.
Proof. We can consider the Cantor set S being embedded in R, such coding is
described in Section 2.2.1. We approximate S by periodic orbits On, n ? N, in R,
in the following way. Let On = {on1,on2,...,onmn}, where the last lower index mn =
(ki ?1)?ki?1 ?ki?2 ?...?k2 ?k1 with the notation from the algorithm in Section 2.2.1.
The set On is a subset of the union of the intervals that are removed at i?th step
(there are exactly m intervals removed at this step), every point from On lying in a
different of these intervals. Hence, the sets On, n ? N are pairwise disjoint.
It means that there is a homeomorphism hprime?k : S ?uniontext?n=1 On ? S ?uniontext?n=1 On
such that hprime?k|S = h?k, and hprime?k|O
n
= On, for each n. Then (S ?uniontext?n=1 On) ? D is
a Cantor set that can be embedded in R2. By Lemma 3.17, hprime?k ? hr? has an ex-
tension Fprime : R2 ? R2 which is also a homeomorphism. We notice that Fprime is also
an extension of F. Therefore, ? is a subset of the mapping torus ?prime of Fprime, and the
suspension pi?prime of Fprime on ?prime is an extension of pi?. The verification of the fact that
? is approximated by pairwise disjoint invariant Denjoy?like sets Dn is immediate
from the construction. Finally, we remark that it is possible to extend the flow pi?prime
extended onto the whole R3 so that the properties of the embedding mentioned in
this theorem are preserved. a50
20
The next result by J. Krasinkiewicz [9] and also R. D. McMillan [11] generalize a
theorem of K. Borsuk [4] on movability of plane continua. By a surface we understand
a compact two dimensional manifold.
Theorem 3.19 (Krasinkiewicz, McMillan) Every continuum that can be embed-
ded in a surface is movable.
In the following, the Denjoy?like sets Dn, n ? N, are the sets constructed in the
proof of Theorem 3.18.
Theorem 3.20 Every Denjoy?like set Dn, n ? N, is movable.
Proof. We construct an embedding of Dn in a surface. Let hr? : S1a ? S1a be as in
Section 2.2.2. Consider n copies of S1a, i.e. the product S1a?On, where On is a periodic
orbit as in the proof of Theorem 3.18. We define a map g : S1a ?On ? S1a ?On to
be the product of the corresponding maps on S1a and On, respectively. The mapping
torus of the homeomorphism g is a surface homeomorphic to a surface of a torus
which is wrapped n?times. It is easy to see that this surface is homeomorphic to Dn.
By Theorem 3.19, Dn is movable. a50
The following definitions and Theorem 3.22 are necessary for the proof of The-
orem 3.24.
Definition 3.21 Let X and Y be topological spaces and let f0 and f1 be continuous
maps of X to Y. If there is a continuous map h : X?I ? Y such that h(x,i) = fi(x)
for i = 0,1, then we say that the maps f0 and f1 are homotopic. The map h is called
a homotopy between f0 and f1.
21
Theorem 3.22 (Borsuk?s homotopy extension theorem) Let M be a closed
subspace of a metrizable space X and f0 and f1 two homotopic maps of M to an
ANR. Then if f0 is continuously extendable over X, then f1 is also continuously
extendable over X. Moreover, for every extension of f0 one can find an extension of
f1 homotopic to it.
The proof of Borsuk?s homotopy extension theorem can be found, e.g., in [3].
Definition 3.23 The map pn : ? ? Dn, n ? N, defined bellow is called the n?th
projection of ? on Dn. For any ? = ((x,y),t) ? ? ? R3 with x ? S, y ? D and
t ? [0,1], the n?th projection is defined by pn(?) = ((onl ,y),t) . See the proof of
Theorem 3.18 for the construction of periodic points onl . The index l ? {1,2,...,mn}
is such that the point x = (i1,i2,...,in,...) ? S is mapped by pn to the closest point
onl ? Iin?1n?1 on the right of x, or if there is no such point on the right then to the left.
The intervals Iin?1n?1 are described in Section 2.2.1. By construction, pn is continuous.
Similarly are defined continuous projection of Dq on Dn, q > n. Let pqn : Dq ? Dn
be such that pqn((oqk,y),t) = ((onl ,y),t). For any index k ? {1,2,...,mq} the index
l ? {1,2,...,mn} is such that the point oqk is mapped by pqn to the closest point onl
on the right of oqk, or if there is no such point on the right then to the left. Therefore,
for any q > n,
pqn ?pq = pn.
Theorem 3.24 Let Dprime =uniontext?n=1Dn. The union of ? and Dprime is movable.
Proof. By definition of movability, we have to prove the following statement. For
every neighbourhood U of ? ?Dprime there is a neighbourhood U0 ? U of ? ?Dprime such
22
that for each neighbourhood W of ??Dprime, there is a continuous map ? satisfying the
conditions
? : U0 ?I ? U,?(x,0) = x and ?(x,1) ? W for every point x ? U0. (3.2)
We say in this case that U0 can be deformed to W within U.
Actually, we will prove a stronger statement: For every neighbourhood U of
??Dprime there is a number N ? N and a neighbourhood U0 ? U of ??Dprime such that
for every neighbourhood W of uniontextNj=1Dj, there is a continuous map ? satisfying the
conditions (3.2).
For a given neighbourhood U of ? ? Dprime we will construct the neighbourhood
U0 of ? ? Dprime as a finite union of pairwise disjoint neighbourhoods U1,U2,...,UN,
where Uj is a neighbourhood of Dj, j < N, and UN is a neighbourhood of the set
??uniontext?j=N Dj. Then we deform each set Uj, 1 ? j ? N, into W within U.
Let U be a neighbourhood of ? ?Dprime. Then there is an ? > 0 such that every
open ball with radius at most ? centered at a point from ??Dprime is contained in U.
By Theorem 3.18, ? is approximated, in the sense of Hausdorff metric, by pair-
wise disjoint Denjoy?like sets Dn, n ? N. Therefore, there exists a number Nprime ? N
such that dH(?,DNprime) < ? and d(pNprime(?),?) < ?, for each ? ? ?. By definition, the
projection pNprime : ? ? U satisfies pNprime(?) = DNprime. Note that U is an open set in R3, and
therefore an ANR (see [8]). Hence, the identity on ? is homotopic within U to pNprime .
The corresponding homotopy h : ??I ? U is given by h(?,t) = (1?t)?+tpNprime(?),
where h(?,t) ? U for each ? ? ? and t ? I. By Borsuk?s homotopy extension
23
theorem 3.22 there is an extension PNprime : U ? U of pNprime homotopic to the identity on
U. Hence, we have an extension H : U ?I ? U of h.
Theorem 3.20 provides movability of all Denjoy?like sets Dn, n ? N. Therefore,
by definition of movability, for every neighbourhood U ofDn there is a neighbourhood
Vn ? U of Dn such that for each neighbourhood W of Dn, there is a map ?n satisfying
the conditions (3.2) with U0 replaced by Vn and ? replaced by ?n. Because the sets
? and Di, i ? N, are pairwise disjoint, we can assume that Vn negationslash= Vm, for n negationslash= m, and
that Vn ?? = ?, for every n ? N.
Now we will construct a neighbourhood UN, for some N ? N, of ? ?uniontext?j=N Dj
that is disjoint with every Vj, j < N.
For the rest of the proof, we use the following notation. If f : X ?I ? X is a
map, we denote by ?f : X ? X the map given by ?f(x) = f(x,1), for each x ? X.
Let Uprime = U ? ?H?1(VNprime), and further let UN = Uprime \uniontextj