Normat 59:2, 59–66 (2011) 59
What is the most important aspect of Smale’s
horseshoe?
Pernille Hviid Petersen
Department of Science, Systems and Models
Roskilde University
php@ruc.dk
This paper is based on a project I did in the spring 2010 at Roskilde University as
part of my Master’s degree. The aim of the project was to investigate the influence
of Smales horseshoe on the developement of the theory of dynamical systems. I
analysed three cases where Smales horseshoe had been included in the study of
both the Hénon mapping and the restricted three body problem. From the analysis
of these cases you get the distinct impression, that the most important aspe ct of the
horseshoe is that it is top ologically conjugate to a shift autophism and so can be
used as a translator b e tween the shift automorphism and another dynamical system
which has a horseshoe imbedded in its dynamics. In contrast to this Stephen Smale
himself states in a paper from 1980 [Smale, 1980] that the most important aspect
of the horseshoe is not the connection to the shift automorphisms, but the fact
that the horseshoe is structurally stable.
The aim of this paper is to promote a point of view giving a possible explation-
ation of this disagreement. The structure of the paper is as follows: In the first
section Smales horseshoe is presented. In the second section the three cases that I
have analysed are briefly introduced. In the third section a possible explaination
is given for why Smale in the paper from 1980 wievs the structural stability of
the horseshoe as the most important aspect of it. In the fourth sec tion a possible
reason for the disagremeent between the cases and Smales paper will be discussed.
Smales horseshoe
In this section the main idea behind Smales horseshoe is introduced. I am only
going to write about Smales original version of the horseshoe mapping, which he
presented in two article from 1965 and 1967 respectively [Smale, 1965], [Smale,
1967]. At the end of this section I will comment on the relation between this version
of the horseshoe and the three cases that I have analysed.
Let g be a diffeomorphism of a subset Q of R
2
to R
2
. Smale defines Q to be
the square Q = {(x, y)||x|Æ1, |y|Æ1} in R
2
[Smale, 1967, p. 770]. The mapping
g contracts the square Q in the vertical direction, e xpands it in the horizontal
direction and folds it back over itself, so that g(Q) is shaped like a horseshoe that
lies over Q as in figure 1. In other words g is a mapping that maps Q onto the
60 Pernille Hviid Petersen Normat 2/2011
Figure 1 : The square Q and the horseshoe shaped image of Q under g.
Figure 2 : The square Q and the set P
1
fi P
2
= g
≠1
(g(Q) fl Q).
horseshoe shaped region of figure 1 in such a way that g(A)=A
Õ
, g(B)=B
Õ
,
g(C)=C
Õ
and g(D)=D
Õ
. From figure 1 it is easy to see that the intersection
Qflg(Q) consists of the two rectangles Q
1
and Q
2
. According to Smale any mapping
that has the following two prop erties can be used as the mapping g [Smale, 1967,
p. 771]:
1. The mapping g is a diffeomorphism of Q onto the horse shoe shaped region
of figure 1, that is defined so that g(A)=A
Õ
, g(B)=B
Õ
, g(C)=C
Õ
and
g(D)=D
Õ
.
2. On each of the components P
1
and P
2
of the set g
≠1
(Q fl g(Q)) (see figure 2)
g is an affine mapping.
According to Smale a consequence of the second property is that P
1
and P
2
will be
as in figure 2 and g(P
1
)=Q
1
and g(P
2
)=Q
2
[Smale, 1967, p. 771]. The mapping
g is what we will call a horseshoe map.
If g is used repeatedly on Q more and more of the points that starts in Q is
mapped outside of Q. What is interesting regarding Smales horseshoe is the set
of points which stays in Q under infinitely many both forwards and backwards
Normat 2/2011 Pernille Hviid Petersen 61
iterations with g. This set of points is denoted and is given by [Smale, 1967, p.
771]
=
‹
mœZ
g
m
(Q
(m)
) (1)
where Q
(m)
is the intersection of Q and g
m≠1
(Q) when m>0, Q
0
= Q and Q
(m)
=
g
m
(Q
(m+1)
) when m<0. It is easy to see that is invariant under g. Futhermore
on the map g is topologically conjugate to the s hift automorphism – : X
S
‘æ X
S
,
where S is a set that contains two elements and X
S
= S
Z
[Smale, 1967, p. 771].
The elements in X
S
can be written as biinfinite strings. The action of – on a string
is that it shifts all entries one place to the right. The shift automorphism possess es
a countably infinite set of periodic orbits, a uncountable set of aperiodic orbits and
a dense orbit, and since the shift automorphism is topologically conjugate to the
horseshoe, the horseshoe too has these properties [Wiggins, 1990, p. 436].
In his article from 1965 Smale shows, that the horseshoe map g is structurally
stable [Smale, 1965, p. 63 and p. 74-77]. This means that in the space of diffeomor-
phisms on R
2
there is an open neighborhood of g such that every diffeomorphism
in that neighborhood is topologically conjugate to g.
As mentioned above the horseshoe which I have presented in this section is
Smales original version of the horseshoe. When it comes to the application of Smales
horseshoe in the study of other dynamical systems, this version is not very useful.
This is because of the two requirements to the horseshoe map that Smale outlines
and that many dynamical systems fail to live up to. Indeed the version of the
horseshoe that is included in two of my three cases is not Smales original version,
but a more generalised version, that is desciped e.g. by Moser [Moser, 1973]. When
it comes to the connection between the horseshoe and the shift automorphism and
the structural stability of the horseshoe however there is no difference at all between
Smales original version of the horseshoe and the generalised version. Therefore I
will not say anymore about the different versions of the horseshoe.
The three cases
The aim of this section is to give a brief presentation of the three cases and to
explain how Smales horseshoe is included in each of them.
The first case is from a book called Stable and random motions in dynamical
systems written by Jürgen Moser in 1973 [Moser, 1973]. In the third chapter of the
book Moser presents a proof of a fascinating theorem about the dynamics in the
restricted three body problem. The version of the restricted three body problem
which he is working with consists of two particles, m
1
and m
2
, of equal masses
that are moving along two identical elliptic orbits placed symmetrically around the
center of mass of the two paticles. These orbits are contained in a fixed plane s in
space. The third particle, m
3
, has a mass so small compared to the masses of m
1
and m
2
that it can be ignored. This m eans that m
1
and m
2
are acting on m
3
,but
m
3
is not acting on m
1
and m
2
. The particle m
3
is moving along a line, which is
perpendicular to s and which passes through the center of mass of m
1
and m
2
.The
restricted three body problem is to determine the motion of m
3
when the initial
62 Pernille Hviid Petersen Normat 2/2011
Figure 3 : The restricted three body problem. The figure is borrowed from [Moser,
1973, p. 84] and modified by me.
conditions of the three particles are known. The scenario of the restricted three
body problem is shown in figure 3.
The theorem which Moser proves is concerned with the possible motions of the
third particle in the restricted three body problem. Let z(t) denote a particular
such motion that intersects the plane s infinitely many times. These intersections
occur at the times given by the biinfinite sequence {t
k
}. Now choose the units
in such a way that the time it takes m
1
and m
2
to complete one revolution in
their orbits equals 2fi. Then the number of complete revolutions that m
1
and m
2
performs between two successive intersections between m
3
and s is given by
‡
k
=
5
t
k+1
≠ t
k
2fi
6
(2)
where [x] denotes the biggest integer that is less than or equal to x [Moser, 1973,
p. 85]. Cle arly any motion of m
3
having an infinite sequence of passing times of s
in both future and past corresponds to a biinfinite sequence of integers {‡
k
},such
that the numbers of the sequence describe the number of complete revolutions that
m
1
and m
2
performs between two successive passing times of s. Mosers theorem
states the opposite, namely that given a sufficiently small eccentricity ‘>0 of the
orbits of m
1
and m
2
there is a integer n = n(‘) for w hich for every sequence {‡
k
}
that satisfies the requirement that ‡
k
Ø n, there is a motion of m
3
such that the
numbers of the sequence describe the number of complete revolutions which m
1
and m
2
performs between two successive passing times of s of that motion [Moser,
Normat 2/2011 Pernille Hviid Petersen 63
1973, p. 85]. Moser proves this theorem by finding a horseshoe imbedded in the
dynamical system that describes the motion of m
3
. Having found this horseshoe he
uses the connection between the horseshoe and the shift automorphisms to argue
for the theorem. For Moser the horseshoe is included as a tool by means of which
he can prove his theorem and in this context the most important aspect of the
horseshoe is its connection with the shift automorphisms.
The second case is from an article called On the Hénon Transformation written
by James H. Curry in 1979 [Curry, 1979]. In this article Curry presents the results
of some numerical studies of the Hénon mapping which Hénon presented in an
article written in 1976 [Hénon, 1976]. The aim of some of these numerical studies
was to produce evidence that there is a Cantor set in the trapping region of the
state space of the Hénon mapping with the parameter values used by Hénon. The
trapping region is a region of the state space that is mapped to itself under the
Hénon mapping [Hénon, 1976, p. 75-76]. This means that trajectories which enters
this region is trapped inside it. Curry argues for the existence a Cantor set in
the trapping region of the state space of the Hénon mapping by finding a strong
indication of the existence of a horseshoe in the trapping region. The reason that
the existence of a horseshoe implies the existence of a Cantor set is that the subset
of the domain of the horseshoe map g, on which g is topologically conjugate
to a shift automorphism, is homeomorphic to a Cantor set. This follows from the
fact that the domain of the shift automorphism is homeomorphic to a Cantor set
[Smale, 1967, p. 770]. Like Moser Curry too uses the horseshoe as a tool by means
of which he provides evidence, that there is a Cantor set in the trapping region
of the Hénon mapping. Even though Curry is not using the connection between
the horseshoe and the shift automorphism as explicitly as Moser, he is using the
horseshoe in a way which he would not have been able to if the connection had not
been there.
The third case is from an article called Shift Automorphisms in the Hénon
Mapping written by Robert Devaney and Zbigneiw Nitecki in 1979 [Devaney and
Nitecki, 1979]. In this article Devaney and Nitecki presents a proof of a theorem
that among other things states, that it is possible to choose the parameters of the
Hénon mapping in s uch a way, that the Hénon mapping restricted to a subset of
its domain is topologically conjugate to the shift automorphism. This statement
is proved by finding a horseshoe imbedded in the Hénon mapping. Unlike Moser
and Curry Devaney and Nitecki do not use the existence of a horseshoe and hence
a shift automorphism in the Hénon mapping to prove another result. The finding
of a horseshoe is for them not a tool, but an end. Like Moser and Curry however
Devaney and Nitecki emphasises the connection between the horseshoe and the
shift automorphism.
In all of the three cases Smales horseshoe is included as either a tool or an end of
a proof. Furthermore, even though the aims of the three cases are different they are
all emphasising the connection between the horseshoe and the shift automorphisms
more than they emphasise any other aspect of the horseshoe, thus leaving the
impression that this aspect of the horseshoe is the most important one .
64 Pernille Hviid Petersen Normat 2/2011
Smales paper from 1980
Smales paper from 1980 is called On how I got started in dynamical systems and
consists of a rather informal description of his early work in dynamical systems,
which among other things includes the discovery of the horseshoe. As me ntioned
in the beginning Smale does not in his 1980 paper consider the connection between
the horse shoe and the shift automorphism to be the most important aspect of the
horseshoe. Actually he does not even mention this connection. Instead he more
or less explicitly states that the most important aspect of the horseshoe is its
structural stability. I believe, that the reason for this is that the 1980 paper mainly
is concerned with the development which led to the discovery of the horseshoe. The
reason for the view of the 1980 paper is in other words the context of the paper.
In 1956 Smale finished his thesis in topology, and according to himself he consid-
ered himself mainly a topologist [Smale, 1980, p. 150]. Nevertheless he became more
and more interested in ordinary differential equations and dynamical systems, and
in 1959 he wrote his first paper on dynamical systems [Smale, 1960]. This paper
was about Morse inequalities for a class of dynamical systems, that later has been
named Morse-Smale dynamical systems, and which among other things is charac-
terised by having a finite number of fixed points and periodic orbits. In the paper
Smale conjectured, that the Morse-Smale dynamical systems form an open dense
set in the space of all ordinary differential equations [Smale, 1960, p. 43], [Smale,
1980, p. 148].
After his article on the Morse-Smale systems was published in 1960 Smale re-
cieved a letter from Norman Levison. In this letter Levinson pointed out, that
Smales conjecture about the Morse-Smale systems being open and dense in the
space of all ordinary differential equations could not be true, since an earlier paper
of Levinson contained a c ounterexample to it. Levinsons paper was a simplification
of the work on the Van der Pol equation done by Cartwright and Littlewood [Smale,
1980, p. 149], [Guckenheimer, 1980, p. 987]. In an article from 1998 Smale says, that
he had to work hard to translate Levinsons counterexample into a more geometric
form before he was convinced that Levinson was right and that his own conjecture
was wrong [Smale, 1998, p. 41]. The geometric version of Levinsons counterex-
ample which came out of Smales struggle was the horse shoe . The horseshoe is a
counterexample to Smales early conjecture because of its structural stability, which
as mentioned above means that there is an open neighborhood of the horseshoe
mapping such that every mapping in this neighborhood is topologically conjugate
to the horseshoe. Since the horseshoe possesses a countable infinity of periodic
orbits, Smales early conjecture thus can not be right.
It would be fair to say, that one of the main developments that Smale describes
in his 1980 paper is the development of his own understanding of the space con-
sisting of all ordinary differential equations. In this development the discovery of
the horseshoe played a central role, because it was this discovery that made Smale
realise, that his early comprehension of the structure of this space was wrong. And
regarding this realisation the structural stability of the horseshoe is completely
indispensable.
Normat 2/2011 Pernille Hviid Petersen 65
Conclusion
Why is it that in the three cases the most important aspec t of the horseshoe is its
connection to the shift automorphisms, whereas in the 1980 paper Smale empha-
sises the structural stability of the horseshoe as the most important aspect? If one
considers the way the horseshoe is included in the three cases, it seems reasonable
to say, that they do not emphasise the structural stability of the horseshoe, because
this aspect is not important to their work. The structural stability can not be used
to make their proofs any easier, prettier or more ingenious. The connection between
the horseshoe and the shift automorphisms on the other hand can be used to do
this, which is especially clear from the cases of Moser and Curry. This seemes to me
to be a possible explanation of why the three cases emphasises the connection be-
tween the horseshoe and the shiftautomorphism instead of the structural stability
of the horseshoe.
Now let us focus our attention on the 1980 paper of Smale. The reason why this
paper nominates the structural stability of the horseshoe to be the most important
aspect of the horseshoe is, that this aspect played a decisive role in the development
of Smales understanding of the space of all ordinary differential equations, which
is one of the main themes in the paper. The connection between the horseshoe
and the shift automorphism on the other hand was of no real importance to this
development, and thus there is no reason for mentioning it in this paper.
In the light of these reflections a poss ible explanation of the disagreement be-
tween the three cases and the 1980 paper of Smale is that the cases and the paper
speak about the horseshoe from two different contexts - and seen from these two
contexts the most important aspect of Smales horseshoe is not the s ame. Accord-
ing to Tinne Hoff Kjeldsen the context in which a mathematical result or concept
is considered often has a great impact on how the result or concept is regarded
[Kjeldsen, 2009, p. 110]. Therefore I find my explanation of the disagreement not
only possible, but also plausible.
From the above discussion it must be concluded, that the question of the title of
this paper can not be given an unequivocal answer. The reason of the disagreement
between the three cases and the 1980 paper is not that one of them is wrong, but
instead that they are looking at Smales horseshoe from different contexts. This
reflection must entail, that the answer to the question of what the most important
aspect of Smales horseshoe is will depend on the context that one considers the
horseshoe from. This point becomes especially clear, if we look at the article called
Finding a Horseshoe on the Beaches of Rio that Smale wrote in 1998 [Smale, 1998].
In this article Smales emphasises the chaotic behaviour of the horseshoe more than
any other aspect of the horseshoe. This aspect though is not mentioned a singel
time neither in the three cas es nor in the 1980 paper, even though this paper too is
written by Smale. That Smale himself in two different papers implicitly nominates
two different aspects of the horseshoe to be the most important one clearly shows,
that you can not say what the most important aspect the horseshoe is. The only
thing which you can say is, what the most important aspect is seen from the context
that you are working in.
66 Pernille Hviid Petersen Normat 2/2011
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