Periodicity generated by adding machines
We show that a homeomorphism of the plane $\mathbb{R}^2$ with an invariant Cantor set $\mathbf {C}$, on which the homeomorphism acts as an adding machine, possesses periodic points arbitrarily close to $\mathbf {C}$. The existence of periodic points near an invariant Cantor set is related to a shape...
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| author | Kuperberg, K. Kuperberg, K. |
| author_facet | Kuperberg, K. Kuperberg, K. |
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{
"author": "K. Kuperberg",
"institution": "Department of Mathematics, Auburn University"
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| description | We show that a homeomorphism of the plane $\mathbb{R}^2$ with an invariant Cantor set $\mathbf {C}$, on which the homeomorphism acts as an adding machine, possesses periodic points arbitrarily close to $\mathbf {C}$. The existence of periodic points near an invariant Cantor set is related to a shape theory question whether a solenoid invariant in a flow defined on $\mathbb{R}^3$ must be contained in a larger movable invariant compactum. |
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Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 140–147
Krystyna Kuperberg
(Department of Mathematics, Auburn University, Auburn, AL
36849-5310, USA)
Periodicity generated by adding
machines
kuperkm@auburn.edu
We show that a homeomorphism of the plane R2 with an invariant Cantor
set C, on which the homeomorphism acts as an adding machine, possesses
periodic points arbitrarily close to C. The existence of periodic points
near an invariant Cantor set is related to a shape theory question whether
a solenoid invariant in a flow defined on R3 must be contained in a larger
movable invariant compactum.
1 Introduction
Let φ : X → X be a homeomorphism (Z-action) or a flow (R-action)
defined on a metric space X. A set A ⊂ X invariant under φ is Lyapunov
stable if for every neighborhood U of A there is a neighborhood V of
A such that for every p ∈ V , the forward orbit of p is contained in
U . J. Buescu and I. Stewart proved in [8] (see also [9]) that if h :
R2 → R2 is a homeomorphism with an invariant Lyapunov stable Cantor
set C, and h|C is an adding machine, then every neighborhood of C
contains a periodic orbit of h. The theorem was also proved by H. Bell
and K. Meyer in [2]. In addition, the authors construct in this paper a
specific example of a Lyapunov stable adding machine in R2 invariant
under a C1 homeomorphism h of R2 and show that the theorem does
not hold for a homeomorphism H on R3 and a Lyapunov stable adding
c© Krystyna Kuperberg, 2013
Krystyna Kuperberg 141
machine invariant under H. We give a simple proof that without the
assumption of Lyapunov stability a weaker version of the theorem holds:
Every neighborhood of C contains a periodic point of h. The proof bears
a similarity to the proof of the Cartwright-Littlewood Theorem given
Morton Brown in [7]. The Cartwright-Littlewood Theorem asserts that
if planar continuum ∆ does not separate the plane R2 and is invariant
under an orientation preserving homeomorphism h : R2 → R2, then h
has a fixed point p ∈ ∆.
Much earlier E.S. Thomas considered in [18] one-dimensional
solenoids invariant in a C1 flow on a 3-manifold. A solenoid in this case
is the inverse limit of circles with bonding maps being group homomor-
phisms. If almost all bonding maps are of degree one, then the solenoid
is said to be trivial. Assuming that the flow on a non-trivial solenoid
is minimal, the Poincaré first-return map on a local cross-section of the
solenoid is an adding machine. The flow restricted to an invariant set is
minimal on this set if every orbit is dense in the set. In case of a solenoid,
this is equivalent to the fact that there are no fixed points in the solenoid,
i.e., the flow is non-singular.
A compact invariant set is isolated if in some compact neighborhood it
is the largest invariant set. The notion applies to both homeomorphisms
and flows. Thomas uses isolating blocks, considered by C. Conley and
R. W. Easton in [10] and previously by T. Ważewski in [20], in order
to establish an Alexander-Spanier cohomology exact sequence involving
the solenoid. He then shows that an invariant non-trivial solenoid in a
nonsingular flow on a 3-manifold is not isolated. M. Kulczycki proved
in [14] that under certain conditions, a planar adding machine is not
isolated.
2 Adding machine
For a sequence of integers (k1, k2, k3, . . .), each greater than one, de-
note by C(k1, k2, k3, . . .), or shortly by C, the Cantor set Π∞n=1Z/knZ.
Definition 1. An adding machine is a homeomorphism α : C→ C such
that if
α(i1, i2, i3, . . .) = (j1, j2, j3, . . .)
then
142 Periodicity generated by adding machines
1. if there is an m ≥ 1 such that in = kn − 1 for n < m and im <
km − 1, then jn = 0 for n < m, jm = im + 1, and jn = in for
n > m,
2. otherwise jm = 0 for all m, i.e., if im = km − 1 for m ≥ 1, then
jm = 0 for m ≥ 1.
The map α is an adding machine with base (k1, k2, k3, . . .) acting onC.
The Cantor set itself is ofter referred to as an adding machine; precisely,
an adding machine is the pair (C, α).
Definition 2. Let α be an adding machine with base (k1, k2, k3, . . .) act-
ing on C. For a finite sequence of integers i1, . . . , in, 0 ≤ ij < kj for
j ≤ n, define a cylinder of length n as the set
Ci1,...,in = {(x1, x2, . . .) |x1 = i1, . . . , xn = in}.
Note that the cylinder Ci1,...,in is invariant under αs , where s is a
multiple of the product k1 · · · kn.
3 Periodic points near a planar adding ma-
chine
Let h : R2 → R2 be a homeomorphism andC = C(k1, k2, k3, . . .) ⊂ R2
an invariant Cantor set. Assume that h|C is an adding machine with base
(k1, k2, k3, . . .). Let P be the set of periodic points of h in R2, including
the fixed points although clearly the fixed points of h are away from C.
Let Cl(P ) be the closure of P . Each of the sets P and Cl(P ) is invariant
under h.
The theorem below shows that in every neighborhood of C, there is
a periodic point of h. Stability is not assumed.
Theorem. C ∩ Cl(P ) 6= ∅.
Proof. Suppose that C∩Cl(P ) = ∅. Let U be a component of R2 \Cl(P )
intersecting C. Thus U contains a cylinder invariant under hs, some
power hs of h. If U is simply connected, then by Brouwer’s Theorem [6]
we arrive at a contradiction that there is a fixed point of the orientation
preserving homeomorphism hs ◦ hs in U , a periodic point of h outside
P . The Brouwer Translation Theorem asserts that for a fixed point free,
Krystyna Kuperberg 143
orientation preserving homeomorphism of the plane no orbit of a point
is bounded, hence there are no non-empty, compact, invariant sets.
In general, let Ũ be the universal cover of U with π : Ũ → U the
covering map. There is a cylinder Ci1,...,in contained in an open, evenly
covered disk D ⊂ U . Since Ci1,...,in is invariant under hk1···kn , so is U .
Let f = hk1···kn
|U . Since h has no periodic points in U , f as well as f2,
which is an orientation preserving homeomorphism of U , have no fixed
points in U .
By composing a lift of f2 with an appropriate deck transformation,
we obtain an orientation preserving homeomorphism f̃ : Ũ → Ũ with an
invariant compactum C̃, a copy of Ci1,...,in mapped homeomorphically
by the projection π onto Ci1,...,in . Since Ũ is homeomorphic to R2, by
144 Periodicity generated by adding machines
cylinder
periodic points
Brouwer’s theorem, f̃ has a fixed point a. On the other hand since f2
has no fixed points, no fiber π−1(p) is invariant under f̃ . Hence a cannot
be a fixed point of f̃ .
Therefore the assumption that C ∩ Cl(P ) = ∅ is not valid. There are
periodic points of h arbitrarily close to the Cantor set C.
Remark. The above theorem does not address the periods of the pe-
riodic points that are close to the Cantor set equipped with the adding
machine with base (k1, k2, k3, . . .). The almost periodicity of the adding
machine yields natural relations of these periods to the products of num-
bers k1, k2, . . . multiplied by the number 2 in case of orientation reversing
homeomorphisms.
4 Shape theory
The notion of movability is one of the most important concepts of
shape theory. A compact subset F of the Hilbert cube Q is movable [4]
if for every neighborhood U of F there exists a neighborhood V of F
such that for every neighborhood W of F there is a deformation of V
into W within U . This property does not depend on the embedding of
F in Q and the Hilbert cube can be replaced in the definition by any
metric ANR. For the basic notions of the theory of shape the reader is
referred to [3] and K. Borsuk’s monograph [5]. The notion of movability
seems closely related to notion of Lyapunov stability and thus it is of
importance in dynamics.
Non-trivial solenoids were the first and most obvious examples of non-
movable compacta. On the other hand, the Denjoy continua [11], which
by construction are in a natural manner embedded in the surface of a
torus, are movable. A description of a C1 Denjoy set (conitnuum) is
easily accessible in [15] or [16]. Denjoy continua are completely classified
in [1] and [12]. Let D be a Denjoy continuum embedded in the surface
of a torus S1 × S1. Let π : S1 × S1 → S1 × S1 be a covering projection
with finite fibers. The set π−1(D) is Denjoy-like. The complement of a
Krystyna Kuperberg 145
Denjoy continuum in S1 × S1 is connected, whereas the complement of
a Denjoy-like continuum in S1 × S1 may have several components.
Let φ : R×M →M be a non-singular flow on a 3-manifold M . Let Σ
be a solenoid in M approximated in terms of the Hausdorff distance by a
sequence of pairwise disjoint simple closed curves {Cn}∞n=1 disjoint from
the solenoid. It is easy to show that the compactum X = Σ ∪
⋃∞
n=1 Cn
is movable.
Question 1. If a solenoid Σ is invariant under φ, is Σ contained in a
larger movable compact set invariant under φ?
The next question is a slight variation of Question 1.
Question 2. If a solenoid Σ is invariant under φ and U is a neighborhood
of Σ, is Σ contained in a larger movable compact set invariant under φ
and contained in U?
Question 3. Could the larger movable invariant set in Questions 2 al-
ways consist of Σ and a sequence of invariant approximating circles?
In [17], P. Šindelářová constructed a flow on R3 with an invariant non-
movable one-dimensional continuum Ω. The continuum is not a solenoid,
but maps continuously onto a non-trivial solenoid and therefore by [19]
or [13] it is not movable. In Šindelářová’s flow, Ω is approximated by
invariant Denjoy-like continua {Dn}∞n=1 and the union Ω ∪
⋃∞
n=1Dn is
movable.
Question 4. Is every compact invariant set in flow on a 3-manifold
contained in a movable invariant set?
Question 5. If a compactum Y is invariant under a flow φ on a 3-
manifold and U is a neighborhood of Y , is Σ contained in a movable
compact set invariant under φ and contained in U?
Question 6. Would Questions 4 and 5 pose a different challenge if one
assumed that the flow φ on the non-movable invariant set were minimal?
Let D be a Cantor set in R2 invariant under an orientation preserving
homeomorphism g : R2 → R2. By Brouwer’s theorem, g has a fixed point
p ∈ R2. (It is easy to construct an example such that g|D is a Denjoy
homeomorphism and g has no periodic points other than one fixed point.)
This suggest the following:
146 Periodicity generated by adding machines
Question 7. Let Z be a compact invariant set in a flow on R3 such that
there exists a sequence of invariant Denjoy-like continua {Dn}∞n=1 so that
the union Z ∪
⋃∞
n=1Dn is movable. Does there exist an invariant simple
closed curve? Do there exist invariant simple closed curves arbitrarily
close to Z?
Finally let’s recall the main problem:
Question 8. Let h : R2 → R2 be a homeomorphism and let C be a
Cantor set invariant under h. If h|C is an adding machine, does there
exist a periodic orbit in every neighborhood of C?
References
[1] M. Barge and R.F. Williams, Classification of Denjoy continua,
Topology Appl. 106 (2000), 77–89.
[2] H. Bell and K. Meyer, Limit periodic functions, adding machines,
and solenoids, J. Dynam. Differential Equations 7 (1995), 409–422.
[3] K. Borsuk, Concerning homotopy properties of compacta, Fund.
Math. 62 (1968), 223-254.
[4] K. Borsuk, On movable compacta, Fund. Math. 66 (1969/1970), 137–
146.
[5] K. Borsuk, Theory of Shape, Monografie Matematyczne 59, Warsaw
1975.
[6] L. E. J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann.
72 (1912), 37–54.
[7] Morton Brown, A short short proof of the Carthwright-littlewood the-
orem, Proc. Amer. Math. Soc. 65 (1977), 372.
[8] J. Buescu and I. Stewart, Liapunov stability and adding machines,
Ergodic Theory Dynam. Systems 15 (1995), 271–290.
[9] J. Buescu, M. Kulczycki, and I. Stewart, Liapunov stability and
adding machines revisited , Dyn. Syst. 21 (2006), 379–384.
[10] C. Conley and R. Easton, Isolated invariant sets and isolating blocks,
Trans. Amer. Math. Soc. 158 (1971), 35–36.
Krystyna Kuperberg 147
[11] A. Denjoy, Sur les courbes définies par les équations differentielles à
la surface du tore, J. Math. Pures Appl. 11 (1932), 333–375.
[12] R. Fokkink, The structure of trajectories, Thesis, Technische Uni-
versiteit Delft, 1991.
[13] J. Krasinkiewicz, Continuous images of continua and 1-movability ,
Fund. Math. 98 (1978), 141–164.
[14] M. Kulczycki, Adding machines as invariant sets for homeomor-
phisms of a disk into the plane, Ergodic Theory Dynam. Systems 26
(2006), 783–786.
[15] G. Kuperberg, A volume-preserving counterexample to the Seifert
conjecture, Comment. Math. Helv. 71 (1996), 70–97.
[16] P.A. Schweitzer, Counterexamples to the Seifert conjecture and open-
ing closed leaves of foliations, Ann. of Math. 100 (1974), 386–400.
[17] P. Šindelářová, An example on movable approximations of a minimal
set in a continuous flow , Topology Appl. 154 (2007), 1097–1106.
[18] E.S. Thomas, Jr., One-dimensional minimal sets, Topology 12
(1973), 233–242.
[19] A. Trybulec, On the movable continua, Dissertation, Inst. of Math.,
Pol. Acad. of Sci., Warsaw, 1974.
[20] T. Ważewski, Sur une méthode topologique de l’examen de l’allure
asymptotique des intégrales des équations différentielles, Proc. In-
ternat. Congress Math. (Amsterdam, 1954) vol. III, 132–139; No-
ordhoff, Groningen and North-Holland, Amsterdam, 1956.
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| spelling | oai:trim.imath.kiev.ua:article-3102018-02-10T20:56:26Z Periodicity generated by adding machines Періодичність породжена підсумовуючою машиною Kuperberg, K. Kuperberg, K. We show that a homeomorphism of the plane $\mathbb{R}^2$ with an invariant Cantor set $\mathbf {C}$, on which the homeomorphism acts as an adding machine, possesses periodic points arbitrarily close to $\mathbf {C}$. The existence of periodic points near an invariant Cantor set is related to a shape theory question whether a solenoid invariant in a flow defined on $\mathbb{R}^3$ must be contained in a larger movable invariant compactum. Ми встановлюємо, що гомеоморфізм площини $\mathbb{R}^2$ з інваріантною множиною Кантора $\mathbf {C}$, на якій гомеоморфізм діє як підсумовуюча машина, має періодичні точки як завгодно близькі до&nbsp; $\mathbf {C}$.&nbsp; Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/310 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 140-147 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 140-147 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 140-147 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/310/289 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Kuperberg, K. Kuperberg, K. Periodicity generated by adding machines |
| title | Periodicity generated by adding machines |
| title_alt | Періодичність породжена підсумовуючою машиною |
| title_full | Periodicity generated by adding machines |
| title_fullStr | Periodicity generated by adding machines |
| title_full_unstemmed | Periodicity generated by adding machines |
| title_short | Periodicity generated by adding machines |
| title_sort | periodicity generated by adding machines |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/310 |
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