A COUNTEREXAMPLE TO A STATEMENT CONCERNING LYAPUNOV STABILITY

Abstract

We find a class of weakly unimodal C ∞ maps of an interval with zero topological entropy such that no such map f is Lyapunov stable on the set Per(f) of its periodic points. This disproves a statement published in several books and papers, e.g., by V.

Full text

Acta Math. Univ. Comenianae
Vol. LXX, 2(2001), pp. 265–268
265
A COUNTEREXAMPLE TO A STATEMENT CONCERNING
LYAPUNOV STABILITY
P. ˇ
SINDEL ´
Aˇ
ROV ´
A
Abstract. We find a class of weakly unimodal C∞maps of an interval with zero
topological entropy such that no such map fis Lyapunov stable on the set Per(f)
of its periodic points. This disproves a statement published in several books and
papers, e.g., by V. V. Fedorenko, S. F. Kolyada, A. N. Sharkovsky, A. G. Sivak and
J. Sm´ıtal.
1. Introduction and Preliminaries
In a series of papers and books (cf., e.g., [3], [4], [7], [8], [11]) it is stated that a
function f∈C(I, I) has zero topological entropy if and only if fis stable in the
sense of Lyapunov on the set Per(f) of its periodic points. However, this statement
is false, see Theorems A and B below. It seems that this false result appeared for
the first time, without proof, in [11] and then it was only cited in the other papers.
Actually, the above quoted papers and books contain long lists of conditions
for continuous maps of the interval, which are equivalent to the condition that f
has zero topological entropy. However, another counterexample disproving any of
these equivalences is given in [12].
By the period of a periodic point we mean its smallest period. If the periods
of points in Per(f) are the powers of 2, then fis of type 2∞. Recall [9] that a
function fin the class C(I, I) of continuous maps of the compact unit interval I
has zero topological entropy if and only if it is of type 2∞. A map f∈C(I , I)
is unimodal if there exists c∈(0,1) such that fis strictly increasing on [0, c]
and strictly decreasing on [c, 1]. A map fis weakly unimodal if there exists
c∈(0,1) such that fis non-decreasing on [0, c] and non-increasing on [c, 1].
Let fbe weakly unimodal. We shall say that x,y∈Iare equivalent (denoted
by x∼y) if there exists n≥1 such that fnis constant on [x, y]. Clearly, ∼is
an equivalence relation. Let ˜
I=I/ ∼be the factor space obtained by identifying
to a point each equivalence class. These classes are closed intervals (possibly
Received March 16, 2001.
2000 Mathematics Subject Classification. Primary 26A18, 37E05.
Key words and phrases. Topological entropy, Lyapunov stability, periodic points, weakly
unimodal maps.
This research was supported, in part, by the contracts 201/00/0859 from the Grant Agency
of Czech Republic, and CEZ:J10/98:192400002 from the Czech Ministry of Education. Support
of these institutions is gratefully acknowledged.

266 P. ˇ
SINDEL ´
Aˇ
ROV ´
A
degenerated). The natural projection π:I→˜
Iis continuous and non-decreasing.
Since fis continuous, x∼yimplies f(x)∼f(y). Therefore there exists a unique
map ˜
f:˜
I→˜
Isuch that
f◦π=π◦˜
f.(1)
This ˜
fis continuous and either monotone or unimodal.
A map f∈C(I, I) is Lyapunov stable on a set A⊆Iif for any ε > 0 there
exists δ > 0 such that if |x−y|< δ for xand yin Athen |fn(x)−fn(y)|< ε for
any n.
Finally, we recall the Feigenbaum map Φ: I→Iwhich we use as the main tool
in our argument. It is the unique unimodal map of type 2∞vanishing at the end-
points of I, with a critical point c, a continuous derivative and a compact periodic
interval Jof period 2, containing cin its interior and such that the composition
of Φ2|Jwith an affine scaling is the original map Φ. The properties of Φ are
well-known, cf., e.g., [2], or [6]. In particular, we have the following result:
Lemma 1. Let anbe the periodic point of Φof period 2nwith the largest image
under Φ, and let cbe the critical point of Φ. Then
a1< a3<· · · < c < · · · < a4< a2< a0<Φ(c),
and
Φ(a0)<Φ(a1)<Φ(a2)<Φ(a3)<· · · <Φ(c).
To prove our results we use the approach from [10]; in particular, we use the
following two lemmas.
Lemma 2. (a) If x∈Iis a periodic point of fof period kthen π(x)is a
periodic point of ˜
fof period k. (b) If y∈˜
Iis a periodic point of ˜
fof period kthen
there exists a unique periodic point x∈Iof ffor which π(x) = y. The period of
xis k.
For a map flet Jf={x∈I;f(x)≥f(y) for all y∈I}, and let Fbe the class
of all weakly unimodal maps f, for which
the set Jfcontains more than one point,(2)
fis of type 2∞.(3)
Lemma 3. If f∈ F then ˜
fis unimodal and satisfies (3).
The following result is well known. See, e.g., [5, Proposition 4.3 and 1.3].
Lemma 4. If a map fis unimodal and satisfies (3), then the relative position
of the critical point, its images and the periodic points are the same for Φand f.
2. Main Results
Theorem A. No f∈ F is Lyapunov stable on Per(f). On the other hand, F
consists of mappings with zero topological entropy and contains a C∞map.

A COUNTEREXAMPLE CONCERNING LYAPUNOV STABILITY 267
Proof. Let f∈ F. By (3), fhas zero topological entropy. By Lemma 3, ˜
f
is unimodal and satisfies (3). By Lemma 4, the relative position of the critical
point, its images and the periodic points are the same for Φ and ˜
f. Since πis
non-decreasing, by Lemma 2 it is the same also for f. Thus, by Lemma 1, we have
b1< b3<· · · < d < · · · < b4< b2< b0< f(d),(4)
and
f(b0)< f(b1)< f (b2)< f (b3)<· · · < f (d),(5)
where d∈Jfand bnis the periodic point of fof period 2nwith the largest image
under f. Let ε > 0 be the length of interval Jf,δ > 0, and let p=f(b2n+1),
q=f(b2n). Then p, q are periodic points and by (5), 0 < p −q < δ for any
sufficiently large n. But f22n+1 −1(p) = b2n+1 and f22n+1−1(q) = b2n. Thus, by
(5), f22n+1−1(p)−f22n+1 −1(q)> ε. To finish the proof note that Fcontains a C∞
map [10]. 
For completeness, we show that the condition of zero topological entropy is
necessary for Lyapunov stability on the set of periodic points. The proof is based
on a standard argument.
Theorem B. Let f∈C(I, I ). If fis Lyapunov stable on Per(f)then fhas
zero topological entropy.
Proof. Assume, contrary to what we wish to show, that fhas a positive topo-
logical entropy. Apply a known result (cf., e.g., [1]) that f∈C(I, I) has positive
topological entropy if and only if, for some positive integer nthere exist compact
disjoint subintervals J, K of Isuch that J∪K⊆fn(J)∩fn(K). Without loss of
generality assume that n= 1 (otherwise replace fby fn). Let K, L be compact
disjoint intervals in Isuch that K∪L⊆f(K)∩f(L). Put ε= dist(K, L). Choose
an arbitrary δ > 0. By induction we can see that, for any positive integer m, there
exist 2mdisjoint compact intervals Km
1, Km
2, . . . , K m
2mcontained in Ksuch that
fm(Km
i) = Kfor 1 ≤i≤2m−1and fm(Km
i) = Lfor 2m−1< i ≤2m.(6)
For a sufficiently large m, there are i, j such that 1 ≤i≤2m−1< j ≤2mand
the diameter of the set Km
i∪Km
jis less then δ. SinceKm
i∪Km
j⊂fm+1(Km
i)∩
fm+1(Km
j), there exist periodic points p∈Km
iand q∈Km
jsuch that fm+1(p) = p
and fm+1(q) = q. By (6), |fm(p)−fm(q)|> ε, i.e., fis not Lyapunov stable on
Per(f). 
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P. ˇ
Sindel´aˇrov´a, Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic,
e-mail: Petra.Sindelarova@math.slu.cz