On minimality of nonautonomous dynamical systems
The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a sequence of continuous selfmaps of X is studied. A sufficient condition for nonminimality of such a system is formulated. A special attention is paid to the particular case when X is a real compact interv...
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irk-123456789-1769952021-02-10T01:26:09Z On minimality of nonautonomous dynamical systems Kolyada, S.F. Snoha, Ľ. Trofimchuk, S.I. The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a sequence of continuous selfmaps of X is studied. A sufficient condition for nonminimality of such a system is formulated. A special attention is paid to the particular case when X is a real compact interval I. A sequence of continuous selfmaps of I forming a minimal nonautonomous system may uniformly converge. For instance, the limit may be any topologically transitive map. But if all the maps in the sequence are surjective then the limit is necessarily monotone. An example is given when the limit is the identity. As an application, in a simple way we construct a triangular map in the square I² with the property that every point except of those in the leftmost fibre has an orbit whose ω-limit set coincides with the leftmost fibre. Вивчається мiнiмальнiсть неавтономної динамiчної системи, що задається компактним хаусдорфовим простором X та послiдовнiстю неперервних вiдображень на ньому. Сформульовано достатню умову для немiнiмальностi таких систем. Особливу увагу придiлено випадку, коли X є вiдрiзком прямої I. Послiдовнiсть неперервних вiдображень на I, що формує мiнiмальну неавтономну динамiчну систему, може рiвномiрно збiгатись. Наприклад, границею може бути будь-яке транзитивне вiдображення. Але якщо всi вiдображення з цiєї послiдовностi є сюр’- єктивними, тодi границею є необхiдно монотонне вiдображення. Наведено приклад, коли границею є тотожне вiдображення. Як деяку аплiкацiю наведено просту конструкцiю трикутного вiдображення в квадратi I² з властивiстю, що довiльна точка, за винятком точок iз крайнього лiвого вертикального шару, має орбiту, ω-гранична множина якої збiгається з цим шаром. 2004 Article On minimality of nonautonomous dynamical systems / S.F. Kolyada, Ľ. Snoha, S.I. Trofimchuk // Нелінійні коливання. — 2004. — Т. 7, № 1. — С. 86-92. — Бібліогр.: 16 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176995 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a
sequence of continuous selfmaps of X is studied. A sufficient condition for nonminimality of such a
system is formulated. A special attention is paid to the particular case when X is a real compact interval
I. A sequence of continuous selfmaps of I forming a minimal nonautonomous system may uniformly
converge. For instance, the limit may be any topologically transitive map. But if all the maps in the sequence
are surjective then the limit is necessarily monotone. An example is given when the limit is the identity. As
an application, in a simple way we construct a triangular map in the square I² with the property that every
point except of those in the leftmost fibre has an orbit whose ω-limit set coincides with the leftmost fibre. |
| format |
Article |
| author |
Kolyada, S.F. Snoha, Ľ. Trofimchuk, S.I. |
| spellingShingle |
Kolyada, S.F. Snoha, Ľ. Trofimchuk, S.I. On minimality of nonautonomous dynamical systems Нелінійні коливання |
| author_facet |
Kolyada, S.F. Snoha, Ľ. Trofimchuk, S.I. |
| author_sort |
Kolyada, S.F. |
| title |
On minimality of nonautonomous dynamical systems |
| title_short |
On minimality of nonautonomous dynamical systems |
| title_full |
On minimality of nonautonomous dynamical systems |
| title_fullStr |
On minimality of nonautonomous dynamical systems |
| title_full_unstemmed |
On minimality of nonautonomous dynamical systems |
| title_sort |
on minimality of nonautonomous dynamical systems |
| publisher |
Інститут математики НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176995 |
| citation_txt |
On minimality of nonautonomous dynamical systems / S.F. Kolyada, Ľ. Snoha, S.I. Trofimchuk // Нелінійні коливання. — 2004. — Т. 7, № 1. — С. 86-92. — Бібліогр.: 16 назв. — англ. |
| series |
Нелінійні коливання |
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AT kolyadasf onminimalityofnonautonomousdynamicalsystems AT snohal onminimalityofnonautonomousdynamicalsystems AT trofimchuksi onminimalityofnonautonomousdynamicalsystems |
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2025-07-15T14:57:15Z |
| last_indexed |
2025-07-15T14:57:15Z |
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1837725317297340416 |
| fulltext |
UDC 517.9
ON MINIMALITY OF NONAUTONOMOUS DYNAMICAL SYSTEMS*
ПРО МIНIМАЛЬНIСТЬ НЕАВТОНОМНИХ ДИНАМIЧНИХ СИСТЕМ
S. Kolyada
Inst. Math. Acad. Sci. Ukraine
Tereshchenkivs’ka Str., 3, Kyiv, 01601, Ukraine
e-mail: skolyada@imath.kiev.ua
L’. Snoha
Matej Bel Univ.
Tajovského 40, 974 01 Banská Bystrica, Slovakia
e-mail: snoha@fpv.umb.sk
S. Trofimchuk
Univ. de Chile
Las Palmeras, 3425, Santiago, Chile
e-mail: trofimch@uchile.cl
The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a
sequence of continuous selfmaps of X is studied. A sufficient condition for nonminimality of such a
system is formulated. A special attention is paid to the particular case when X is a real compact interval
I . A sequence of continuous selfmaps of I forming a minimal nonautonomous system may uniformly
converge. For instance, the limit may be any topologically transitive map. But if all the maps in the sequence
are surjective then the limit is necessarily monotone. An example is given when the limit is the identity. As
an application, in a simple way we construct a triangular map in the square I2 with the property that every
point except of those in the leftmost fibre has an orbit whose ω-limit set coincides with the leftmost fibre.
Вивчається мiнiмальнiсть неавтономної динамiчної системи, що задається компактним хаус-
дорфовим простором X та послiдовнiстю неперервних вiдображень на ньому. Сформульовано
достатню умову для немiнiмальностi таких систем. Особливу увагу придiлено випадку, коли
X є вiдрiзком прямої I . Послiдовнiсть неперервних вiдображень на I , що формує мiнiмальну
неавтономну динамiчну систему, може рiвномiрно збiгатись. Наприклад, границею може бути
будь-яке транзитивне вiдображення. Але якщо всi вiдображення з цiєї послiдовностi є сюр’-
єктивними, тодi границею є необхiдно монотонне вiдображення. Наведено приклад, коли гра-
ницею є тотожне вiдображення. Як деяку аплiкацiю наведено просту конструкцiю трикутного
вiдображення в квадратi I2 з властивiстю, що довiльна точка, за винятком точок iз крайнього
лiвого вертикального шару, має орбiту, ω-гранична множина якої збiгається з цим шаром.
1. Introduction. Minimality is a central topic in topological dynamics (see, e.g., [1 – 3]). A dynami-
cal system (X, f) whereX is a topological space and f : X → X continuous, is called (topologi-
cally) minimal if there is no proper subset M ⊆ X which is nonempty, closed and f -invariant
* The first author was supported by Fundacion Seneca (Murcia) (grant number IV 00210/CV/03), the second
author was supported by the Slovak grant agency (grant number 1/0265/03) and the third author by FONDECYT
(project 1030992).
c© S. Kolyada, L’. Snoha, and S. Trofimchuk, 2004
86 ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
ON MINIMALITY OF NONAUTONOMOUS DYNAMICAL SYSTEMS 87
(i.e., f(M) ⊆ M). In such a case we also say that the map f itself is minimal. Clearly, the system
(X, f) is minimal if and only if the (forward) orbit of every point x ∈ X is dense inX . Recently
the authors proved (see [4]) that the minimal maps send open sets to sets with nonempty interi-
ors (so called feebly open or almost open maps) and, moreover, they are almost one-to-one (a
typical point has just one preimage). To show this, among others a simple but useful sufficient
condition for nonminimality of a map (see Lemma 2.1 below) was used.
We will call a space X minimal if it admits a minimal map f : X → X . An important and
old question is which compact Hausdorff spaces (compact metric spaces, continua, etc.) admit
minimal maps. This problem is investigated for instance in [5] and [6]. In particular, from [5]
we know that if we are interested in whether a space admits a minimal homeomorphism or not
and whether it admits a minimal nonivertible map or not, then all four possibilities can occur
(examples are interval (no, no), circle (yes, no), torus (yes, yes), pinched torus (no, no)). In [6]
it is proved that the only 2-manifolds (compact or not) which admit minimal maps are either
finite unions of tori or finite unions of Klein bottles.
In this paper we will study a nonautonomous discrete dynamical system (X; f1,∞) given
by a compact Hausdorff topological space X and a sequence f1,∞ := (fi)∞i=1 of continuous
selfmaps of X . The trajectory of a point x is defined to be the sequence x, f1(x), f2(f1(x)), . . .
and its orbit is the set of values of this sequence. The system is minimal if every orbit is dense in
X . Easy examples (see, e.g., the sequence of constant maps from the example in Remark after
Lemma 2.3) show that from minimality of such a system one cannot deduce the feeble openness
of fn or fn ◦ · · · ◦ f2 ◦ f1. Neither one can deduce that these maps are almost one-to-one. This
indicates that there is a larger variety of nonautonomous minimal systems than autonomous
ones. (One of our aims is to illustrate this observation by showing less degenerate examples.)
The paper is organized as follows. We first generalize the above mentioned sufficient condi-
tion for nonminimality to the case of nonautonomous dynamical systems on a compact Haus-
dorff space (see Section 2). A special attention is paid to the particular case when X is a real
compact interval I (see Section 3). We show that the interval admits a nonautonomous dynami-
cal system given by a sequence of continuous surjective maps. This sequence may even uni-
formly converge. For instance, the limit may be any topologically transitive map and we discuss
the question whether the minimality of the sequence of maps always implies the topological
transitivity of its uniform limit. We show that, rather surprisingly, this is not necessarily the
case. If all the maps in the sequence are surjective then the uniform limit of such a sequence is
definitely not transitive — in fact, it is necessarily monotone (see Proposition 3.1). An example
is given when the limit is the identity (see Theorem 3.1). As an application, we construct a tri-
angular map in the square I2 with the property that every point except of those in the leftmost
fibre has an orbit whose ω-limit set coincides with the leftmost fibre. To show this property for
our triangular map is much simpler than in [7] where this is proved for a map from [8].
2. Sufficient conditions for nonminimality of a system. Let X be a Hausdorff topological
space and f : X → X continuous (written f ∈ C(X)). If Y ⊆ X is nonempty, closed and
f -invariant then Y is called a minimal set of the system (X; f) if the system (Y ; f |Y ) is minimal.
Recall also that the system (X; f) or the map f itself is called (topologically) transitive if
for every pair of nonempty open sets U and V in X , there is a positive integer n such that
fn(U) ∩ V 6= ∅. Clearly, minimality implies transitivity. If f is transitive then f(X) is obviously
dense in X and since we additionally assume X to be compact, f(X) is also compact. Hence
f(X) = X .
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
88 S. KOLYADA, L’. SNOHA, S. TROFIMCHUK
If X has an isolated point and f is transitive then X is just a periodic orbit of f . Hence,
if X admits a transitive map then either it has no isolated point or otherwise it is finite. If X
is a compact metric space without isolated points, then the above definition of transitivity is
equivalent to the existence of a dense orbit (for a survey of results on transitivity see, e.g., [9]).
Lemma 2.1 [4]. Let X be a compact Hausdorff space and f ∈ C(X). Then the following two
conditions are equivalent and each of them is sufficient for f not to be minimal:
1) there is a closed set A 6= X in X with f(A) = f(X);
2) there is an open set B 6= ∅ in X with f(B) ⊆ f(X \B).
Nonautonomous dynamical systems (in fact their topological entropy) have been studied in
[10] and [11]. Let us recall the definitions and some notations.
Let f1,∞ := (fi)∞i=1 be a sequence of continuous selfmaps of X . For any positive integers
i, n set fni = fi+(n−1) ◦ · · · ◦ fi+1 ◦ fi and additionally f0
i = idX . We will call (X; f1,∞) a
nonautonomous discrete dynamical system. The trajectory and the orbit of a point x ∈ X will
be the sequence (fn1 (x))∞n=0 and the set {fn1 (x) : n = 0, 1, 2, . . . }, respectively. If k is a positive
integer, the sequence (fn1 (x))k−1
n=0 is said to be a finite trajectory of x (with length k).
We define the minimality of a nonautonomous system as the density of all orbits. Again, this
is equivalent to the nonexistence of any proper subset M ⊆ X which is nonempty, closed and
f1,∞-invariant (i.e., fn1 (M) ⊆ M for n = 1, 2, . . . ). Note that for autonomous systems (when
fn = f for every n) this agrees with the usual definition.
In the previous lemma we worked with compact Hausdorff spaces. In the proof of the
next lemma we need that a sequence of points in the space have a convergent subsequence.
Therefore we additionally assume metrizability (in fact to add 1st countability instead of metri-
zability would be sufficient, see [12, p. 229, 217]).
Recall also that for autonomous systems the next lemma is equivalent to the well known
fact that f is minimal if and only if for any open set B there is a k such that every point visits
the set B not later than in time k.
Lemma 2.2. Let (X, ρ) be a compact metric space and let (X; f1,∞) be a nonautonomous
dynamical system. Then the following are equivalent:
1) (X; f1,∞) is not minimal;
2) there is a nonempty open set B ⊆ X such that (X; f1,∞) has arbitrarily long finite trajec-
tories disjoint with B.
Proof. 2) follows from 1) trivially. So let 2) hold. Then there is an increasing sequence of
positive integers (nk)∞k=1 and a sequence of points (xk)∞k=1 such that for every k, the f1,∞-
trajectory of xk of length nk is disjoint with B. Without loss of generality we may assume that
the sequence xk converges. Denote its limit by y. Obviously, y /∈ B. Due to continuity, for any
positive integer r, f r1 (y) is the limit of f r1 (xk) when k tends to infinity. For all sufficiently large k
we have nk > r and so f r1 (xk) /∈ B. Hence f r1 (y) /∈ B. Thus the f1,∞-trajectory of y is disjoint
with B and so (X; f1,∞) is not minimal.
Corollary 2.1. Let (X, ρ) be a compact metric space with no isolated points and let (X; f1,∞)
be a nonautonomous dynamical system. Suppose that there is a nonempty open set B ⊆ X and
n0 ∈ N such that:
1) fn0−1
1 is onto;
2) (X; fn0,∞) has arbitrarily long finite trajectories disjoint with B.
Then (X; f1,∞) is not minimal.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
ON MINIMALITY OF NONAUTONOMOUS DYNAMICAL SYSTEMS 89
Proof. By Lemma 2.2, there is a point y ∈ X whose fn0,∞-trajectory is not dense in X , i.e.,
does not intersect some nonempty open set in X (in fact the set B as it can be seen from the
proof of Lemma 2.2). By 1) there is a point z with fn0−1
1 (z) = y. Then the f1,∞-trajectory of z
contains at most n0 − 1 points from the set B and so it cannot be dense in B (note that X has
no isolated point). Thus (X; f1,∞) is not minimal.
Remark 2.1. The assumption 1) in Corollary 2.1 cannot be removed. To see it, take X =
= I = [0, 1]. For n ≥ 2 let fn be the standard tent map and let f1(x) = d, x ∈ I , where d is
a point whose trajectory under the tent map is dense in I . Then 2) is fulfilled with, say, n0 = 2
and B =
(
0,
1
2
)
but (X; f1,∞) is minimal. (One can construct an analogous example with any
other topologically transitive map in place of the tent map.)
Lemma 2.3. Let (X, ρ) be a compact metric space with no isolated points and let (X; f1,∞)
be a nonautonomous dynamical system. Suppose that there is a nonempty open set B ⊆ X and
n0 ∈ N such that:
1) fn0−1
1 as well as the maps fn, n ≥ n0, are onto;
2) for every n ≥ n0, fn(B) ⊆ fn(X \B).
Then (X; f1,∞) is not minimal.
Proof. In view of Corollary 2.1 it is sufficient to find arbitrarily long finite trajectories of the
system (X; fn0,∞) disjoint withB. To this end, fix k ∈ N and p ∈ X \B. For any n ≥ n0 we have
fn(X \B) = fn(X) = X and so any point has an fn-preimage in X \B. Thus there are points
xk, xk−1, . . . , x0 ∈ X \ B such that fn0+k(xk) = p, fn0+(k−1)(xk−1) = xk, . . . , fn0(x0) = x1.
Then {x0, x1, . . . , xk, p} is an fn0,∞-trajectory of x0 of length k + 2 disjoint with X \B.
Remark 2.2. The assumption 1) in Lemma 2.3 cannot be removed. To see this, let X = I =
= [0, 1] and let {qn : n ∈ N} be an enumeration of the rationals from [0, 1]. Put fn(x) = qn for
all x ∈ I and n ∈ N and takeB =
(
0,
1
2
)
. Then (I; f1,∞) is minimal though all the assumptions
of Lemma 2.3 except for 1) are satisfied.
Lemma 2.4. Let (X, ρ) be a compact metric space with no isolated points and let (X; f1,∞)
be a nonautonomous dynamical system. Suppose that the sequence (fn)∞n=1 uniformly converges
to a map f . If f is not onto then (X; f1,∞) is not minimal (even no f1,∞-trajectory is dense).
Proof. Since f is not onto, there is a nonempty open set B with ρ(B, f(X)) > 0. Then for
all sufficiently large n, fn(X)∩B = ∅. This implies that every trajectory of the system (X; f1,∞)
has only finitely many points in B and so is not dense in X (note that X has no isolated points).
3. Minimality of nonautonomous systems on the interval and an application. In C(I) where
I is a real compact interval, say I = [0, 1], there are no minimal maps. Nevertheless, a sequence
of maps fn ∈ C(I) may be minimal, see for instance the remark after Lemma 2.3. One can even
find an example with surjective maps. Let {Qn : n ∈ N} be an enumeration of open intervals
with rational endpoints in [0, 1]. For every n choose a point qn ∈ Qn. Let fn be any continuous
selfmap of I such that fn(I \ Qn) = {qn} and fn(Qn) = I . For every x ∈ I and every n ∈ N
we have that x ∈ Qn or fn(x) ∈ Qn. Hence the system (I; f1,∞) is minimal.
But what happens if fn uniformly converges to a map f ? Since f cannot be minimal, does
this mean that the sequence (fn)∞n=1 itself cannot be minimal ? The answer is negative as we
saw in the remark after Corollary 2.1. But notice that in this example f1 was not onto. If all the
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
90 S. KOLYADA, L’. SNOHA, S. TROFIMCHUK
maps fn are onto then still (fn)∞n=1 may be minimal and convergent (see Theorem 3.1 below)
but the limit function must be monotone. This is rather paradoxical: a minimal sequence of onto
maps (i.e., a complicated sequence in a sense) may uniformly converge but only to a monotone
map (i.e., to a very simple map which is even far from being topologically transitive).
Proposition 3.1. Let (I; f1,∞) be a minimal nonautonomous dynamical system such that the
sequence (fn)∞n=1 uniformly converges to a map f and all the maps fn are onto. Then f is (not
necessarily strictly) monotone.
Proof. Suppose on the contrary that f is not monotone. Then there are points a < b < c in
I with f(a) < f(b) > f(c) or f(a) > f(b) < f(c). In either case, one can find a nondegenerate
open interval B ⊆ (a, b) such that for all sufficiently large n, fn(B) ⊆ fn(I \ B). Hence by
Lemma 2.3 the system is not minimal, a contradiction.
In what follows, by a u-homeomorphism or a d-homeomorphism we will mean any increa-
sing homeomorphism f : I → I such that for all x ∈ (0, 1), f(x) > x or f(x) < x, respectively.
It is not difficult to see that one can define a minimal sequence (fn)∞n=1 uniformly converging
to the identity, of the form
f1, U1, D1, . . . , Un, Dn, . . .
where f1(x) =
1
2
andUi orDi is a block of u-homeomorphisms or d-homeomorphisms, respecti-
vely. On the other hand, (fn)∞n=1 cannot be minimal if fn, n ≥ n0, are homeomorphisms and
fn0−1
1 is onto (for there is a point x with fn0−1
1 (x) ∈ {0, 1} and g({0, 1}) = {0, 1} whenever g
is a homeomorphisms). Nevertheless, we have the following theorem.
Theorem 3.1. There exists a minimal nonautonomous dynamical system (I; f1,∞) such that
the sequence (fn)∞n=1 uniformly converges to the identity and all the maps fn are onto. Moreover,
all fn can be chosen piecewise linear with nonzero slopes and at most three pieces of linearity,
and even for every n the system (I; fn,∞) is minimal.
Proof. First introduce some notation and terminology. For ε > 0 let Fε denote the set of
maps which are ε-close to the identity, onto, piecewise linear with nonzero slopes and at most
three pieces of linearity.
If S = (f1, . . . , fn) is a finite sequence of selfmaps of I and x ∈ I , we will say that the
trajectory of x under S is ε-dense in I if for every y ∈ I there is i ∈ {0, 1, . . . , n} with |f i1(x)−
−y| ≤ ε.
Finally, if f is a selfmap of I and Ai, Bi, i = 1, . . . , r, are subsets of I with f(Ai) = Bi,
i = 1, . . . , r, we will write f〈A1, . . . , Ar〉 = 〈B1, . . . , Br〉.
Now we are ready to prove the theorem. Suppose for a moment that for any small ε > 0 we
are able to define a finite sequence Sε of maps from Fε such that the trajectory under Sε of any
point from I is ε-dense in I . Then, whenever ε(n) → 0, the infinite sequence
Sε(1), Sε(2), . . . , Sε(n), . . .
fulfills all the properties required in the theorem.
Thus it remains only to show how one can construct Sε. So fix a small ε > 0
(
say, ε <
1
4
)
and put A =
[
0,
ε
2
]
, B =
[ε
2
, 1− ε
2
]
, C =
[
1− ε
2
, 1
]
, A+ =
[ε
2
, ε
]
and C− =
[
1− ε, 1− ε
2
]
. It
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ON MINIMALITY OF NONAUTONOMOUS DYNAMICAL SYSTEMS 91
is clear that there are increasing homeomorphisms fε, gε, hε ∈ Fε and positive integers n1(ε),
n2(ε), n3(ε) such that
(fε)n1(ε)〈A,B,C〉 = 〈A,A+, I \ (A ∪A+)〉 ,
(gε)n2(ε)〈A,A+, I \ (A ∪A+)〉 = 〈I \ (C− ∪ C), C−, C〉 ,
(hε)n3(ε)〈I \ (C− ∪ C), C−, C〉 = 〈A,B,C〉
(fε is the identity on A and a d-homeomorphism on I \ int(A), gε is a u-homeomorphism and
hε is the identity on C and a d-homeomorphism on I \ int(C)). Consider the finite sequence
Tε = fε, . . . , fε︸ ︷︷ ︸
n1(ε)
, gε, . . . , gε︸ ︷︷ ︸
n2(ε)
, hε, . . . , hε︸ ︷︷ ︸
n3(ε)
.
It is easy to check that the trajectory under Tε of any point x ∈ I ends in the interval A,B or C
in which it started and, moreover, if x ∈ B then the trajectory of x is ε-dense in I . Nevertheless,
the Tε-trajectory of a point from A ∪ C may not be ε-dense in I . Therefore take a piecewise
linear map ϕε having three pieces of linearity such that ϕε(0) = ε, ϕε(ε) = 0, ϕε(1 − ε) = 1,
ϕε(1) = 1 − ε (notice that ϕε ∈ Fε). Since ϕε(A ∪ C) ⊆ B, the trajectory of any point from
A ∪ C under the finite sequence ϕε, Tε is ε-dense in I . Hence the sequence
Sε = Tε, ϕε, Tε
has all the desired properties.
The theorem is proved.
A continuous map F of the space X × Y into itself is called skew product or triangular (see
e.g. [8, 13]) if it is of the form F (x, y) = (f(x), g(x, y)). The continuous map f : X → X is
called the basis map of F . Instead of g(x, y) we also write gx(y). Then we shortly write F =
= (f, gx). Here gx, x ∈ X , is a family of continuous maps Y → Y depending continuously on
x. The maps gx are called fibre maps, the sets Yx = {x} × Y are called fibres (Yx is the fibre
over x). Since πX ◦ F = f ◦ πX , where πX : (x, y) 7→ x is the X-projection, the map F is an
extension of f . The Y -projection πY : (x, y) 7→ y is also called the 2nd projection. We denote
by C4(X × Y ) the set of all continuous triangular maps from X × Y into itself.
Consider a map F ∈ C4(I2). It can happen as well that an orbit of F is not dense in I × I
(it may even be nowhere dense) and yet its 2nd projection is dense in I . It is not possible that
all orbits of F have dense 2nd projections because the square has the fixed point property.
Nevertheless, as an application of Theorem 3.1 we prove the following theorem.
Theorem 3.2. There is a triangular map F = (f, gx) ∈ C4(I2) such that:
1) all points of the form (0, y) are fixed;
2) limn→∞ f
n(x) = 0 for every x;
3) every point from I × I which is not of the form (0, y) has the orbit whose 2nd projection is
dense in I (hence, in view of (2), its ω-limit set equals {0} × I);
4) the map F has zero topological entropy.
Proof. First we define the basis map f on I = [0, 1]. Put f(0) = 0 and f(1) = 1/2. Further,
for any n = 1, 2, . . . let f(1/2n) = f(1/2n + 1/2n+1) = 1/2n+1. To finish the definition of
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 1
92 S. KOLYADA, L’. SNOHA, S. TROFIMCHUK
f let f be linear on each closed interval J having the property that f has been defined at the
endpoints of J but has not been defined in the interior points of J . Notice that for every x ∈ I ,
limn→∞ f
n(x) = 0.
Now, for each x ∈ I , we need to define the fibre map gx(y), y ∈ I . We use the system
(I; f1,∞) from the previous theorem. Put g0(y) = y and for every n ∈ N∪{0} let g1/2n = fn+1.
So far we have defined a map F0 ∈ C4({1/2n : n = 0, 1, . . . } × I). Now use Extension lemma
from [14] (cf. [15]) to extend F0 to a map F ∈ C4(I2).
It remains to show that F satisfies 3) and 4). The map f is piecewise linear with infini-
tely many pieces, each of them having the slope zero or one. For every x ∈ I \ {0} there is
a nonnegative integer k (depending on x) such that fk(x) belongs to the set {1/2n : n =
= 0, 1, . . . }. From this and the fact that the system (I; fk,∞) is minimal for every k, it easily
follows that every point from I × I which is not of the form (0, y) has the orbit whose 2nd
projection is dense in I . Since the set Ω(F ) of nonwandering points of F is the fibre I0 where F
is the identity, for the entropy h(F ) of F we get (see [16]) h(F ) = h(F|Ω(F )) = 0.
The theorem is proved.
Let us remark that the existence of such maps as in Theorem 3.2 has been established in [7]
(by using a map from [8]), but the proof for our map is simpler.
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Received 10.10.2003
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