Download F-limit points in dynamical systems defined on the interval

Cent. Eur. J. Math. • 11(1) • 2013 • 170-176
DOI: 10.2478/s11533-012-0056-0
Central European Journal of Mathematics
F-limit points in dynamical systems
defined on the interval
Research Article
Piotr Szuca1∗
1 Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland
Received 30 October 2011; accepted 8 March 2012
Abstract: Given a free ultrafilter p on N we say that x ∈ [0, 1] is the p-limit point of a sequence (xn )n∈N ⊂ [0, 1] (in symbols,
x = p -limn∈N xn ) if for every neighbourhood V of x, {n ∈ N : xn ∈ V } ∈ p. For a function f : [0, 1] → [0, 1] the
function f p : [0, 1] → [0, 1] is defined by f p (x) = p -limn∈N f n (x) for each x ∈ [0, 1]. This map is rarely continuous. In
this note we study properties which are equivalent to the continuity of f p . For a filter F we also define the ωF -limit
set of f at x. We consider a question about continuity of the multivalued map x → ωfF (x). We point out some
connections between the Baire class of f p and tame dynamical systems, and give some open problems.
MSC:
40A35, 40A30, 26A21, 03E15, 26A03, 54A20
Keywords: Ideal convergence • Filter convergence • Interval maps • ω-limit sets • Hausdorff metric • Ellis semigroup •
Enveloping semigroup • Tame dynamical system
© Versita Sp. z o.o.
Let I = [0, 1]. In what follows, a function is understood to be a continuous function from I into I unless clearly specified
otherwise. For f : I → I and x ∈ I we define f 0 (x) = x, f 1 (x) = f(x) and f n+1 (x) = f(f n (x)), n ∈ N. The ω-limit set, ωf (x),
is the set of all subsequential limits of the sequence {f n (x)}n∈N , i.e.
ωf (x) =
\
N∈N
{f n (x) : n ≥ N}.
We furnish the family of ω-limit sets of a continuous function f with the Hausdorff metric. Then it becomes a subspace
of the compact metric space K of all closed non-empty subsets of I.
If Φ is a function from I into the class of non-empty subsets of I, then we say that Φ is lower semi-continuous or lsc
(upper semi-continuous or usc, respectively) if for each closed (resp. open) subset V ⊂ I the set {x : Φ(x) ⊂ V } is
∗
170
E-mail: [email protected]
P. Szuca
closed (resp. open) in I. It is well known that Φ : I → K is continuous if and only if it is both usc and lsc. Note that if
Φ(x) = {φ(x)} for all x, then Φ is continuous if and only if φ is continuous.
Bruckner and Ceder in [4] considered some questions related to the continuity of the map ωf : x → ωf (x). They proved
the following theorem.
Theorem 1 ([4, Theorem 1.2]).
For a continuous function f : I → I the following conditions are equivalent:
(1) ωf is continuous;
(2) {f n }n∈N is equicontinuous;
(3) ωf 2 is continuous;
T
(4) Fix(f 2 ) = n∈N f n (I) (Fix(f) denotes the set of all fixed points of f);
(5) Fix(f 2 ) is connected and for all x, {f 2n (x)}n∈N converges to a point of Fix(f 2 );
(6) Fix(f 2 ) is connected;
(7) ωf is lower semi-continuous;
(8) ωf is upper semi-continuous.
This shows that the map x 7→ ωf (x) is rarely continuous. (For example, from (5) it follows that such an f has only
periodic points of periods 1 and 2. As is well known, such function has zero topological entropy.)
A filter on N is a family of subsets of natural numbers N closed under taking finite intersections and supersets of its
elements. A filter is proper if it does not contain the empty set. A filter is free if it is proper and contains all co-finite
sets. If F is a filter on N, then the set F? = {N \ A : A ∈ F} is called the dual ideal.
Fix a proper filter F. A sequence (xn )n∈N of reals is said to be F-convergent to x (x = F-limn∈N xn ) if for each ε > 0
{n ∈ N : |xn − x| < ε} ∈ F.
(Recall that x = limn→∞ xn if and only if x = F-limn∈N xn for the filter F of all cofinite subsets of N.) We say that x is
an F-cluster point of (xn )n∈N if for each ε > 0
{n ∈ N : |xn − x| < ε} ∈
/ F? .
Observe that if x = F-limn∈N xn then x is the only F-cluster point of (xn )n∈N . For f : I → I define the ωF -limit set of f
at x as
ωfF (x) = y ∈ I : y is an F-cluster point of (f n (x))n∈N .
Any maximal proper filter is called an ultrafilter. We will denote by the symbol βN the set of all ultrafilters. (In fact,
the space of all ultrafilters with the appropriate topology is a Čech–Stone compactification of N. See e.g. [13]). For
an ultrafilter p ∈ βN and f : I → I, the function f p : I → I is defined by f p (x) = p -lim f n (x) for each x ∈ I. Such
functions have been studied by Blass in [2], where the author establishes the connection between the algebra of βN and
an arbitrary dynamical system. It is important for us that
• f p is well-defined on I for each f;
• y ∈ ωf (x) if and only if there is a p ∈ βN \ N such that y = f p (x), see Corollary 5;
p
p
• ωf (x) = {f p (x)}, so the continuity of ωf is equivalent to the continuity of f p ; and
• g ∈ E(f) if and only if there is a p ∈ βN such that g = f p , where the enveloping semigroup E(f) is defined as
the closure in II (with its compact, non-metrizable, pointwise convergence topology) of the set {f n : n ∈ N}.
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F-limit points in dynamical systems defined on the interval
García-Ferreira and Sanchis in [6] analyzed the continuity of f p at a point x. In particular, they showed that if
{f n : n ∈ N} is equicontinuous at x then f p is continuous at x for each p [6, Theorem 3.22]. Our main result, Theorem 2,
is an extension of their results in the case of a dynamical system defined on an interval.
Theorem 2.
For a continuous function f : I → I the conditions (1)–(8) from Theorem 1 are equivalent to the conditions
(9) {f p : p ∈ βN} is equicontinuous;
(10) there is a p ∈ βN \ N such that f p is continuous;
(11) {ωfF : F is a proper filter} is equicontinuous;
(12) there is a free filter F such that ωfF is continuous.
Remark.
• For other conditions equivalent to the equicontinuity of the family {f n : n ∈ N} (for f defined on a compact metric
space) see [6, Theorems 3.29, 4.6], [7].
• In [6, Theorem 3.27] it is proved that if X is a compact metric space with only one non-isolated point, and f : X → X
is continuous, then either f p is continuous for all p ∈ βN, or f p is discontinuous for all p ∈ βN \ N.
• For continuous f : I → I it is not known whether f p can be discontinuous at x if we assume that f q is continuous
at x for f p 6= (f q )n , n = 0, 1, . . ., see [6, Question 5.2].
To prove Theorem 2 we will need the following facts.
Theorem 3 ([10, Theorem 12]).
Let F be a filter. A point x is an F-cluster point of (xn )n if and only if G-lim xn = x for some filter G ⊃ F.
Corollary 4.
ωfF (x0 ) ⊂ ωf (x0 ) for any free filter F.
Corollary 5.
y ∈ ωfF (x) if and only if there is a p ∈ βN, p ⊃ F, such that y = f p (x).
Corollary 6 follows from Corollary 4 and the well-known fact that if ωf N (x0 ) = A then ωf (x0 ) =
Corollary 6.
Suppose that x0 ∈ I, N ∈ N and lim f nN (x0 ) = a. Then, for each proper filter F
n→∞
Proof of Theorem 2.
ωfF (x0 ) ⊂ {f k (a) : 0 ≤ k < N}.
We will show the following chain of implications:
Theorem 1
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k=0
f k (A).
P. Szuca
(2) ⇒ (9). (The idea of the proof of this implication is borrowed from the proof of [6, Theorem 3.22].) Since for functions
defined on a compact space equicontinuity and uniform equicontinuity are equivalent, then for any ε > 0 there exists
δ(ε) > 0 such that for all x1 , x2 ∈ I and n ∈ N,
|f n (x1 ) − f n (x2 )| < ε
whenever |x1 − x2 | < δ(ε). For each p ∈ βN \ N, x ∈ I and ε > 0 define
p
Apε (x) = {n ∈ N : |f p (x) − f n (x)| < ε}.
p
p
p
p
p
Let |x1 − x2 | < δ(ε/3). Since Aε/3 (x1 ) ∈ p and Aε/3 (x2 ) ∈ p, Aε/3 (x1 ) ∩ Aε/3 (x2 ) 6= ∅. Then, for any n ∈ Aε/3 (x1 ) ∩ Aε/3 (x2 ),
|f p (x1 ) − f p (x2 )| ≤ |f p (x1 ) − f n (x1 )| + |f n (x1 ) − f n (x2 )| + |f n (x2 ) − f p (x2 )| < ε.
Thus, the family {f p : p ∈ βN} is equicontinuous.
(9) ⇒ (10) and (11) ⇒ (12) are obvious.
(10) ⇒ (12).
p
p
Since ωf (x) = {f p (x)}, the continuity of f p is equivalent to the continuity of ωf , for any p ∈ βN.
(12) ⇒ (6). (The idea of the proof of this implication is borrowed from the proof of [4, Theorem 1.2].) Suppose that ωfF
is continuous and Fix(f 2 ) is not connected. Since Fix(f 2 ) is a non-empty closed set, there exist a, b ∈ Fix(f 2 ) such that
(a, b)∩Fix(f 2 ) = ∅. We may assume that f 2 (x) > x for all x ∈ (a, b). First observe that by Corollary 6, ωfF (a) ⊂ {a, f(a)}
and ωfF (b) ⊂ {b, f(b)}. We claim that {a, f(a)} ∩ {b, f(b)} 6= ∅. We have two cases.
Case 1. f 2 (x) < b for each x ∈ (a, b). Then for every x ∈ (a, b), (f 2n (x))n is an increasing sequence converging to b.
Hence, by Corollary 6, for each x ∈ (a, b), ωfF (x) ⊂ {b, f(b)}. Since ωfF is lower semi-continuous at a, there is an
x ∈ (a, b) such that ∅ 6= ωfF (x) ∩ ωfF (a) ⊂ {a, f(a)} ∩ {b, f(b)}.
Case 2. There exists c ∈ (a, b) such that f 2 (c) = b. Choose x1 ∈ (a, b) such that f 2 (x1 ) = b. Choose x2 ∈ (a, x1 )
such that f 2 (x2 ) = x1 . Continuing in this way we obtain a decreasing sequence (xn )n converging to a for which
f 2 (xn+1 ) = xn for each n. Hence, by Corollary 6, ωfF (xn ) ⊂ {b, f(b)}. Thus, from the continuity of ωfF at a, it follows that
{a, f(a)} ∩ {b, f(b)} 6= ∅.
This concludes the proof of the claim. Then, using the fact that a 6= b, f 2 (a) = a, and f 2 (b) = b, we obtain a = f(b) and
b = f(a), and so [a, b] ⊂ f([a, b]). Therefore f, and f 2 too, has a fixed point in (a, b), a contradiction.
(9) ⇒ (11).
Fix an ε > 0 and take δ(ε) > 0 such that for all p ∈ βN and x1 , x2 ∈ I we have
|f p (x1 ) − f p (x2 )| < ε
whenever |x1 − x2 | < δ(ε). Fix a free filter F and x1 , x2 ∈ I with |x1 − x2 | < δ(ε). For any y1 ∈ ωfF (x1 ) there is a
py1 ∈ βN \ N such that F ⊂ py1 and f py1 (x1 ) = y1 . Set y2 = f py1 (x2 ). Then y2 ∈ ωfF (x2 ) and |y1 − y2 | < ε. Using the
same argument one may show that for any y2 ∈ ωfF (x2 ), there is a y ∈ ωfF (x1 ) with |y − y2 | < ε. Thus, the Hausdorff
distance between ωfF (x1 ) and ωfF (x2 ) is less than ε, and so ωfF is continuous. Moreover, since δ(ε) does not depend
on F, the family {ωfF : F is a proper filter} is equicontinuous.
Questions: some forms of chaos and its characterizations
A function f is of the first Baire class if it is a pointwise limit of a sequence of continuous functions. It is well known
that for a Polish space X and a complete metric space Y , a function f : X → Y is Baire 1 if and only if fP has a point
of continuity for every non-empty closed set P ⊂ X . By B1 (X ) we will denote the set of all functions of the first Baire
class.
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F-limit points in dynamical systems defined on the interval
A function f is of the second Baire class if it is a pointwise limit of a sequence of functions of the first Baire class.
It is known that for every continuous f the function ωf : I → K is of Baire class 2 [4]. However, in general, f p
can be non-measurable. Consider, for example, the shift function σ : 2N → 2N (this function is given by the formula
σ ((ai )i∈N ) = (ai+1 )i∈N ). Since 2N is homeomorphic to the Cantor set, this function is an example of a continuous
dynamical system defined on the Cantor set. Let Ah1i = {a = (a0 , a1 , . . .) ∈ 2N : a0 = 1} (this is an open subset of 2N ).
Then, for each p ∈ βN,
a ∈ 2N : σ p (a) ∈ Ah1i = a = (a0 , a1 , . . .) ∈ 2N : {i : ai = 1} ∈ p = p,
and, as is well known, the last set is non-measurable, see e.g. [1, Theorem. 4.1.1]. By the Tietze extension theorem σ
can be extended to a continuous function f : I → I.
A topological space K is Rosenthal compact if it is homeomorphic to a compact subset of the space B1 (X ) endowed with
the pointwise topology for some Polish space X , see e.g. [13]. All compact metric spaces are Rosenthal. A dynamical
system (I, f) is tame if E(f) is Rosenthal compact.
Theorem 7.
Let X be compact and metric. For a continuous function f : X → X the following conditions are equivalent:
(1’) (X , f) is tame;
(2’) E(f) ⊂ B1 (X );
(3’) the cardinality of E(f) is less or equal to c;
(4’) E(f) does not contain a homeomorphic copy of βN;
(5’) for each g in E(f) there exists a non-decreasing sequence of naturals (nk )k such that f nk → g as k → ∞;
(6’) f p is Baire 1 for each p ∈ βN.
For the proof, see [3, 7–9, 11].
Remark.
By the equivalence of (3’) and (4’) the cardinality of E(f) is either less or equal to c, or equal to 2c .
Bruckner and Ceder in [4] show that the sentence ωf is not Baire 1 defines some form of chaos, which lies between
positive topological entropy and chaos in the sense of Li–Yorke. They have given some characterizations of functions f
for which ωf is Baire 1. For example, ωf is Baire 1 if and only if each infinite ω-limit set of f is homeomorphic to the
Cantor set.
Problem 8.
Let f : I → I. Are the conditions from Theorem 7 equivalent to the conditions
(7’) ωf is Baire class 1;
(8’) each infinite ω-limit set of f is homeomorphic to the Cantor set?
Problem 9.
Let f : I → I. Are the conditions from Theorem 7 equivalent to the condition
(9’) ωfF is Baire class 1 for each free filter F?
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P. Szuca
We say that f is a 2∞ -function if f has cycles of order equal to each power of 2 and no others. We say that f is a
2n -function if f has cycles of order equal to each 2k for k ≤ n and no others. If f has zero topological entropy, we
write h(f) = 0. The reader may refer to the literature for the definition. There are known many useful characterizations
of zero topological entropy, see e.g. [5, Theorem A]. For our purposes we find it convenient to use the terminology of
entropy and we mention only the following characterization: h(f) = 0 if and only if f is a 2∞ -function or a 2n -function
for some n.
Recall that by [4], if ωf is Baire 1 then f is either a 2∞ -function or 2n -function, and, for every 2n -function f, ωf is Baire 1.
p
Recall also that in the case of p being an ultrafilter, ωf is Baire 1 if and only if f p is Baire 1.
Propositions 10 and 11 are motivated by Problem 8 and 9, see Figure 1.
Proposition 10.
For any free filter F, if ωfF is Baire 1 then f is either a 2∞ -function or a 2n -function (i.e. f is a function with zero
topological entropy).
The proof is very similar to that in [4, Theorem 2.2]; we give it here for completeness. Suppose that h(f) > 0.
By [5, Theorem A (xii)] there exists an infinite ω-limit set M for f which contains a periodic point. Let K ⊃ M be a
maximal ω-limit set.
Proof.
(i) By [12, Theorem 3.1 (iii)] K is perfect.
(ii) By [12, Theorem 3.7 (v)] the set of periodic points in K is dense in K .
(iii) By [12, Theorem 3.1 (v)], if L ⊂ K is an ω-limit set for f, then {x ∈ K : ωf (x) = L} is dense in K .
Then there are two different periodic orbits O1 = {x1 , f(x1 ), . . . , f n1 (x1 )} ⊂ K and O2 = {x2 , f(x2 ), . . . , f n2 (x2 )} ⊂ K with
both A1 = {x ∈ K : ωf (x) = O1 } and A2 = {x ∈ K : ωf (x) = O2 } being dense in K . Let ε > 0 be a distance (in the
Hausdorff metric) between O1 and O2 . Observe that for every x 0 ∈ A1 and x 00 ∈ A2 , ωfF (x 0 ) ⊂ O1 and ωfF (x 00 ) ⊂ O2 .
Since O1 and O2 are disjoint, the distance between ωfF (x 0 ) and ωfF (x 00 ) is not less than ε. Thus ωfF K is everywhere
discontinuous, hence not Baire 1.
Proposition 11.
If f is a 2n -function, then ωfF is Baire 1, for each filter F.
First we claim that for each x, ωf (x) is a 2k -cycle for some k ≤ n. (It is enough to check that there are no infinite
ω-limit sets for f.) Assume that M = ωf (x) is infinite and that K ⊃ M is a maximal ω-limit set for f. Since h(f) = 0,
by [5, Theorem A (xii)], K does not contain any periodic point. By [12, Theorem 3.6] K has periodic decompositions of
arbitrarily high period, which implies that f has periodic points of arbitrarily high period – a contradiction.
Proof.
n ·N
It follows that ωf 2n (x) = {g(x)} for some g : I → I. Then g = lim f 2
N→∞
n
ωf (x) = g(x), f(g(x)), . . . , f 2 −1 (g(x))
and therefore g is Baire 1. Moreover, for any x
and
n
f 2 (g(x)) = g(x).
(It is possible that there are redundant entries in the above sequence in the case ωf (x) is not a 2n -cycle.) Then, since f
is continuous, for each k ∈ N,
n
n
lim f 2 ·N+k (x) = f k (g(x)) = f k mod 2 (g(x)),
N→∞
where k mod 2n denotes the remainder of division of k by 2n . Let
B = k ∈ {0, 1, . . . , 2n − 1} : {2n · N + k : N ∈ N} ∈
/ F? .
The set B is non-empty and finite, and for each x ∈ [0, 1]
ωfF (x) = {f k (g(x)) : k ∈ B}.
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F-limit points in dynamical systems defined on the interval
Fix a closed non-empty set P. Since g is Baire 1, there is x0 ∈ P such that gP is continuous at x0 . Then, since f k is
continuous for each k ∈ B, ωfF P is continuous at x0 . Thus, ωfF is Baire 1.
(70 ) ωf ∈ B1
by [4, Th. 2.8]
(∃free F )(ωfF ∈ B1 )
f ∈ 2n
by Prop. 11
Figure 1.
(60 ) (∀p∈βN )(f p ∈ B1 )
by [4, Th. 2.2]
by Prop. 10
h(f) = 0
(90 ) (∀F )(ωfF ∈ B1 )
Dependencies between notions used in Theorem 7, Problems 8 and 9, and Propositions 10 and 11. Arrows denote implications.
Problem 12.
Suppose that all f p are Baire 1 (i.e. for each non-empty perfect set P and p ∈ βN there is x ∈ P such that f p P is
continuous at x). Does, for each non-empty perfect set P, there exist x such that all functions f p P are equicontinuous
at x?
Remark.
It is possible to prove that a positive answer to Problem 12 gives a positive answer to Problem 9.
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