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ANNALES

POLONICI MATHEMATICI LXV.3 (1997)

Turbulent maps and their ω-limit sets by F. Balibrea and C. La Paz (Murcia)

Abstract. One-dimensional turbulent maps can be characterized via their ω-limit sets [1]. We give a direct proof of this characterization and get stronger results, which allows us to obtain some other results on ω-limit sets, which previously were difficult to prove.

Let I = [0, 1] and denote by C(I) the set of continuous maps from I into I. A map f ∈ C(I) is turbulent if there are closed intervals J and K, with at most one point in common, such that f (J ) ∩ f (K) ⊇ J ∪ K [3].

These maps were also called L-schemes [2].

It is not difficult to prove the more operative characterization of such maps [3]: “A map f ∈ C(I) is turbulent if and only if there are points a, b, c ∈ I such that: a < c < b; f (a) = f (b) = a; f (c) = b and f (x) > a for a < x < b; x < f (x) < b for a < x < c, or the same with all inequalities reversed”.

Let f ∈ C(I). The orbit Orbf(x) of a point x ∈ I is the sequence (fn(x))n=0 where f0(x) = x and fn(x) = f ◦ fn−1(x) for every n > 0.

We say that Orbf(x) is periodic of order k if fk(x) = x and fi(x) 6= fj(x) for all 0 ≤ i 6= j ≤ k − 1. If f (x) = x, then x is a fixed point of f . We define the ω-limit set ω(x, f ) of a point x ∈ I to be the set of limit points of Orbf(x). For every x ∈ I this is a non-empty, closed and invariant set (f (ω(x, f )) = ω(x, f )). On the other hand, it is well known that if ω(x, f ) = A ∪ B and A, B are closed and disjoint sets then f (A) ∩ B 6= ∅.

Turbulent maps have complicated dynamics as far as periodic struc- ture is concerned. They have periodic points of period 3 and according to Sharkovski˘ı’s Theorem [2] have periodic points of all periods. But they can also produce nowhere dense ω-limit sets and in some cases ω-limit sets with non-empty interior.

1991 Mathematics Subject Classification: 26A18, 58F13; Secondary 58F15.

Key words and phrases: turbulent, ω-limit, one-sided fixed point.

This work has been partially supported by the DGICYT PB94-1159.

[223]

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224 F. B a l i b r e a and C. L a P a z

In [1] a characterization of turbulent maps is given in terms of the ω-limit sets produced by them: “The map f is turbulent if and only if there exists x0 ∈ I such that ω(x0, f ) is a unilaterally convergent sequence of points, that is, a monotone convergent sequence and its limit”.

The aim of this note is to prove Theorem 1 which establishes a general condition on some ω-limit sets produced by turbulent maps. As a particu- lar case we obtain the characterization given in [1] and some other results contained in [3].

Theorem 1. A map f ∈ C(I) is turbulent if and only if for some x0∈ I, ω(x0, f ) is an infinite set and there exist x < y ≤ z (resp. x > y ≥ z) such that f (y) ≥ z (resp. f (y) ≤ z), x ∈ [min ω(x0, f ), max ω(x0, f )] being a fixed point of f and z ∈ ω(x0, f ).

P r o o f. “If”. Supposing f (z) 6= z let

a = max{x ∈ [x, y] : f (x) = x} and b = min{x > y : f (x) = a}.

There exists c ∈ (a, b) such that f (c) ≥ b. If not, ω(x0, f ) ∩ (a, b) 6= ∅ and f (a, b) ⊂ (a, b), which is not possible. Therefore f is a turbulent map.

“Only if”. We will prove the existence of uncountable and infinite count- able ω-limit sets.

Let [a, b] ⊂ I be the interval where f is turbulent and let {An} be the sequence of interiors of the maximal closed intervals J included in [a, b] and not equal to [a, b], satisfying any of the following conditions:

(i) The image of J by an iterate of f is outside of [a, b];

(ii) J is periodic or mapped to a periodic interval or a point by an iterate of f ;

(iii) fn(J ) converges to a point or fm(J ) ∩ fn(J ) = ∅ if m 6= n.

Now, we consider the perfect and invariant set K = [a, b] \S

nAn. This set contains cycles of orders n or 2n for all n.

Since all the intervals in (i), (ii), or (iii) are disjoint and maximal, any open set with some of them in its interior contains two points whose images by iterates of f are a, b respectively, and thus contains a periodic point.

Therefore, there exists a countable set {Ci} of cycles such thatS Ci= K.

For Ci, a cycle of order m, we pick pi∈ Ciand the decreasing sequences (xin), (yin) satisfying xi1= pi, f (xin) = xin−1, lim(xin) = a; y1i = b, fm(yni) = yn−1i , lim(yni) = pi.

Let (zni) be an ordering of {(xin), (yni)}. Then the set

M =[

i

n[m

j=1

fj(zni) o

⊂ K

is countable and M = K.

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Turbulent maps and their ω-limit sets 225

Let (xn) be an ordering of M and (Vnj)n,j=1a countable basis of neigh- borhoods of (xn) whose diameters are respectively less than 1/j, and B1, B2, . . . an ordering of them. Then given Bi and Bi+1 it is not difficult to prove the existence of compact intervals Ji ⊂ Ki ⊂ Bi and mi ∈ N such that fmi(Ji) ⊂ Bi+1 and (fmi(Ji)) ∩ (xn) 6= ∅.

If (Jn), (Kn) are sequences of compact intervals recurrently defined in such a way that Kn+1 = fmn(Jn) and Jn+1 ⊂ Kn+1 ⊂ Bn+1, then the sequence of compact intervals (fPni=1mi(Ji+1)) is decreasing. If x0 ∈ T

n(fPni=1mi(Ji+1)) it is easy to check that ω(x0, f ) = K.

Now we will prove the existence of an infinite countable ω-limit set.

If f is a turbulent map with a < c < b, then the sequence (xn) recur- rently defined by: x1 = b, x2 = c, xn+1 = inf{x ∈ (a, xn) : f (x) = xn} is decreasing. Moreover, f (xn) = xn−1, f (x1) = a. By similar arguments to those above we get the result.

Corollary 2. If f ∈ C(I) and there exists x0 ∈ I such that ω(x0, f ) is an infinite set with a one-sided fixed point x ((x, x) ∩ ω(x0, f ) = ∅ or (x, x) ∩ ω(x0, f ) = ∅), then f is turbulent.

P r o o f. If x is an isolated fixed point from the left, then there exists x < z ∈ ω(x0, f ) such that f (z) > z. Otherwise the closed sets A = [x, 1) ∩ ω(x0, f ) and B = ω(x0, f ) \ A satisfy f (A) ∩ B = ∅.

Theorem 3. A map f ∈ C(I) is turbulent if and only if for some x0∈ I, ω(x0, f ) is an infinite set and there exist fixed points x < y of f such that

[x, y] ⊆ [min ω(x0, f ), max ω(x0, f )] and [x, y] ∩ ω(x0, f ) 6= ∅.

P r o o f. If f is a turbulent map, then there exists an infinite ω-limit set with a one-sided isolated limit point, which yields the existence of such an interval [x, y].

If there exists an interval [x, y] ⊆ [min ω(x0, f ), max ω(x0, f )] with [x, y]∩

ω(x0, f ) 6= ∅ and x, y are not one-sided isolated points belonging to ω(x0, f ), then there exists z ∈ [x, y] ∩ ω(x0, f ) such that f (z) > z or f (z) < z. Hence the map f is turbulent.

Corollary 4. If f ∈ C(I) and there exists x0∈ I such that ω(x0, f ) is an infinite set with two fixed points x < y, then f is turbulent.

Corollary 5. Suppose f ∈ C(I) is not a turbulent map. The map f2 is turbulent if there exists a point x0∈ I such that ω(x0, f ) is an infinite set satisfying at least one of the conditions: (i) it possesses a fixed point ; (ii) it possesses a two-periodic point x and there exist x < y ≤ z (resp. x > y ≥ z) with z ∈ ω(x0, f ) such that f2(y) ≥ z (resp. f2(y) ≤ z).

P r o o f. If (ii) holds, then Theorem 1 ends the proof. If (i) holds, then ω(x0, f ) possesses only a fixed point x and two points x1 < x < x2 such

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226 F. B a l i b r e a and C. L a P a z

that f (x1) = max ω(x0, f ) and f (x2) = min ω(x0, f ). Therefore, there exists y ∈ [x, max ω(x0, f2)] such that f2(y) ≥ z with z ∈ ω(x0, f2).

Corollary 6. If ω(x0, f ) = I and f is not turbulent , then (i) it pos- sesses a unique fixed point x ∈ (0, 1) and f2 is a turbulent map; (ii) either ω(x0, f2) = [0, x], or ω(f (x0), f2) = [0, x], or ω(x0, f2) = I.

P r o o f. If ω(x0, f ) = I then f possesses a fixed point x and since it is not turbulent, x is unique, not an end-point and there exist z1 < x < z2

such that f2(z1) = 0 and f2(z2) = 1. The sets ω(x0, f2) and ω(f (x0), f2) are intervals and if none of them is [0, 1], then ω(x0, f2) or ω(f (x0), f2) must be [0, x].

The following result, proved in a difficult way in the literature, can now be proved in an easy way, using the characterization of turbulent maps.

Theorem 7. If f ∈ 2 (it has only periodic points of period 2n for each n ∈ N), then ω(x0, f ) is finite or an uncountable nowhere dense set for any x0∈ I.

P r o o f. If Int(ω(x0, f )) 6= ∅ for some x0∈ I, then there exist an interval J and n ∈ N such that ω(x0, fn|J) = J . If ω(x0, f ) were an infinite countable set then a fixed point would belong to ω(x0, fn) for some n.

In both cases there would exist m ∈ N such that fmwould be a turbulent map and f would not be a 2-map.

References

[1] M. J. E v a n s, P. D. H u m k e, C. M. L e e and R. J. O ’ M a l l e y, Characterizations of turbulent one-dimensional mappings via ω-limit sets, Trans. Amer. Math. Soc. 326 (1991), 261–280.

[2] A. N. S h a r k o v s k i˘ı, Coexistence of cycles of a continuous mapping of the line into itself , Ukrain. Mat. Zh. 16 (1964), 61–71 (in Russian). MR 32#4213.

[3] L. S. B l o c k and W. A. C o p p e l, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992.

Departments of Mathematics and Applied Mathematics and Statistics University of Murcia

Murcia, Spain

E-mail: balibrea@gaia.fcu.um.es

Re¸cu par la R´edaction le 8.2.1996 evis´e le 30.5.1996

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