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On the spectrum of dynamical systems on trees
Topology and its Applications 222 (2017) 227–237
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Topology and its Applications www.elsevier.com/locate/topol
On the spectrum of dynamical systems on trees Jan Tesarčík Mathematical Institute in Opava, Silesian University in Opava, Na Rybníčku 1, 74601, Opava, Czech Republic
a r t i c l e
i n f o
Article history: Received 13 April 2016 Received in revised form 10 March 2017 Accepted 12 March 2017 Available online xxxx MSC: 37B40 37E25 Keywords: Dynamical system Tree Distributional functions Spectrum Weak spectrum Basic omega-limit sets
a b s t r a c t In their paper, Schweizer and Smítal (1994) [10] introduced the notions of distributional chaos for continuous maps of the interval, spectrum and weak spectrum of a dynamical system. Among other things , they have proved that in the case of continuous interval maps, both the spectrum and the weak spectrum are finite and generated by points from the basic sets. Here we generalize the mentioned results for the case of continuous maps of a finite tree. While the results are similar, the original argument is not applicable directly and needs essential modifications. In particular, it was necessary to resolve the problem of intersection of basic sets, which was a crucial point. An example of one-dimensional dynamical system with an infinite spectrum is presented. © 2017 Elsevier B.V. All rights reserved.
1. Introduction and the main results Let X be a compact metric space and d be its metric. We consider a continuous map f : X → X (we denote the set of such maps by C(X, X)), in this setting we talk about a dynamical system. Let N be the set of positive integers, taking an x ∈ X, we define recursively an nth iteration by putting f 0 (x) = x and f n (x) = f (f n−1 (x)), for any n ∈ N. The sequence (f n (x))∞ n=0 is called a trajectory of point x and denoted by trf (x). We define an orbit of a point x ∈ X by putting Orbf (x) = {f n (x); n ∈ N ∪ {0}}, similarly, we define an orbit of a set A ⊂ X as Orbf (A) = {f n (x); x ∈ A and n ∈ N ∪ {0}}. If we have f p (x) = x for some x ∈ X and some positive integer p, then we call the point x periodic and the smallest p of such a kind is called the period of x. We say that f is transitive, if for every two nonempty open sets U and V in X there is an integer n, such that f n (U ) ∩ V = ∅. Let K be a nonempty subset of X, we say that f |K is invariant if f (K) ⊂ K. E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.topol.2017.03.007 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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In addition, if f (K) = K holds, then we say that f |K is strongly invariant. Let A be a subset of X, then diam A = sup {d(x, y); x, y ∈ A}. Let f : X → Y be a map between two metric spaces, then we say that f is monotone on X if for any y ∈ Y the set f −1 (y) is connected. Let f ∈ C(X, X) and g ∈ C(Y, Y ) be maps of compact metric spaces, a continuous surjection ϕ : X → Y is called semiconjugacy if ϕ ◦ f = g ◦ ϕ. Moreover, if there exists a k ∈ N such that #ϕ−1 (y) ≤ k for each y ∈ Y , then we say that ϕ semiconjugates f and g almost exactly. For a given x ∈ X we define the omega-limit set of x as a set of accumulation points of trajectory trf (x) and we denote it as ωf (x). We say that ωf (x) is maximal if and only if there is no ωf (y), such that ωf (x) ⊂ ωf (y), ωf (x) = ωf (y). In their paper [10], Schweizer and Smítal have defined the notion of distributional chaos. Several authors have continued exploring this notion in various settings. The notion is based on distributional functions of pairs of points. For any x, y ∈ X, a positive integer n and real t put ξ(x, y, t, n) = #{i; 0 ≤ i ≤ n − 1 and δx,y (i) < t}
(1)
where δx,y (i) = d f i (x) , f i (y) is the distance of ith iteration of the points x and y. Put ∗ Fx,y (t) = lim sup
1 ξ(x, y, t, n), n
(2)
Fx,y (t) = lim inf
1 ξ(x, y, t, n). n
(3)
n→∞
n→∞
∗ Function Fx,y is called an upper distributional function and Fx,y is called a lower distributional function. ∗ ∗ ∗ Both Fx,y and Fx,y are nondecreasing functions with Fx,y (t) = Fx,y (t) = 0 for every t < 0 and Fx,y (t) = Fx,y (t) = 1 for every t > diam X. An arc is a topological space homeomorphic to the compact interval [0, 1]. A graph G is a continuum (a nonempty compact connected metric space) which can be written as the union of finitely many arcs any two of which can intersect only in their endpoints. If a graph does not contain a set homeomorphic to the circle, it is called a tree. A subgraph is a subset of a graph which itself is a graph. By a periodic subgraph we mean a subgraph H ⊂ G, such that there is an n ≥ 1 for which H, f (H), . . . , f n−1 (H) have pairwise disjoint interiors and f n (H) = H; in this case, we also speak about a periodic orbit of subgraphs. A nonempty intersection of three or more arcs, with pairwise disjoint interiors, is called a vertex, where the maximal number of such arcs is called the degree of the vertex. The maximal order of a graph is the maximum number of degrees of vertices on the graph. We say that two points u, v ∈ G are synchronous if both ωf (u) and ωf (v) are contained in the same maximal omega-limit set ω, and if, for any periodic subgraph H such that Orbf (H) ⊃ ω, there is a j ≥ 0 such that f j (u), f j (v) ∈ H. The spectrum of f , denoted by Σ(f ), is a set of minimal elements of the set D(f ) = {Fu,v ; u and v are synchronous}. And the weak spectrum Σw (f ) of f is a set of minimal elements of the set Dw (f ) = {Fu,v ; lim inf i→∞ δu,v (i) = 0}. The relation between Σ(f ) and Σw (f ) for the case of continuous interval self maps is discussed in Remark 2.6 [10], and the authors have shown that in this case there holds Σ(f ) ⊂ Dw (f ), but we can not say anything about Σ(f ) ∩ Σw (f ). In the case of continuous tree self maps, like in the case of continuous interval self maps, there is no direct relation between Σ(f ) and Σw (f ), but the arguments from [10] can be applied to this case and, therefore, it is true that Σ(f ) ⊂ Dw (f ). The following theorem is our main result. It generalizes the result in [10] for the interval maps and in [5] for the circle maps to the case of tree maps.
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Theorem 1. Let f be a continuous self map of a tree. (1) If the topological entropy of f is zero, then Σ(f ) = Σw (f ) = {χ(0,∞) }. (2) If the topological entropy of f is positive, then both the spectrum Σ(f ) and the weak spectrum Σw (f ) are finite and nonempty. Specifically, Σ(f ) = {F1 , . . . , Fm } for some m ≥ 1, and Σw (f ) \ Σ(f ) = {Fm+1 , . . . , Fn } for some n ≥ m. Furthermore, for each i there is an εi > 0, such that Fi (εi ) = 0. Even though the main results are formulated for the case of continuous tree maps, the results needed for their proof (mainly the dynamics of basic sets) can be proven in more general space. Therefore, we prove them for the case of graph maps. 2. The basic sets The omega-limit sets play an important role in the theory of discrete dynamical systems. They give us useful information about the behavior of the system. Special attention must be paid to the basic sets. As one can see below, the basic sets possess interesting properties and even their presence ensures attractive phenomena. In the sixties, A.N. Sharkovsky systematically studied the properties of the omega-limit sets of the continuous maps of the interval (e.g. [8] and [9]). He showed that any such set is contained in a maximal omega-limit set and introduced three types of maximal omega-limit sets. More precisely the cycle, if it is finite; the infinite of the first type (lately known as a solenoid), and the infinite of the second type (lately known as a basic set). In the eighties, A. Blokh widely extended the results on the properties of the basic sets but using a different definition (see [2] and [3]). In the paper [4], Sharkovsky’s classification of maximal omega-limit sets was extended to the case of graph maps. It was necessary to add a fourth type of the maximal omega-limit sets to cover the graph specific case of the omega-limit sets (singular sets). Let G be a graph and f ∈ C(G, G). For an omega-limit set ω ⊂ G we define Pω = Orbf U , (4) where the intersection is taken over all neighborhood U intersecting ω. If ω is finite, then it is a cycle. Now consider ω to be infinite. If Pω is a nowhere dense set, then we call ω a solenoid (this terminology is used in [6]). If Pω consists of finitely many connected components and ω contains a periodic point then ω is called a basic set. Let Pω = H1 ∪ · · · ∪ Hn , where H1 , · · · , Hn are minimal periodic subgraphs with disjoint interiors, then ω ∩ Hi is called a portion of the basic set ω. Note that if we consider f n , then ω ∩ Hi is a basic set for f n and Pω ∩ Hi does not admit a proper periodic subgraph, i.e., ω ∩ Hi is indecomposable. If Pω consists of finitely many connected components and ω contains no periodic point, then we call ω a singular set. Note that the case of a singular set is only possible on a finite union of subgraphs that are homeomorphic with a circle. In the paper [7] it was shown that Sharkovsky’s definition of the basic sets, Blokh’s definition of the basic sets and the definition of the basic sets from [4] are mutually equivalent. Example 2. Consider the following interval map fI : [0, 1] → [0, 1] ⎧ 3x for x ∈ [0, 13 ); ⎪ ⎨ 1 for x ∈ [ 13 , 23 ]; fI (x) = ⎪ ⎩ 3 − 3x for x ∈ ( 23 , 1]. Note that the Cantor middle third set C is a basic set for fI and that PC = [0, 1].
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The proofs of many results of properties of the basic sets in the interval case use the convex hull of the basic sets. This method cannot be used in the case of graph maps because the convex hull may not be defined (consider a cantor-like basic set on the circle). Therefore, we used a workaround, considering Pω instead of a convex hull. Contrary to the case of an interval, where the convex hulls of two basic sets cannot overlap (see [10, Theorem 3.7]), in the case of graph maps, the situation is more complicated when we consider the intersection of Pω of two basic sets as shown in the following example. Example 3. Let G = S 16 ( 16 , 0) ∪ ([ 13 , 23 ] × {0}) ∪ S 16 ( 56 , 0), where Sr (x, y) is the circle with the center at (x, y) and radius r.
Graph G. Defining f : G → G, for any (x, y) ∈ G with y ≥ 0 we put f (x, y) = (u, v), where u = fI (x), fI (x) is defined as in Example 2, and v ≥ 0, so that (u, v) ∈ G; similarly, for (x, y) ∈ G with y ≤ 0 we put f (x, y) = (u, v), where u = fI (x) and v ≤ 0, so that (u, v) ∈ G. It is easy to see that if a point (u1 , v1 ) ∈ G where v1 > 0 generates the basic set ω1 and some other point (u2 , v2 ) ∈ G where v2 < 0 generates the basic set ω2 , then pr1 (ω1 ) = pr1 (ω2 ) = C and Pω1 ∩ Pω2 = {(0, 0), (1, 0)} ∪ ([ 13 , 23 ] × {0}). The problem of intersection of Pω for the basic sets on graphs is resolved by Lemma 7. In the following theorem we summarize the known properties of the basic set, which will be used in the rest of the paper. Theorem 4 ([4]). Let f ∈ C(G, G) and ω ˜ be its basic set, then (i) (ii) (iii) (iv)
ω ˜ is a perfect set; The system of all basic sets is countable; ˜; The set of periodic points is dense in ω Let Pω˜ = H ∪ f (H) ∪ · · · ∪ f n−1 (H) and K, J be subgraphs, such that J ∩ ω ˜ is infinite and K ⊂ int(H). i+jn Then f (J) ⊃ K for some i and sufficiently large j; (v) The topological entropy of f is positive (h (f ) > 0). We describe the situation from (iv) by saying that f |ω˜ is strongly transitive in H.
3. The zero entropy case By Theorem 4, if f has a zero topological entropy, then it has no basic sets. The following lemma shows that a nontrivial distributional function can be generated by points with the omega-limit sets contained in the basic sets only. Lemma 5. Let f be a continuous map of a tree T , such that f has no basic set. Then Σ(f ) = Σw (f ) = {χ(0,∞) }. Proof. Let u, v ∈ T . If both ωf (u) and ωf (v) are cycles, then Fu,v = χ(0,∞) in both cases when u and v are synchronous, or when lim inf i→∞ δu,v (i) = 0.
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If one of the sets ωf (u) and ωf (v) is a cycle and the other one is contained in a solenoid, then in both cases when u and v are synchronous, or when lim inf i→∞ δu,v (i) = 0 , we get that the solenoid contains a periodic point, which is impossible. Suppose that ωf (u) ⊂ ωu and ωf (v) ⊂ ωv , where ωu and ωv are solenoids. Since Pωu is nowhere dense for a given ε > 0, there is a periodic orbit H, f (H), . . . , f n−1 (H) of subgraphs covering ωu each of f i (H) with a diameter less than ε. Similarly, we get a periodic orbit K, f (K), . . . , f m−1 (K) of subgraphs for ωv . Now if u and v are synchronous, then there is an integer j, such that f j+i(u), f j+i (v) ∈ f i mod n (H) for all i. Since ε was chosen arbitrarily, we get Fu,v = χ(0,∞) . On the other hand, if lim inf i→∞ δu,v (i) = 0, then we get n = m and f i (H) ∩ f i (K) = ∅, for all i = 0, 1, . . . , n − 1. Since diam(f i (H) ∪ f i (K)) ≤ 2ε, we similarly get Fu,v = χ(0,∞) . 2 4. The proof of the main results The following itinerary lemma comes from [10]; here, we generalize it for the graph case. ∞
∞
Lemma 6. Let f ∈ C (G, G) and u, v be in G. Let (Ui )i=1 and (Vi )i=1 be closed subgraphs with the property ∞ ∞ that the sequence (Ui )i=1 is eventually in every neighborhood of u and the sequence (Vi )i=1 is eventually in every neighborhood of v, and such that for any i and j there are positive integers u (i, j) and v (i, j) such that f u(i,j) (Ui ) ⊃ Vj and f v(i,j) (Vi ) ⊃ Uj . Then {u, v} ⊂ ωf (y), for some y in G. ∞
∞
Proof. Define a decreasing sequence (Ji )i=1 of compact sets and an increasing sequence (n (i))i=1 of positive integers as follows: J1 = U1 and n (1) = u (1, 2). Then f n(1) (J1 ) ⊃ V2 , choose J2 ⊂ J1 , such that f n(1) (J2 ) = V2 . Take n (2) = n (1) + v (2, 3). Then f n(2) (J2 ) = f v(2,3) (V2 ) ⊃ U3 and there is a J3 ⊂ J2 , ∞ such that f n(2) (J3 ) = U3 . Then there are J4 and n (3), such that f n(3) (J4 ) = V4 , etc. Let y ∈ i=1 Ji . Since the trajectory of y visits every neighborhood of u and every neighborhood of v, the result follows. 2 The next lemma resolves the problem of how a Pω of two basic sets can intersect. It substitutes the non-overlap rule for convex hulls of the basic sets in the interval case. Lemma 7. Let f ∈ C (G, G), ω1 , ω2 be two different basic sets, Pω1 and Pω2 be defined by (4). Then at least one Pω1 ∩ Pω2 ∩ ω1 , Pω1 ∩ Pω2 ∩ ω2 must be finite. Proof. We assume that both Pω1 ∩ Pω2 ∩ ω1 and Pω1 ∩ Pω2 ∩ ω2 are infinite, then take an arc J ⊂ int (Pω1 ), such that J ∩ω2 is infinite, and an arc K ⊂ int (Pω2 ), such that K ∩ω1 is infinite. This is possible because, by Theorem 4, ω1 , ω2 are perfect. Now by Theorem 4 f i (J) ⊃ K, for any sufficiently large i, and f j (K) ⊃ J, for any sufficiently large j. Using this together with Lemma 6, we get that both ω1 , ω2 are contained in an ω-limit set ωf (z) for some z ∈ G, which is a contradiction with the maximality of ω1 , ω2 . 2 We formulate the following lemma to substitute the wrong Sharkovsky result, claiming that the trajectory of each point with omega-limit sets contained in the basic sets eventually enters a basic set. Lemma 8. Let f ∈ C (G, G), ω ˜ be its basic set and x ∈ G be such that ωf (x) ⊂ ω ˜ . Then there is an x ˜∈ω ˜ such that limn→∞ δx,˜x (n) = 0. Proof. Let x be such that ωf (x) ⊂ ω ˜ . If for some n ∈ N f n (x) ∈ ω ˜ ; then, since ω ˜ is strongly invariant, we ∞
n n can find x ˜∈ω ˜ , such that f (˜ x) = f (x). Otherwise, suppose that Orbf (x) ⊂ i An(i) where An(i) i=0 is the sequence of open subgraphs contiguous to ω ˜ , such that f i (x) ∈ An(i) , for any nonnegative integer i. Since f |ω˜ is almost exactly semiconjugated to a transitive map of a graph (see Theorem 2 in [2] for details) we obtain that the set A¯n(i) , i = 1, 2, . . . is invariant under f . We distinguish two cases.
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∞ (1) If the sequence An(i) i=0 is eventually periodic, then, since ωf (x) ⊂ ω ˜ , the set ωf (x) must be finite i.e. a cycle and the result easily follows. ∞ ∞ (2) The sequence An(i) i=0 is wandering. In this case, since An(i) i=0 are nonoverlapping, we have limi→∞ diam An(i) = 0 and x ˜ is an endpoint of An(0) belonging to ω ˜. 2 The following five lemmas and their proofs are modifications of the ones from [10], we use them in proof of our main theorem. Lemma 9. Let f ∈ C (G, G), then for any λ ∈ (0, 1) and t ∈ (0, diam G) there is an integer n (t, λ) with the following property: If ωi are portions of a basic set ω with a period m, where m ≥ n (t, λ) and Pωi (defined by (4) with f replaced by f m ) are nonoverlapping for i = 0, 1, . . . , m − 1, then for all u, v ∈ ω Fuv (t) > λ.
p Proof. Let G = i=1 Gi where Gi are arcs with pairwise disjoint interiors, let d be the maximal degree of
p the point of G. Put c = i=1 diam Gi . Fix t and λ, let n (t, λ) be such that (n (t, λ) − c · d/t)/n (t, λ) > λ. Since there are at most c · d/t distinct sets f s (ω) with a diam f s (ω) ≥ t (this follows from Lemma 7), we have Fu,v (t) ≥ (1/m) · # {s < m; diam f s (ω) < t} ≥ (m − c · d/t)/m > λ. 2 ∗ be continuous at t. Then, for any ε > 0, there are Lemma 10. Let f ∈ C(G, G) and let both Fx,y and Fx,y positive integers k, q, arbitrarily large, and δ > 0, such that
1 ξ(u, v, t, k) < Fx,y (t) + ε k and 1 ∗ ξ(u, v, t, q) > Fx,y (t) − ε q whenever d(u, x) < δ and d(v, y) < δ. ∗ ∗ Proof. Choose an ε1 > 0 such that Fx,y (t + 2ε1 ) < Fx,y (t) + ε/2 and Fx,y (t − 2ε1 ) > Fx,y (t) − ε/2. Then 1 choose a k such that k ξ(x, y, k, t + 2ε1 ) < Fx,y (t + 2ε1 ) + ε/2. The first inequality follows from the fact that ξ(u, v, k, t) < ξ(x, y, k, t + 2ε1 ) whenever δ > 0 is sufficiently small. The argument for the second inequality is similar. 2
Lemma 11. Let f ∈ C (G, G), ω1 , ω2 be its basic sets, Pω1 = H1 ∪ · · · ∪ Hn , Pω2 = K1 ∪ · · · ∪ Km be minimal periodic decompositions and Q ⊂ G × G be a countable set of pairs (u, v), such that u ∈ ω1 ∩ H1 and v ∈ ω 2 ∩ K1 . Then there are points x ∈ ω1 ∩ H1 and y ∈ ω2 ∩ K1 , such that for any t > 0: Fx,y (t) ≤ inf {Fu,v (t) ; (u, v) ∈ Q}
(5)
∗ ∗ Fx,y (t) ≥ sup Fu,v (t) ; (u, v) ∈ Q .
(6)
and
Proof. Let R be a countable set, dense in [0, diam G], and such that for any (u, v) ∈ Q and any t ∈ R, both ∞ ∞ ∗ Fu,v (t) and Fu,v (t) are continuous at t. Let (tj )j=1 and (uj , vj )j=1 be sequences of points where tj ∈ R and (uj , vj ) ∈ Q, such that for any t ∈ R and any (u, v) ∈ Q, t = tj , u = uj and v = vj for infinitely many j.
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Next, using induction, we define the positive integers k (1) < q (1) < k (2) < q (2) < ... < k (i) < q (i) < ... ∞
∞
and decreasing sequences (Ui )i=1 and (Vi )i=1 of closed subgraphs with lim diam (Ui ) = lim diam (Vi ) = 0
i→∞
i→∞
and for any u ∈ Un , v ∈ Vn and any j ≤ n, 1 1 ξ (u, v, tj , k (j)) ≤ Fu(j),v(j) (tj ) + k (j) j
(7)
1 1 ∗ ξ (u, v, tj , q (j)) ≥ Fu(j),v(j) (tj ) − . q (j) j
(8)
and
We take U1 = H1 and V1 = K1 , k (1) = 1, q (1) = 2 and assume that Un , Vn , k (n) and q(n) have been defined such that f k(j) (Un ) ∩ ω1 and f q(j) (Vn ) ∩ ω2 are infinite whenever j is sufficiently large. By (iv) of Theorem 4, f |ω1 is strongly transitive on H1 , f |ω2 is strongly transitive on K1 and u ∈ ω1 ∩ int(H1 ), v ∈ ω2 ∩ int(K1 ) if (u, v) ∈ Q. So, there is some s > q (n), such that u (n + 1) ∈ f s (Un ) and v (n + 1) ∈ f s (Vn ). Let a ∈ Un and b ∈ Vn be such that f s (a) = u (n + 1) and f s (b) = v (n + 1). Then ∗ ∗ Fa,b = Fu(n+1),v(n+1) and Fa,b = Fu(n+1),v(n+1) . The existence of Un+1 ⊂ Un , Vn+1 ⊂ Vn , k (n + 1) and q (n + 1) follows by Lemma 10. (We take Un+1 and Vn+1 as compact neighborhoods of a and b, with diam (Un ) > 2 diam (Un+1 ) and diam (Vn ) > 2 diam (Vn+1 ) .) Note that (by (i) of Theorem 4) a, b, Un+1 , Vn+1 can be chosen such that both f s (Un+1 ) ∩ ω1 and f s (Vn+1 ) ∩ ω2 are infinite. ∞ ∞ Take x ∈ j=1 Uj and y ∈ j=1 Vj . For any t ∈ R and any (u, v) ∈ Q, take j such that t = tj , u = u (j) and v = v (j). Since x ∈ Uj and y ∈ Vj , we will apply (7) with u = x and v = y. Since j can be arbitrarily large, we have Fx,y (t) ≤ Fu,v (t). This remains true for any t, since R is dense in [0, diam G], so this implies (5). We get Equation (6) similarly. We have x ∈ H1 and y ∈ K1 . We need to show that x can be chosen in ω1 and y in ω2 . Let ∞ w ∈ ω1 ∩ H1 be such that ωf (w) = ω1 and let (Wi )i=1 be a decreasing sequence of compact neighborhoods of w with limi→∞ Wi = w. Now apply Lemma 6 with u replaced by x, v by w and Vi by Wi (Vi from Lemma 6). Since ω1 is maximal, we obtain ωf (x) ⊂ ω1 . Similarly, we have ωf (y) ⊂ ω2 . Now, by Lemma 8, there are x ˜ ∈ ω1 and y˜ ∈ ω2 , such that limi→∞ δx,˜x (i) = 0 and limi→∞ δy,˜y (i) = 0. Replace x by x ˜ and y by y˜. Clearly (5) and (6) remain valid. 2 ∞
Lemma 12. Let f ∈ C (G, G) and (ωi )i=1 be a sequence of a distinct indecomposable portion of the basic omega-limit sets. Then limi→∞ diam ωi = 0.
n Proof. Let G = k=1 Gk be the decomposition of G into arcs, such that the interior of each Gk contains no vertex and any two of them can intersect only in their endpoints. By Lemma 7 and the fact that G has finitely many vertexes, there are only finitely many ωi that intersect more the one Gk . Therefore, we can assume that every ωi is contained in only one arc. It is sufficient to verify the assertion for the case when ∞ all ωi are contained in one Gk . Suppose, on the contrary, that there is a subsequence (ωni )i=1 , such that diam ωni > ε > 0 for any i. Then for some ni and nj both Pωni ∩ Pωnj ∩ ωni and Pωni ∩ Pωnj ∩ ωnj must be infinite. Which is in contradiction with the Lemma 7. 2
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∞
Lemma 13. Let f ∈ C (G, G) and let (ωi )i=1 be the minimal periodic portions of basic sets of f . For any i, j, set Gi,j = inf {Fu,v ; u ∈ ωi , v ∈ ωj }. Then: (i) Each Gi,j is zero on interval [0, ε (i, j)] where ε (i, j) is a positive number. (ii) The set {Gi,j ; ωi ∩ ωj = ∅} has a finite number of minimal elements. ∞ (iii) The set {Gi,i }i=1 has a finite number of minimal elements. Proof. (i) By (iii) of Theorem 4, there are distinct periodic points p ∈ ωi and q ∈ ωj . Since mins δp,q (s) = ε > 0, we have Fp,q (t) = 0, and, subsequently, Gi,j (t) = 0, for t ≤ ε, since Gi,j (t) ≤ Fp,q (t). Take ε (i, j) = ε. (ii) We may assume that ωi = ωj for any i = j. Let Pωi be the cover of ωi . Let ε = ε (1, 1). We say that an ωi is extremal, if diam ωi > ε/2 and if for any ωj there exists no arc A ⊂ Pωj with endpoints in ωj , such that ωi ⊂ A. By Lemma 7, there are finitely many extremal ωi ’s. We may assume that ω1 , . . . , ωn(1) are all extremal sets of f ; note that n (1) ≥ 1, since f has a positive topological entropy and therefore at least one maximal omega-limit set is a basic set (cf. e.g. Theorem 3 in [4]). Let m > 0 be an integer. We say that a set ωi is significant if diam ωi > ε/2 and the period of ωi is less than m. By Lemma 12, there are finitely many significant ωi ’s. Thus we may assume without the loss of generality, that there are integers n (3) ≥ n (2) ≥ n (1) > 0, such that ω1 , . . . , ωn(2) is the minimal system invariant under f , containing all extremal and all significant sets, and that ωn(2)+1 , . . . , ωn(3) is the system of all portions that have a point in common with some of ωi , i ≤ n (2). We show that the minimal elements of {Gi,j ; ωi ∩ ωj = ∅} are in the set M = {Gi,j ; i, j ≤ n (3)} whenever m is sufficiently large, depending on n (1). Assume that i > n (3) and ωi ∩ ωj = ∅. Then ωj cannot be significant, consequently j > n (2). Let u ∈ ωi and v ∈ ωj . It suffices to show that Fu,v ≥ G for some G in M . If the diam f s (ωi ∪ ωj ) ≤ ε, for any s, then Fu,v (t) = 1 for t > ε, and, therefore, Fu,v ≥ G1,1 . So it suffices to consider the case when the diam f s (ωi ∪ ωj ) > ε for some s. Since the set ω1 , . . . , ωn(2) is invariant, we may assume without any loss of the generality that diam (ωi ∪ ωj ) > ε and that diam f s (ωi ∪ ωj ) ≤ diam (ωi ∪ ωj ) for any s.
(9)
Since one of the sets ωi , ωj , say ωi , has a diameter > ε/2, there is an extremal ωr (ωi cannot be extremal since i > n (1)) and an arc Au,v ⊂ Pωr with endpoints u, v ∈ ωr , such that ωi ⊂ Au,v ; similarly, there is an arc Aw,z ⊂ Pωr with endpoints w, z ∈ ωr , such that ωj ⊂ Aw,z , since ωi ∩ ωj = ∅. By Theorem 4, there are periodic points a, b in ωr with the property: If H is an subgraph such that H ∩ ωr is finite, the diam H > ε and H ⊂ Pωr ,
(10)
then H ⊂ Sa,b where Sa,b is a subgraph containing points a, b ∈ ωr , Sa,b ⊂ Pωr , which satisfies diam Sa,b ≥ max{d (α, β) ; α, β ∈ {u, v, w, z}}. Now, by Lemma 7, one of the sets Pωr ∩ Pωi ∩ ωr , Pωr ∩ Pωi ∩ ωi must be finite. If Pωr ∩ Pωi ∩ ωi is a finite set, then Pωi ∩ ωi is finite, since Pωi ⊂ Pωr and then ωi is finite, since ωi ⊂ Pωi , which is impossible. Thus Pωr ∩ Pωi ∩ ωr is finite, since Pωi ⊂ Pωr , we have Pωi ∩ ωr is finite. Similarly, Pωj ∩ ωr is finite. Since H = Pωi ∪ Pωj satisfies (10), we obtain ωi ∪ ωj ⊂ Sa,b . Take t0 such that diam (ωi ∪ ωj ) < t0 < diam Sa,b . Let Fa,b (t0 ) = λr . Clearly λr < 1. Let ε (r, r) be as in (i). We have Gr,r (t) = 0 ≤ Fu,v (t) for t ≤ ε (r, r), and by (9) Gr,r (t) ≤ 1 = Fu,v (t) for t > t0 . And if ε (r, r) < t0 , then, by Lemma 9, Gr,r (t) ≤ Fa,b (t) ≤ λr < Fu,v (t) whenever ε (r, r) < t ≤ t0 and m ≥ n (ε (r, r) , λr ). Thus Gr,r ≤ Fu,v if m ≥ n (ε (r, r) , λr ). It follows that when the parameter m = max {n (ε (r, r) , λr ) ; 1 ≤ r ≤ n (1)}, then {Gi,j ; i, j ≤ n (3)} contains the minimal elements of {Gi,j ; ωi ∩ ωj = ∅}. ∞ (iii) It suffices to show that the minimal elements of {Gi,i }i=1 are contained in {Gi,i ; i ≤ n (2)}. But this follows from the above argument. 2
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Proof of Theorem 1. (1) In the case of a tree map with a zero entropy all maximal omega-limit sets are either periodic orbits or solenoids. Applying Lemma 5, we get Σ (f ) = Σw (f ) = {χ(0,∞) }. ∞ (2) By Theorem 4, the system of all basic sets (ωi )i=1 is countable and since the entropy is positive, it is not empty. Spectrum. Let u, v ∈ T be such that u and v are synchronous, ωf (u) , ωf (v) ⊂ ω ˜, ω ˜ be a maximal omega-limit set. If ω ˜ is a periodic set or solenoid, then by Lemma 5 Fu,v = χ(0,∞) . Therefore, we can assume that the omega-limit sets of all synchronous points are contained in the basic sets. Put D = {Fu,v ; u, v are synchronous} and E = {Fu,v ; u, v ∈ ωi , for some i}. Clearly E ⊂ D. Now, if u and v are synchronous, then ωf (u) and ωf (v) are contained in the same maximal basic set ω ˜ . By Lemma 8, there are u ˜, v˜ in ω ˜ such that limn→∞ δu,˜u (n) = 0 and limn→∞ δv,˜v (n) = 0. The points u ˜, v˜ are contained in ω ˜ and hence in some ωi . Thus E and D must have the same system of Σ (f ) of minimal elements and Lemma 13 gives the result. Weak spectrum. Let u, v ∈ T , denote ωu as a maximal omega-limit set containing ωf (u) (similarly ωv ). Suppose lim inf δu,v (i) = 0, i→∞
(11)
which gives us ωu ∩ ωv = ∅. If both ωu , ωv are cycles, then by (11) u, v are synchronous and Lemma 5 gives us the result. The case when one of ωu , ωv is a cycle and the other is a solenoid is impossible. If one of ωu , ωv , say ωu , is a solenoid, then (2) of Theorem 1 in [2] gives ωv ⊂ ωu . Since the basic set contains a periodic point, ωv must be a solenoid, moreover, ωu = ωv and Lemma 5 gives us the result. Therefore, all non-trivial distributions generated by the points satisfying (11) have their omega-limit sets contained in the basic sets. Let Dw = {Fu,v ; u, v such that lim inf n→∞ δu,v (n) = 0} and Ew = {Fu,v ; u ∈ ωi , v ∈ ωj , ωi ∩ ωj = ∅}, and let Fu,v ∈ Dw . Then there are i, j, k such that f k (u) ∈ ωi and f k (v) ∈ ωj and (11) implies ωi ∩ ωj = ∅. Thus Fu,v ∈ Ew and Dw ⊂ Ew . Conversely, let Fu,v ∈ Ew . Let u ∈ ωi , v ∈ ωj and w ∈ ωi ∩ ωj . Take ∗ Q = {(u, v) , (w, w)} and apply Lemma 11 to get x, y such that Fx,y ≤ Fu,v and Fx,y = χ(0,∞) . Therefore lim inf n→∞ δu,v (n) = 0, which implies Fx,y ∈ Dw . Thus we have proved that Dw ⊂ Ew and that Ew has the lower bounds in Dw . Consequently, Ew and Dw must have the same system Σw (f ) of the minimal elements and Lemma 13 gives the result. 2 5. An example with the infinite spectrum In this section we show that the results on the finite spectrum are not true any more when we consider more general but still one dimensional spaces. Namely, we present an example of a dendroid (an arcwise connected, hereditarily unicoherent continuum) and a continuous map on it, such that there is an infinite family of an incomparable distributional function generated by synchronous points (i.e., it has the infinite spectrum). A triangular map of the square with the infinite spectrum was constructed in [1]. Here we show that this is possible even in the spaces with the dimension less than two. Before we give the mentioned example, we turn our attention to a certain class of continuous maps of compact intervals. Call this class tent-like maps. For a given b ∈ [ 34 , 1] and l ∈ [0, 11 32 ], we define a function fb,l : [0, b] → [0, b] (see Fig. 1) ⎧ ⎪ ⎪ ⎨
2b x, 2l + b fb,l (x) = ⎪ 2b 2b2 ⎪ ⎩ x+ , 2l − b b − 2l
for x ∈ [0, (b + 2l)/2]; for x ∈ ((b + 2l)/2, b].
It is easy to compute that xb,l = 2b2 /(3b − 2l) is a fixed point for fb,l .
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Fig. 1. A tent-like map fb,l .
Fig. 2. Space X from Example 14.
In [10] it has been shown that for a function from such class there is a unique minimal distribution (a stepfunction) where the positions of jumps are given by the positions of the periodic points (see Remark 2.5 in [10] for details). In particular, if Fb,l is the minimal distribution for fb,l , then Fb,l (x) = 0 on (−∞, xb,l ],
and Fb,l (x) = 1 on (b, ∞].
(12)
Example 14. There is a space (a dendroid) X ⊂ R2 and a continuous map f of X such that its spectrum Σ(f ) is not finite. Proof. First, we define the sequences bn = (3 · 2n + 1)/2n+2 and ln = (11 · 2n − 11)/2n+5 , for n = 0, 1, 2, . . . . Space. First, put In = {1/2n } × [0, bn ], for each n = 0, 1, 2, . . . and I∞ = {0} × [0, 34 ]. Let X = ([0, 1] × {0}) ∪ I∞ ∪
∞
In .
n=0
See Fig. 2. Map. We define a continuous map f : X → X as follows. The map f restricted to ([0, 1] × {0}) is equal to the identity. On each In we put f |In : In → In f |In (x, y) = (x, fbn ,ln (y)). The continuity gives us f |I∞ (x, y) = (x, f 34 , 11 (y)). 32
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Spectrum Σ(f ) of f . The map f |In is a tent-like map, which we described above. In particular, In is a basic set for f and the minimal distribution on In satisfies (12) for b = bn and l = ln . It is easy to verify that any two different Fbn ,ln are incomparable. Therefore, the spectrum Σ (f ) must be infinite. Weak spectrum Σw (f ) of f . Since the map f restricted to ([0, 1] × {0}) is equal to identity, then we can look at f |In as continuous interval self maps. Therefore, we can use the Remark 2.6 from [10], which says that Σ(f |In ) ⊂ Dw (f |In ) and from which it results that the Σw (f ) must be infinite. 2 Acknowledgements I sincerely thank my supervisor, Michal Málek, for his valuable guidance. I am grateful for his constant support and help. I would also like to thank the reviewer for his precious time and important comments. The work was supported, in part, by the grants SGS/1/2014 and SGS/16/2016 from the Silesian University in Opava, RVO funding for [IČ47813059]. This support is gratefully acknowledged. References [1] M. Babilonová, Distributional chaos for triangular maps, Ann. Math. Sil. 13 (1999) 33–38. [2] A. Blokh, On dynamics on one-dimensional branched manifolds 1, Theory of Functions, Funct. Anal. Appl. 46 (1986) 8–18 (in Russian); translation in J. Sov. Math. 48 (1990) 500–508. [3] A. Blokh, On dynamics on one-dimensional branched manifolds 3, Theory of Functions, Funct. Anal. Appl. 48 (1987) 32–46 (in Russian); translation in J. Sov. Math. 49 (1990) 875–883. [4] R. Hric, M. Málek, Omega limit sets and distributional chaos on graphs, Topol. Appl. 153 (2006) 2469–2475. [5] M. Málek, Distributional chaos and spectral decomposition on the circle, Topol. Appl. 135 (2004) 215–229. [6] M. Málek, Distributional chaos for continuous mappings of the circle, Ann. Math. Sil. 13 (1999) 205–210. [7] M. Málek, Omega-limit sets and invariant chaos in dimension one, J. Differ. Equ. Appl. 22 (3) (2016) 468–473. [8] A.N. Sharkovsky, The behavior of a map in neighborhood of an attracting set, Ukr. Mat. Zh. 18 (1966) 60–83 (in Russian). [9] A.N. Sharkovsky, Continuous mappings on the set of ω-limit sets of iterated sequences, Ukr. Mat. Zh. 18 (1966) 127–130 (in Russian). [10] B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Am. Math. Soc. 344 (1994) 737–754.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On the spectrum of dynamical systems on trees Jan Tesarčík Mathematical Institute in Opava, Silesian University in Opava, Na Rybníčku 1, 74601, Opava, Czech Republic
a r t i c l e
i n f o
Article history: Received 13 April 2016 Received in revised form 10 March 2017 Accepted 12 March 2017 Available online xxxx MSC: 37B40 37E25 Keywords: Dynamical system Tree Distributional functions Spectrum Weak spectrum Basic omega-limit sets
a b s t r a c t In their paper, Schweizer and Smítal (1994) [10] introduced the notions of distributional chaos for continuous maps of the interval, spectrum and weak spectrum of a dynamical system. Among other things , they have proved that in the case of continuous interval maps, both the spectrum and the weak spectrum are finite and generated by points from the basic sets. Here we generalize the mentioned results for the case of continuous maps of a finite tree. While the results are similar, the original argument is not applicable directly and needs essential modifications. In particular, it was necessary to resolve the problem of intersection of basic sets, which was a crucial point. An example of one-dimensional dynamical system with an infinite spectrum is presented. © 2017 Elsevier B.V. All rights reserved.
1. Introduction and the main results Let X be a compact metric space and d be its metric. We consider a continuous map f : X → X (we denote the set of such maps by C(X, X)), in this setting we talk about a dynamical system. Let N be the set of positive integers, taking an x ∈ X, we define recursively an nth iteration by putting f 0 (x) = x and f n (x) = f (f n−1 (x)), for any n ∈ N. The sequence (f n (x))∞ n=0 is called a trajectory of point x and denoted by trf (x). We define an orbit of a point x ∈ X by putting Orbf (x) = {f n (x); n ∈ N ∪ {0}}, similarly, we define an orbit of a set A ⊂ X as Orbf (A) = {f n (x); x ∈ A and n ∈ N ∪ {0}}. If we have f p (x) = x for some x ∈ X and some positive integer p, then we call the point x periodic and the smallest p of such a kind is called the period of x. We say that f is transitive, if for every two nonempty open sets U and V in X there is an integer n, such that f n (U ) ∩ V = ∅. Let K be a nonempty subset of X, we say that f |K is invariant if f (K) ⊂ K. E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.topol.2017.03.007 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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In addition, if f (K) = K holds, then we say that f |K is strongly invariant. Let A be a subset of X, then diam A = sup {d(x, y); x, y ∈ A}. Let f : X → Y be a map between two metric spaces, then we say that f is monotone on X if for any y ∈ Y the set f −1 (y) is connected. Let f ∈ C(X, X) and g ∈ C(Y, Y ) be maps of compact metric spaces, a continuous surjection ϕ : X → Y is called semiconjugacy if ϕ ◦ f = g ◦ ϕ. Moreover, if there exists a k ∈ N such that #ϕ−1 (y) ≤ k for each y ∈ Y , then we say that ϕ semiconjugates f and g almost exactly. For a given x ∈ X we define the omega-limit set of x as a set of accumulation points of trajectory trf (x) and we denote it as ωf (x). We say that ωf (x) is maximal if and only if there is no ωf (y), such that ωf (x) ⊂ ωf (y), ωf (x) = ωf (y). In their paper [10], Schweizer and Smítal have defined the notion of distributional chaos. Several authors have continued exploring this notion in various settings. The notion is based on distributional functions of pairs of points. For any x, y ∈ X, a positive integer n and real t put ξ(x, y, t, n) = #{i; 0 ≤ i ≤ n − 1 and δx,y (i) < t}
(1)
where δx,y (i) = d f i (x) , f i (y) is the distance of ith iteration of the points x and y. Put ∗ Fx,y (t) = lim sup
1 ξ(x, y, t, n), n
(2)
Fx,y (t) = lim inf
1 ξ(x, y, t, n). n
(3)
n→∞
n→∞
∗ Function Fx,y is called an upper distributional function and Fx,y is called a lower distributional function. ∗ ∗ ∗ Both Fx,y and Fx,y are nondecreasing functions with Fx,y (t) = Fx,y (t) = 0 for every t < 0 and Fx,y (t) = Fx,y (t) = 1 for every t > diam X. An arc is a topological space homeomorphic to the compact interval [0, 1]. A graph G is a continuum (a nonempty compact connected metric space) which can be written as the union of finitely many arcs any two of which can intersect only in their endpoints. If a graph does not contain a set homeomorphic to the circle, it is called a tree. A subgraph is a subset of a graph which itself is a graph. By a periodic subgraph we mean a subgraph H ⊂ G, such that there is an n ≥ 1 for which H, f (H), . . . , f n−1 (H) have pairwise disjoint interiors and f n (H) = H; in this case, we also speak about a periodic orbit of subgraphs. A nonempty intersection of three or more arcs, with pairwise disjoint interiors, is called a vertex, where the maximal number of such arcs is called the degree of the vertex. The maximal order of a graph is the maximum number of degrees of vertices on the graph. We say that two points u, v ∈ G are synchronous if both ωf (u) and ωf (v) are contained in the same maximal omega-limit set ω, and if, for any periodic subgraph H such that Orbf (H) ⊃ ω, there is a j ≥ 0 such that f j (u), f j (v) ∈ H. The spectrum of f , denoted by Σ(f ), is a set of minimal elements of the set D(f ) = {Fu,v ; u and v are synchronous}. And the weak spectrum Σw (f ) of f is a set of minimal elements of the set Dw (f ) = {Fu,v ; lim inf i→∞ δu,v (i) = 0}. The relation between Σ(f ) and Σw (f ) for the case of continuous interval self maps is discussed in Remark 2.6 [10], and the authors have shown that in this case there holds Σ(f ) ⊂ Dw (f ), but we can not say anything about Σ(f ) ∩ Σw (f ). In the case of continuous tree self maps, like in the case of continuous interval self maps, there is no direct relation between Σ(f ) and Σw (f ), but the arguments from [10] can be applied to this case and, therefore, it is true that Σ(f ) ⊂ Dw (f ). The following theorem is our main result. It generalizes the result in [10] for the interval maps and in [5] for the circle maps to the case of tree maps.
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Theorem 1. Let f be a continuous self map of a tree. (1) If the topological entropy of f is zero, then Σ(f ) = Σw (f ) = {χ(0,∞) }. (2) If the topological entropy of f is positive, then both the spectrum Σ(f ) and the weak spectrum Σw (f ) are finite and nonempty. Specifically, Σ(f ) = {F1 , . . . , Fm } for some m ≥ 1, and Σw (f ) \ Σ(f ) = {Fm+1 , . . . , Fn } for some n ≥ m. Furthermore, for each i there is an εi > 0, such that Fi (εi ) = 0. Even though the main results are formulated for the case of continuous tree maps, the results needed for their proof (mainly the dynamics of basic sets) can be proven in more general space. Therefore, we prove them for the case of graph maps. 2. The basic sets The omega-limit sets play an important role in the theory of discrete dynamical systems. They give us useful information about the behavior of the system. Special attention must be paid to the basic sets. As one can see below, the basic sets possess interesting properties and even their presence ensures attractive phenomena. In the sixties, A.N. Sharkovsky systematically studied the properties of the omega-limit sets of the continuous maps of the interval (e.g. [8] and [9]). He showed that any such set is contained in a maximal omega-limit set and introduced three types of maximal omega-limit sets. More precisely the cycle, if it is finite; the infinite of the first type (lately known as a solenoid), and the infinite of the second type (lately known as a basic set). In the eighties, A. Blokh widely extended the results on the properties of the basic sets but using a different definition (see [2] and [3]). In the paper [4], Sharkovsky’s classification of maximal omega-limit sets was extended to the case of graph maps. It was necessary to add a fourth type of the maximal omega-limit sets to cover the graph specific case of the omega-limit sets (singular sets). Let G be a graph and f ∈ C(G, G). For an omega-limit set ω ⊂ G we define Pω = Orbf U , (4) where the intersection is taken over all neighborhood U intersecting ω. If ω is finite, then it is a cycle. Now consider ω to be infinite. If Pω is a nowhere dense set, then we call ω a solenoid (this terminology is used in [6]). If Pω consists of finitely many connected components and ω contains a periodic point then ω is called a basic set. Let Pω = H1 ∪ · · · ∪ Hn , where H1 , · · · , Hn are minimal periodic subgraphs with disjoint interiors, then ω ∩ Hi is called a portion of the basic set ω. Note that if we consider f n , then ω ∩ Hi is a basic set for f n and Pω ∩ Hi does not admit a proper periodic subgraph, i.e., ω ∩ Hi is indecomposable. If Pω consists of finitely many connected components and ω contains no periodic point, then we call ω a singular set. Note that the case of a singular set is only possible on a finite union of subgraphs that are homeomorphic with a circle. In the paper [7] it was shown that Sharkovsky’s definition of the basic sets, Blokh’s definition of the basic sets and the definition of the basic sets from [4] are mutually equivalent. Example 2. Consider the following interval map fI : [0, 1] → [0, 1] ⎧ 3x for x ∈ [0, 13 ); ⎪ ⎨ 1 for x ∈ [ 13 , 23 ]; fI (x) = ⎪ ⎩ 3 − 3x for x ∈ ( 23 , 1]. Note that the Cantor middle third set C is a basic set for fI and that PC = [0, 1].
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The proofs of many results of properties of the basic sets in the interval case use the convex hull of the basic sets. This method cannot be used in the case of graph maps because the convex hull may not be defined (consider a cantor-like basic set on the circle). Therefore, we used a workaround, considering Pω instead of a convex hull. Contrary to the case of an interval, where the convex hulls of two basic sets cannot overlap (see [10, Theorem 3.7]), in the case of graph maps, the situation is more complicated when we consider the intersection of Pω of two basic sets as shown in the following example. Example 3. Let G = S 16 ( 16 , 0) ∪ ([ 13 , 23 ] × {0}) ∪ S 16 ( 56 , 0), where Sr (x, y) is the circle with the center at (x, y) and radius r.
Graph G. Defining f : G → G, for any (x, y) ∈ G with y ≥ 0 we put f (x, y) = (u, v), where u = fI (x), fI (x) is defined as in Example 2, and v ≥ 0, so that (u, v) ∈ G; similarly, for (x, y) ∈ G with y ≤ 0 we put f (x, y) = (u, v), where u = fI (x) and v ≤ 0, so that (u, v) ∈ G. It is easy to see that if a point (u1 , v1 ) ∈ G where v1 > 0 generates the basic set ω1 and some other point (u2 , v2 ) ∈ G where v2 < 0 generates the basic set ω2 , then pr1 (ω1 ) = pr1 (ω2 ) = C and Pω1 ∩ Pω2 = {(0, 0), (1, 0)} ∪ ([ 13 , 23 ] × {0}). The problem of intersection of Pω for the basic sets on graphs is resolved by Lemma 7. In the following theorem we summarize the known properties of the basic set, which will be used in the rest of the paper. Theorem 4 ([4]). Let f ∈ C(G, G) and ω ˜ be its basic set, then (i) (ii) (iii) (iv)
ω ˜ is a perfect set; The system of all basic sets is countable; ˜; The set of periodic points is dense in ω Let Pω˜ = H ∪ f (H) ∪ · · · ∪ f n−1 (H) and K, J be subgraphs, such that J ∩ ω ˜ is infinite and K ⊂ int(H). i+jn Then f (J) ⊃ K for some i and sufficiently large j; (v) The topological entropy of f is positive (h (f ) > 0). We describe the situation from (iv) by saying that f |ω˜ is strongly transitive in H.
3. The zero entropy case By Theorem 4, if f has a zero topological entropy, then it has no basic sets. The following lemma shows that a nontrivial distributional function can be generated by points with the omega-limit sets contained in the basic sets only. Lemma 5. Let f be a continuous map of a tree T , such that f has no basic set. Then Σ(f ) = Σw (f ) = {χ(0,∞) }. Proof. Let u, v ∈ T . If both ωf (u) and ωf (v) are cycles, then Fu,v = χ(0,∞) in both cases when u and v are synchronous, or when lim inf i→∞ δu,v (i) = 0.
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If one of the sets ωf (u) and ωf (v) is a cycle and the other one is contained in a solenoid, then in both cases when u and v are synchronous, or when lim inf i→∞ δu,v (i) = 0 , we get that the solenoid contains a periodic point, which is impossible. Suppose that ωf (u) ⊂ ωu and ωf (v) ⊂ ωv , where ωu and ωv are solenoids. Since Pωu is nowhere dense for a given ε > 0, there is a periodic orbit H, f (H), . . . , f n−1 (H) of subgraphs covering ωu each of f i (H) with a diameter less than ε. Similarly, we get a periodic orbit K, f (K), . . . , f m−1 (K) of subgraphs for ωv . Now if u and v are synchronous, then there is an integer j, such that f j+i(u), f j+i (v) ∈ f i mod n (H) for all i. Since ε was chosen arbitrarily, we get Fu,v = χ(0,∞) . On the other hand, if lim inf i→∞ δu,v (i) = 0, then we get n = m and f i (H) ∩ f i (K) = ∅, for all i = 0, 1, . . . , n − 1. Since diam(f i (H) ∪ f i (K)) ≤ 2ε, we similarly get Fu,v = χ(0,∞) . 2 4. The proof of the main results The following itinerary lemma comes from [10]; here, we generalize it for the graph case. ∞
∞
Lemma 6. Let f ∈ C (G, G) and u, v be in G. Let (Ui )i=1 and (Vi )i=1 be closed subgraphs with the property ∞ ∞ that the sequence (Ui )i=1 is eventually in every neighborhood of u and the sequence (Vi )i=1 is eventually in every neighborhood of v, and such that for any i and j there are positive integers u (i, j) and v (i, j) such that f u(i,j) (Ui ) ⊃ Vj and f v(i,j) (Vi ) ⊃ Uj . Then {u, v} ⊂ ωf (y), for some y in G. ∞
∞
Proof. Define a decreasing sequence (Ji )i=1 of compact sets and an increasing sequence (n (i))i=1 of positive integers as follows: J1 = U1 and n (1) = u (1, 2). Then f n(1) (J1 ) ⊃ V2 , choose J2 ⊂ J1 , such that f n(1) (J2 ) = V2 . Take n (2) = n (1) + v (2, 3). Then f n(2) (J2 ) = f v(2,3) (V2 ) ⊃ U3 and there is a J3 ⊂ J2 , ∞ such that f n(2) (J3 ) = U3 . Then there are J4 and n (3), such that f n(3) (J4 ) = V4 , etc. Let y ∈ i=1 Ji . Since the trajectory of y visits every neighborhood of u and every neighborhood of v, the result follows. 2 The next lemma resolves the problem of how a Pω of two basic sets can intersect. It substitutes the non-overlap rule for convex hulls of the basic sets in the interval case. Lemma 7. Let f ∈ C (G, G), ω1 , ω2 be two different basic sets, Pω1 and Pω2 be defined by (4). Then at least one Pω1 ∩ Pω2 ∩ ω1 , Pω1 ∩ Pω2 ∩ ω2 must be finite. Proof. We assume that both Pω1 ∩ Pω2 ∩ ω1 and Pω1 ∩ Pω2 ∩ ω2 are infinite, then take an arc J ⊂ int (Pω1 ), such that J ∩ω2 is infinite, and an arc K ⊂ int (Pω2 ), such that K ∩ω1 is infinite. This is possible because, by Theorem 4, ω1 , ω2 are perfect. Now by Theorem 4 f i (J) ⊃ K, for any sufficiently large i, and f j (K) ⊃ J, for any sufficiently large j. Using this together with Lemma 6, we get that both ω1 , ω2 are contained in an ω-limit set ωf (z) for some z ∈ G, which is a contradiction with the maximality of ω1 , ω2 . 2 We formulate the following lemma to substitute the wrong Sharkovsky result, claiming that the trajectory of each point with omega-limit sets contained in the basic sets eventually enters a basic set. Lemma 8. Let f ∈ C (G, G), ω ˜ be its basic set and x ∈ G be such that ωf (x) ⊂ ω ˜ . Then there is an x ˜∈ω ˜ such that limn→∞ δx,˜x (n) = 0. Proof. Let x be such that ωf (x) ⊂ ω ˜ . If for some n ∈ N f n (x) ∈ ω ˜ ; then, since ω ˜ is strongly invariant, we ∞
n n can find x ˜∈ω ˜ , such that f (˜ x) = f (x). Otherwise, suppose that Orbf (x) ⊂ i An(i) where An(i) i=0 is the sequence of open subgraphs contiguous to ω ˜ , such that f i (x) ∈ An(i) , for any nonnegative integer i. Since f |ω˜ is almost exactly semiconjugated to a transitive map of a graph (see Theorem 2 in [2] for details) we obtain that the set A¯n(i) , i = 1, 2, . . . is invariant under f . We distinguish two cases.
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∞ (1) If the sequence An(i) i=0 is eventually periodic, then, since ωf (x) ⊂ ω ˜ , the set ωf (x) must be finite i.e. a cycle and the result easily follows. ∞ ∞ (2) The sequence An(i) i=0 is wandering. In this case, since An(i) i=0 are nonoverlapping, we have limi→∞ diam An(i) = 0 and x ˜ is an endpoint of An(0) belonging to ω ˜. 2 The following five lemmas and their proofs are modifications of the ones from [10], we use them in proof of our main theorem. Lemma 9. Let f ∈ C (G, G), then for any λ ∈ (0, 1) and t ∈ (0, diam G) there is an integer n (t, λ) with the following property: If ωi are portions of a basic set ω with a period m, where m ≥ n (t, λ) and Pωi (defined by (4) with f replaced by f m ) are nonoverlapping for i = 0, 1, . . . , m − 1, then for all u, v ∈ ω Fuv (t) > λ.
p Proof. Let G = i=1 Gi where Gi are arcs with pairwise disjoint interiors, let d be the maximal degree of
p the point of G. Put c = i=1 diam Gi . Fix t and λ, let n (t, λ) be such that (n (t, λ) − c · d/t)/n (t, λ) > λ. Since there are at most c · d/t distinct sets f s (ω) with a diam f s (ω) ≥ t (this follows from Lemma 7), we have Fu,v (t) ≥ (1/m) · # {s < m; diam f s (ω) < t} ≥ (m − c · d/t)/m > λ. 2 ∗ be continuous at t. Then, for any ε > 0, there are Lemma 10. Let f ∈ C(G, G) and let both Fx,y and Fx,y positive integers k, q, arbitrarily large, and δ > 0, such that
1 ξ(u, v, t, k) < Fx,y (t) + ε k and 1 ∗ ξ(u, v, t, q) > Fx,y (t) − ε q whenever d(u, x) < δ and d(v, y) < δ. ∗ ∗ Proof. Choose an ε1 > 0 such that Fx,y (t + 2ε1 ) < Fx,y (t) + ε/2 and Fx,y (t − 2ε1 ) > Fx,y (t) − ε/2. Then 1 choose a k such that k ξ(x, y, k, t + 2ε1 ) < Fx,y (t + 2ε1 ) + ε/2. The first inequality follows from the fact that ξ(u, v, k, t) < ξ(x, y, k, t + 2ε1 ) whenever δ > 0 is sufficiently small. The argument for the second inequality is similar. 2
Lemma 11. Let f ∈ C (G, G), ω1 , ω2 be its basic sets, Pω1 = H1 ∪ · · · ∪ Hn , Pω2 = K1 ∪ · · · ∪ Km be minimal periodic decompositions and Q ⊂ G × G be a countable set of pairs (u, v), such that u ∈ ω1 ∩ H1 and v ∈ ω 2 ∩ K1 . Then there are points x ∈ ω1 ∩ H1 and y ∈ ω2 ∩ K1 , such that for any t > 0: Fx,y (t) ≤ inf {Fu,v (t) ; (u, v) ∈ Q}
(5)
∗ ∗ Fx,y (t) ≥ sup Fu,v (t) ; (u, v) ∈ Q .
(6)
and
Proof. Let R be a countable set, dense in [0, diam G], and such that for any (u, v) ∈ Q and any t ∈ R, both ∞ ∞ ∗ Fu,v (t) and Fu,v (t) are continuous at t. Let (tj )j=1 and (uj , vj )j=1 be sequences of points where tj ∈ R and (uj , vj ) ∈ Q, such that for any t ∈ R and any (u, v) ∈ Q, t = tj , u = uj and v = vj for infinitely many j.
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Next, using induction, we define the positive integers k (1) < q (1) < k (2) < q (2) < ... < k (i) < q (i) < ... ∞
∞
and decreasing sequences (Ui )i=1 and (Vi )i=1 of closed subgraphs with lim diam (Ui ) = lim diam (Vi ) = 0
i→∞
i→∞
and for any u ∈ Un , v ∈ Vn and any j ≤ n, 1 1 ξ (u, v, tj , k (j)) ≤ Fu(j),v(j) (tj ) + k (j) j
(7)
1 1 ∗ ξ (u, v, tj , q (j)) ≥ Fu(j),v(j) (tj ) − . q (j) j
(8)
and
We take U1 = H1 and V1 = K1 , k (1) = 1, q (1) = 2 and assume that Un , Vn , k (n) and q(n) have been defined such that f k(j) (Un ) ∩ ω1 and f q(j) (Vn ) ∩ ω2 are infinite whenever j is sufficiently large. By (iv) of Theorem 4, f |ω1 is strongly transitive on H1 , f |ω2 is strongly transitive on K1 and u ∈ ω1 ∩ int(H1 ), v ∈ ω2 ∩ int(K1 ) if (u, v) ∈ Q. So, there is some s > q (n), such that u (n + 1) ∈ f s (Un ) and v (n + 1) ∈ f s (Vn ). Let a ∈ Un and b ∈ Vn be such that f s (a) = u (n + 1) and f s (b) = v (n + 1). Then ∗ ∗ Fa,b = Fu(n+1),v(n+1) and Fa,b = Fu(n+1),v(n+1) . The existence of Un+1 ⊂ Un , Vn+1 ⊂ Vn , k (n + 1) and q (n + 1) follows by Lemma 10. (We take Un+1 and Vn+1 as compact neighborhoods of a and b, with diam (Un ) > 2 diam (Un+1 ) and diam (Vn ) > 2 diam (Vn+1 ) .) Note that (by (i) of Theorem 4) a, b, Un+1 , Vn+1 can be chosen such that both f s (Un+1 ) ∩ ω1 and f s (Vn+1 ) ∩ ω2 are infinite. ∞ ∞ Take x ∈ j=1 Uj and y ∈ j=1 Vj . For any t ∈ R and any (u, v) ∈ Q, take j such that t = tj , u = u (j) and v = v (j). Since x ∈ Uj and y ∈ Vj , we will apply (7) with u = x and v = y. Since j can be arbitrarily large, we have Fx,y (t) ≤ Fu,v (t). This remains true for any t, since R is dense in [0, diam G], so this implies (5). We get Equation (6) similarly. We have x ∈ H1 and y ∈ K1 . We need to show that x can be chosen in ω1 and y in ω2 . Let ∞ w ∈ ω1 ∩ H1 be such that ωf (w) = ω1 and let (Wi )i=1 be a decreasing sequence of compact neighborhoods of w with limi→∞ Wi = w. Now apply Lemma 6 with u replaced by x, v by w and Vi by Wi (Vi from Lemma 6). Since ω1 is maximal, we obtain ωf (x) ⊂ ω1 . Similarly, we have ωf (y) ⊂ ω2 . Now, by Lemma 8, there are x ˜ ∈ ω1 and y˜ ∈ ω2 , such that limi→∞ δx,˜x (i) = 0 and limi→∞ δy,˜y (i) = 0. Replace x by x ˜ and y by y˜. Clearly (5) and (6) remain valid. 2 ∞
Lemma 12. Let f ∈ C (G, G) and (ωi )i=1 be a sequence of a distinct indecomposable portion of the basic omega-limit sets. Then limi→∞ diam ωi = 0.
n Proof. Let G = k=1 Gk be the decomposition of G into arcs, such that the interior of each Gk contains no vertex and any two of them can intersect only in their endpoints. By Lemma 7 and the fact that G has finitely many vertexes, there are only finitely many ωi that intersect more the one Gk . Therefore, we can assume that every ωi is contained in only one arc. It is sufficient to verify the assertion for the case when ∞ all ωi are contained in one Gk . Suppose, on the contrary, that there is a subsequence (ωni )i=1 , such that diam ωni > ε > 0 for any i. Then for some ni and nj both Pωni ∩ Pωnj ∩ ωni and Pωni ∩ Pωnj ∩ ωnj must be infinite. Which is in contradiction with the Lemma 7. 2
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∞
Lemma 13. Let f ∈ C (G, G) and let (ωi )i=1 be the minimal periodic portions of basic sets of f . For any i, j, set Gi,j = inf {Fu,v ; u ∈ ωi , v ∈ ωj }. Then: (i) Each Gi,j is zero on interval [0, ε (i, j)] where ε (i, j) is a positive number. (ii) The set {Gi,j ; ωi ∩ ωj = ∅} has a finite number of minimal elements. ∞ (iii) The set {Gi,i }i=1 has a finite number of minimal elements. Proof. (i) By (iii) of Theorem 4, there are distinct periodic points p ∈ ωi and q ∈ ωj . Since mins δp,q (s) = ε > 0, we have Fp,q (t) = 0, and, subsequently, Gi,j (t) = 0, for t ≤ ε, since Gi,j (t) ≤ Fp,q (t). Take ε (i, j) = ε. (ii) We may assume that ωi = ωj for any i = j. Let Pωi be the cover of ωi . Let ε = ε (1, 1). We say that an ωi is extremal, if diam ωi > ε/2 and if for any ωj there exists no arc A ⊂ Pωj with endpoints in ωj , such that ωi ⊂ A. By Lemma 7, there are finitely many extremal ωi ’s. We may assume that ω1 , . . . , ωn(1) are all extremal sets of f ; note that n (1) ≥ 1, since f has a positive topological entropy and therefore at least one maximal omega-limit set is a basic set (cf. e.g. Theorem 3 in [4]). Let m > 0 be an integer. We say that a set ωi is significant if diam ωi > ε/2 and the period of ωi is less than m. By Lemma 12, there are finitely many significant ωi ’s. Thus we may assume without the loss of generality, that there are integers n (3) ≥ n (2) ≥ n (1) > 0, such that ω1 , . . . , ωn(2) is the minimal system invariant under f , containing all extremal and all significant sets, and that ωn(2)+1 , . . . , ωn(3) is the system of all portions that have a point in common with some of ωi , i ≤ n (2). We show that the minimal elements of {Gi,j ; ωi ∩ ωj = ∅} are in the set M = {Gi,j ; i, j ≤ n (3)} whenever m is sufficiently large, depending on n (1). Assume that i > n (3) and ωi ∩ ωj = ∅. Then ωj cannot be significant, consequently j > n (2). Let u ∈ ωi and v ∈ ωj . It suffices to show that Fu,v ≥ G for some G in M . If the diam f s (ωi ∪ ωj ) ≤ ε, for any s, then Fu,v (t) = 1 for t > ε, and, therefore, Fu,v ≥ G1,1 . So it suffices to consider the case when the diam f s (ωi ∪ ωj ) > ε for some s. Since the set ω1 , . . . , ωn(2) is invariant, we may assume without any loss of the generality that diam (ωi ∪ ωj ) > ε and that diam f s (ωi ∪ ωj ) ≤ diam (ωi ∪ ωj ) for any s.
(9)
Since one of the sets ωi , ωj , say ωi , has a diameter > ε/2, there is an extremal ωr (ωi cannot be extremal since i > n (1)) and an arc Au,v ⊂ Pωr with endpoints u, v ∈ ωr , such that ωi ⊂ Au,v ; similarly, there is an arc Aw,z ⊂ Pωr with endpoints w, z ∈ ωr , such that ωj ⊂ Aw,z , since ωi ∩ ωj = ∅. By Theorem 4, there are periodic points a, b in ωr with the property: If H is an subgraph such that H ∩ ωr is finite, the diam H > ε and H ⊂ Pωr ,
(10)
then H ⊂ Sa,b where Sa,b is a subgraph containing points a, b ∈ ωr , Sa,b ⊂ Pωr , which satisfies diam Sa,b ≥ max{d (α, β) ; α, β ∈ {u, v, w, z}}. Now, by Lemma 7, one of the sets Pωr ∩ Pωi ∩ ωr , Pωr ∩ Pωi ∩ ωi must be finite. If Pωr ∩ Pωi ∩ ωi is a finite set, then Pωi ∩ ωi is finite, since Pωi ⊂ Pωr and then ωi is finite, since ωi ⊂ Pωi , which is impossible. Thus Pωr ∩ Pωi ∩ ωr is finite, since Pωi ⊂ Pωr , we have Pωi ∩ ωr is finite. Similarly, Pωj ∩ ωr is finite. Since H = Pωi ∪ Pωj satisfies (10), we obtain ωi ∪ ωj ⊂ Sa,b . Take t0 such that diam (ωi ∪ ωj ) < t0 < diam Sa,b . Let Fa,b (t0 ) = λr . Clearly λr < 1. Let ε (r, r) be as in (i). We have Gr,r (t) = 0 ≤ Fu,v (t) for t ≤ ε (r, r), and by (9) Gr,r (t) ≤ 1 = Fu,v (t) for t > t0 . And if ε (r, r) < t0 , then, by Lemma 9, Gr,r (t) ≤ Fa,b (t) ≤ λr < Fu,v (t) whenever ε (r, r) < t ≤ t0 and m ≥ n (ε (r, r) , λr ). Thus Gr,r ≤ Fu,v if m ≥ n (ε (r, r) , λr ). It follows that when the parameter m = max {n (ε (r, r) , λr ) ; 1 ≤ r ≤ n (1)}, then {Gi,j ; i, j ≤ n (3)} contains the minimal elements of {Gi,j ; ωi ∩ ωj = ∅}. ∞ (iii) It suffices to show that the minimal elements of {Gi,i }i=1 are contained in {Gi,i ; i ≤ n (2)}. But this follows from the above argument. 2
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Proof of Theorem 1. (1) In the case of a tree map with a zero entropy all maximal omega-limit sets are either periodic orbits or solenoids. Applying Lemma 5, we get Σ (f ) = Σw (f ) = {χ(0,∞) }. ∞ (2) By Theorem 4, the system of all basic sets (ωi )i=1 is countable and since the entropy is positive, it is not empty. Spectrum. Let u, v ∈ T be such that u and v are synchronous, ωf (u) , ωf (v) ⊂ ω ˜, ω ˜ be a maximal omega-limit set. If ω ˜ is a periodic set or solenoid, then by Lemma 5 Fu,v = χ(0,∞) . Therefore, we can assume that the omega-limit sets of all synchronous points are contained in the basic sets. Put D = {Fu,v ; u, v are synchronous} and E = {Fu,v ; u, v ∈ ωi , for some i}. Clearly E ⊂ D. Now, if u and v are synchronous, then ωf (u) and ωf (v) are contained in the same maximal basic set ω ˜ . By Lemma 8, there are u ˜, v˜ in ω ˜ such that limn→∞ δu,˜u (n) = 0 and limn→∞ δv,˜v (n) = 0. The points u ˜, v˜ are contained in ω ˜ and hence in some ωi . Thus E and D must have the same system of Σ (f ) of minimal elements and Lemma 13 gives the result. Weak spectrum. Let u, v ∈ T , denote ωu as a maximal omega-limit set containing ωf (u) (similarly ωv ). Suppose lim inf δu,v (i) = 0, i→∞
(11)
which gives us ωu ∩ ωv = ∅. If both ωu , ωv are cycles, then by (11) u, v are synchronous and Lemma 5 gives us the result. The case when one of ωu , ωv is a cycle and the other is a solenoid is impossible. If one of ωu , ωv , say ωu , is a solenoid, then (2) of Theorem 1 in [2] gives ωv ⊂ ωu . Since the basic set contains a periodic point, ωv must be a solenoid, moreover, ωu = ωv and Lemma 5 gives us the result. Therefore, all non-trivial distributions generated by the points satisfying (11) have their omega-limit sets contained in the basic sets. Let Dw = {Fu,v ; u, v such that lim inf n→∞ δu,v (n) = 0} and Ew = {Fu,v ; u ∈ ωi , v ∈ ωj , ωi ∩ ωj = ∅}, and let Fu,v ∈ Dw . Then there are i, j, k such that f k (u) ∈ ωi and f k (v) ∈ ωj and (11) implies ωi ∩ ωj = ∅. Thus Fu,v ∈ Ew and Dw ⊂ Ew . Conversely, let Fu,v ∈ Ew . Let u ∈ ωi , v ∈ ωj and w ∈ ωi ∩ ωj . Take ∗ Q = {(u, v) , (w, w)} and apply Lemma 11 to get x, y such that Fx,y ≤ Fu,v and Fx,y = χ(0,∞) . Therefore lim inf n→∞ δu,v (n) = 0, which implies Fx,y ∈ Dw . Thus we have proved that Dw ⊂ Ew and that Ew has the lower bounds in Dw . Consequently, Ew and Dw must have the same system Σw (f ) of the minimal elements and Lemma 13 gives the result. 2 5. An example with the infinite spectrum In this section we show that the results on the finite spectrum are not true any more when we consider more general but still one dimensional spaces. Namely, we present an example of a dendroid (an arcwise connected, hereditarily unicoherent continuum) and a continuous map on it, such that there is an infinite family of an incomparable distributional function generated by synchronous points (i.e., it has the infinite spectrum). A triangular map of the square with the infinite spectrum was constructed in [1]. Here we show that this is possible even in the spaces with the dimension less than two. Before we give the mentioned example, we turn our attention to a certain class of continuous maps of compact intervals. Call this class tent-like maps. For a given b ∈ [ 34 , 1] and l ∈ [0, 11 32 ], we define a function fb,l : [0, b] → [0, b] (see Fig. 1) ⎧ ⎪ ⎪ ⎨
2b x, 2l + b fb,l (x) = ⎪ 2b 2b2 ⎪ ⎩ x+ , 2l − b b − 2l
for x ∈ [0, (b + 2l)/2]; for x ∈ ((b + 2l)/2, b].
It is easy to compute that xb,l = 2b2 /(3b − 2l) is a fixed point for fb,l .
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Fig. 1. A tent-like map fb,l .
Fig. 2. Space X from Example 14.
In [10] it has been shown that for a function from such class there is a unique minimal distribution (a stepfunction) where the positions of jumps are given by the positions of the periodic points (see Remark 2.5 in [10] for details). In particular, if Fb,l is the minimal distribution for fb,l , then Fb,l (x) = 0 on (−∞, xb,l ],
and Fb,l (x) = 1 on (b, ∞].
(12)
Example 14. There is a space (a dendroid) X ⊂ R2 and a continuous map f of X such that its spectrum Σ(f ) is not finite. Proof. First, we define the sequences bn = (3 · 2n + 1)/2n+2 and ln = (11 · 2n − 11)/2n+5 , for n = 0, 1, 2, . . . . Space. First, put In = {1/2n } × [0, bn ], for each n = 0, 1, 2, . . . and I∞ = {0} × [0, 34 ]. Let X = ([0, 1] × {0}) ∪ I∞ ∪
∞
In .
n=0
See Fig. 2. Map. We define a continuous map f : X → X as follows. The map f restricted to ([0, 1] × {0}) is equal to the identity. On each In we put f |In : In → In f |In (x, y) = (x, fbn ,ln (y)). The continuity gives us f |I∞ (x, y) = (x, f 34 , 11 (y)). 32
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Spectrum Σ(f ) of f . The map f |In is a tent-like map, which we described above. In particular, In is a basic set for f and the minimal distribution on In satisfies (12) for b = bn and l = ln . It is easy to verify that any two different Fbn ,ln are incomparable. Therefore, the spectrum Σ (f ) must be infinite. Weak spectrum Σw (f ) of f . Since the map f restricted to ([0, 1] × {0}) is equal to identity, then we can look at f |In as continuous interval self maps. Therefore, we can use the Remark 2.6 from [10], which says that Σ(f |In ) ⊂ Dw (f |In ) and from which it results that the Σw (f ) must be infinite. 2 Acknowledgements I sincerely thank my supervisor, Michal Málek, for his valuable guidance. I am grateful for his constant support and help. I would also like to thank the reviewer for his precious time and important comments. The work was supported, in part, by the grants SGS/1/2014 and SGS/16/2016 from the Silesian University in Opava, RVO funding for [IČ47813059]. This support is gratefully acknowledged. References [1] M. Babilonová, Distributional chaos for triangular maps, Ann. Math. Sil. 13 (1999) 33–38. [2] A. Blokh, On dynamics on one-dimensional branched manifolds 1, Theory of Functions, Funct. Anal. Appl. 46 (1986) 8–18 (in Russian); translation in J. Sov. Math. 48 (1990) 500–508. [3] A. Blokh, On dynamics on one-dimensional branched manifolds 3, Theory of Functions, Funct. Anal. Appl. 48 (1987) 32–46 (in Russian); translation in J. Sov. Math. 49 (1990) 875–883. [4] R. Hric, M. Málek, Omega limit sets and distributional chaos on graphs, Topol. Appl. 153 (2006) 2469–2475. [5] M. Málek, Distributional chaos and spectral decomposition on the circle, Topol. Appl. 135 (2004) 215–229. [6] M. Málek, Distributional chaos for continuous mappings of the circle, Ann. Math. Sil. 13 (1999) 205–210. [7] M. Málek, Omega-limit sets and invariant chaos in dimension one, J. Differ. Equ. Appl. 22 (3) (2016) 468–473. [8] A.N. Sharkovsky, The behavior of a map in neighborhood of an attracting set, Ukr. Mat. Zh. 18 (1966) 60–83 (in Russian). [9] A.N. Sharkovsky, Continuous mappings on the set of ω-limit sets of iterated sequences, Ukr. Mat. Zh. 18 (1966) 127–130 (in Russian). [10] B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Am. Math. Soc. 344 (1994) 737–754.