Hypercyclic and topologically transitive semigroups of composition operators

JID:TOPOL

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.1 (1-8) Topology and its Applications ••• (••••) •••–•••

Contents lists available at ScienceDirect

Topology and its Applications www.elsevier.com/locate/topol

Hypercyclic and topologically transitive semigroups of composition operators Mohammad Javaheri ∗ 515 Loudon Road, Siena College, School of Science, Loudonville, NY 12211, United States

a r t i c l e

i n f o

Article history: Received 20 March 2013 Received in revised form 7 October 2013 Accepted 11 December 2013 MSC: primary 47B33 secondary 54H20

a b s t r a c t We study the dynamical properties of semigroups of composition operators of continuous functions on a separable locally compact metric space X. In particular, we show that if E is a normed vector space and φ : X → X is one-to-one and runaway on X, then the composition operator f → φ ◦ f on C(X, E) is topologically transitive and hypercyclic, where C(X, E) is the space of continuous functions from X into E with the compact-open topology. © 2013 Elsevier B.V. All rights reserved.

Keywords: Topologically transitive Hypercyclic Devaney chaos Semigroup actions Composition operators

1. Introduction In this paper, we study topologically transitive and hypercyclic composition operators on C(X, E), the space of E-valued continuous functions on a separable locally compact metric space (X, d), where (E,  · ) is a finite-dimensional normed vector space. Any continuous self map φ : X → X gives rise to a composition operator Cφ , defined by Cφ (f ) = f ◦ φ,

f ∈ C(X, E).

The topology under consideration on C(X, E) is the compact-open topology: a sequence of functions fn ∈ C(X, E), n  1, is said to converge compactly to f ∈ C(X, E) as n → ∞, if for every compact set K ⊆ X, we have fn |K → f |K uniformly as n → ∞. Recall that a map T on a metric space X is called * Tel.: +1 (518) 783 4201. E-mail address: [email protected]. 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.12.004

JID:TOPOL 2

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.2 (1-8) M. Javaheri / Topology and its Applications ••• (••••) •••–•••

(i) topologically transitive, if for every pair of nonempty open sets U, V ⊆ X, there exists n  1 such that T n (U ) ∩ V = ∅; (ii) mixing, if for every pair of nonempty open sets U, V ⊆ X, one has T n (U )∩V = ∅ for all n large enough; (iii) hypercyclic, if there exists x ∈ X so that {T n (x) | n  1} is dense in X; (iv) Devaney chaotic, if it is topologically transitive and has a dense set of periodic points. Here T n denotes the composition of T with itself n times, and a point x is called a periodic point of T if there exists n  1 such that T n (x) = x. Dynamical properties (i)–(iv) of a composition operator Cφ on certain subspaces of C(X, E) have been extensively studied in connection with the topological properties of the underlying map φ. For example, Shapiro [5,16] proved that Cφ is hypercyclic on H(U) (the space of holomorphic maps on the unit disk U), if and only if it is Devaney chaotic and mixing on H(U), and if and only if φ is univalent with no fixed points in U (a map is called univalent if it is holomorphic and one-to-one). Moreover, it is known that Cφ is hypercyclic if and only if φ is univalent and run-away [11] (a map f : X → X is called run-away if for every compact subset K ⊆ X, there exists n  1 such that f n (K) ∩ K = ∅). Dynamical properties of continuous semigroups of operators are also well-studied (for definitions of hypercyclicity, mixing, and chaos in the context of semigroup actions, see Section 3). For example, let X = C0,ρ ([0, ∞), C) be the space of continuous functions u : [0, ∞) → C with limτ →∞ ρ(τ )u(τ ) = 0, where ρ satisfies the condition ρ(τ )  M eωt ρ(t + τ ),

∀τ  0, ∀t > 0,

for some fixed M  1 and ω ∈ R. Let T (t) be the composition operator associated with translation by t i.e.,   T (t)u (τ ) = u(τ + t). W. Desch et al. [9] studied hypercyclicity for translation semigroups on X and proved that the condition lim inf ρ(t) = 0, t→∞

is the necessary and sufficient condition for the translation semigroup {T (t)}t0 to be hypercyclic on X. The inclusion X → C([0, ∞), C) is continuous with a dense range, where the topology of X is induced by its norm u = supτ |u(τ )|ρ(τ ), and C([0, ∞), C) is endowed with the compact-open topology. Therefore, by the hypercyclic comparison principle [3], the translation semigroup is hypercyclic on C([0, ∞), C) as well; see Theorem 3 for a generalization of this fact. Bermúdez et al. [4] proved that the condition lim ρ(t) = 0,

t→∞

is the necessary and sufficient condition for the translation semigroup {T (t)}t0 to be topologically mixing. For similar results regarding the weighted Lp spaces, see [4,8,9]. For a characterization of hypercyclicity for translation semigroups in spaces of weighted continuous or Lp functions on complex sectors, see [7]. The following theorem is the main result of this paper. Theorem 1. Suppose X is a separable locally compact metric space, E is a finite-dimensional normed vector space, and G is a semigroup of continuous self maps of X with the following properties: (i) Every element of G is one-to-one on X. (ii) The action of G is run-away on X.

JID:TOPOL

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.3 (1-8) M. Javaheri / Topology and its Applications ••• (••••) •••–•••

3

ˆ be the semigroup of composition operators induced by the elements of G i.e., G ˆ = {Cφ | φ ∈ G}. Then Let G ˆ the action of G on C(X, E) is topologically transitive and hypercyclic. Here is another example of how dynamical properties of Cφ are connected to the topological properties of φ. Let A(I) be the space of real analytic functions on an open interval I ⊆ R. Bonet and Domański proved the following theorem in [6]. Theorem 2. Let φ : U → U be a holomorphic map such that φ(−1, 1)  (−1, 1). Suppose that φ (x) = 0 for all x ∈ (−1, 1), and that φ is run-away on (−1, 1). Then Cφ is hypercyclic on A(−1, 1). By the Stone–Weierstrass Theorem [15], polynomials on I are dense in C(X, R) in the compact-open topology. Therefore, if Cφ is hypercyclic on A(I), then it is hypercyclic on C(I, R). We prove the following related theorem in Section 3. Theorem 3. Suppose E is a finite-dimensional normed vector space. Let φ : I → I be a continuous one-to-one map with no fixed points on the open interval I ⊆ R. Then Cφ is hypercyclic, Devaney chaotic, and mixing on C(I, E). We prove Theorem 1 in Section 2. In Section 3, we prove Theorem 3 and the following theorem, which states sufficient conditions on a space X and a self-map φ : X → X that guarantee chaos for the composition operator Cφ on C(X, E). Theorem 4. Suppose X is a separable locally compact metric space and E is a finite-dimensional normed vector space. Let φ : X → X be a one-to-one continuous map. If there exist a map θ : X → (0, 1) and a run-away continuous map ψ : (0, 1) → (0, 1) such that ψ ◦ θ = θ ◦ φ and   θ−1 Im(ψ) ⊆ Im(φ), then Cφ is chaotic on C(X, E). 2. Hypercyclic semigroups of composition operators Examples of hypercyclic semigroup actions on function spaces can be constructed as follows. Let η, μ : I → I be continuous maps from I into I in the compact-open topology, where I = (0, 1) such that the semigroup generated by them is dense in C(I, R) with the compact open topology [12,17]. Now, if f : I → R is a homeomorphism, then the orbit of f under the action of η, μ is dense in C(I, R). To see this, let g ∈ C(I, R) and let θ = f −1 ◦ g : I → I. Let λi : I → I, i  1, be continuous functions in η, μ that converge to θ in the compact-open topology. It follows that f ◦ λi → f ◦ θ = g in the compact-open topology as i → ∞. Therefore, the action of Cη , Cμ on C(I, R) is hypercyclic. Next, we present the proof of Theorem 1. Birkhoff’s transitivity theorem [3,10] states that topological transitivity and hypercyclicity are equivalent for discrete dynamical systems on second countable Baire spaces. However, for semigroup actions on general metric spaces, separate arguments are required for each property. We begin with the following two lemmas. Lemma 5. Let X and E be topological spaces, and W be an open set in the compact-open topology on C(X, E). Then there exists a compact set K ⊆ X such that if f ∈ W and g = f on K, then g ∈ W . Proof. For any pair of a compact set K ⊆ X and an open set U ⊆ E, let BK,U be the set of continuous functions in C(X, E) that map K into U . The set of all BK,U obtained in this way form a sub-basis for

JID:TOPOL 4

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.4 (1-8) M. Javaheri / Topology and its Applications ••• (••••) •••–•••

the compact-open topology on C(X, E). Therefore, given an open set W ⊆ C(X, E) and f ∈ W , there exist compact subsets K1 , . . . , Kn of X and open subsets U1 , . . . , Un of E so that f ∈ BK1 ,U1 ∩ · · · ∩ BKn ,Un ⊆ W. Let K = K1 ∪ · · · ∪ Kn . Now, it follows from g(x) = f (x) for all x ∈ K that g ∈ BKi ,Ui for 1  i  n, and so g ∈ W . 2 Lemma 6. Let X be a separable locally compact metric space. Then there exists an exhaustion of X by compact sets {Lj }j1 such that ∀j  1:

Lj ⊆ Int Lj+1 ;

∞ 

Lj = X,

(2.1)

j=1

where Int Z denotes the interior of Z ⊆ X. For every l, j  1 there exists a finite ordered set Jj,l ⊆ Lj such that Lj ⊆



U (a, 1/l),

(2.2)

a∈Jj,l

where U (a, δ) is the ball of radius δ centered at a. Moreover, every compact set K is included in Lj for some j  1. Finally, if f : X → Y is a function such that for each j  1, f |Lj is continuous on Lj , then f is continuous on X. Proof. The exhaustion in (2.1) exists for any second countable locally compact Hausdorff space [13, Prop. 4.76]. Given l, j  1, consider the open cover {U (x, 1/l): x ∈ Lj } of Lj . By compactness of Lj , there is a finite subcover {U (x, 1/l): x ∈ Jj,l }, where Jj,l is a finite subset of Lj , and (2.2) follows. We give Jj,l an arbitrary ordering for our future purposes. Next, let K be an arbitrary compact subset of X. It follows from (2.1) that {Int Lj : j  1} is an open cover of K. By compactness of K, there is a finite subcover {Int Lj1 , . . . , Int Ljk } for some j1 < · · · < jk and k  1. Then clearly K ⊆ Ljk . Finally, suppose that for every j  1, f |Lj : Lj → Y is continuous. Given an open set U ⊆ Y , the set f −1 (U ) =

∞ 

(fj |Int Lj )−1 (U ),

j=1

is a union of open sets in X, and so f : X → Y is continuous. 2 We are now ready to present the proof of Theorem 1. ˆ is topologically transitive. Let V and W be open Proof of Theorem 1. We first show that the action of G sets in C(X, E). By Lemma 5, there exists a compact set K1 ⊆ I such that if g = f on K1 and f ∈ V then g ∈ V . Similarly, a compact set K2 ⊆ I exists such that if g = f on K2 and f ∈ W then g ∈ W . We let K = K1 ∪ K2 . Since G is run-away on X, there exists φ ∈ G such that φ(K) ∩ K = ∅. Let f ∈ V and g ∈ W be arbitrary. Let  fˆ(x) =

if x ∈ K;

f (x) −1

g◦φ

(x) if x ∈ φ(K),

ˆ is topologically transitive. and extend fˆ to all of X continuously. Then, one has fˆ◦φ ∈ Cφ (V )∩W , and so G

JID:TOPOL

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.5 (1-8) M. Javaheri / Topology and its Applications ••• (••••) •••–•••

5

ˆ is hypercyclic. We need to construct a continuous function f : X → E Next, we prove that the action of G such that for every function g : X → E, any > 0, and any compact set K ⊆ X, there exists φ ∈ G so that f ◦ φ(x) − g(x) < for all x ∈ K. Let E be a countable subset of E. Also, let {Lj }j1 and Jj,l , j, l  1, be as in Lemma 6. Consider all triplets (M, B, l), such that M is one of the compact sets Lj , j  1, l is a positive integer, and B is a finite ordered subset of E of the same size as Jj,l . The set of all such triplets is countable, and so we can choose an ordering: (M1 , B1 , l1 ), (M2 , B2 , l2 ), (M3 , B3 , l3 ), . . . . Therefore, for each i  1, there exists some ti  1 such that Mi = Lti , and we define Ai = Jti ,li . In particular, it follows from (2.2) that Ai ⊆ Mi and Mi ⊆



U (a, 1/li ),

(2.3)

a∈Ai

for all i  1. By induction on n  0, we define functions f1 , . . . , fn , compact sets K1 , . . . , Kn ⊆ X, and φ1 , . . . , φn ∈ G for n  0 so that: (i) Li ⊆ Ki and Ki ⊆ Kj for all 0  i  j  n. (ii) fi is a continuous function on Ki and fi = fj on Ki for all 0  i  j  n. (iii) For each 1  i  n, there exists φi ∈ G such that φi (Ai ) ⊆ Ki and fi ◦ φi (Ai ) = Bi . Let A0 = B0 = L0 = M0 = K0 = ∅, and φ0 and f0 be the empty function. Assuming that we have defined {fi }ni=1 , {Ki }ni=1 , and {φi }ni=1 , satisfying properties (i)–(iii), we define Kn+1 , fn+1 , and φn+1 . Let An+1 = (a1 , . . . , ak ) and Bn+1 = (b1 , . . . , bk ), where k is the common size of An+1 and Bn+1 . By assumption (ii) in Theorem 1, there exists φ = φn+1 ∈ G such that φ(Kn ∪ Mn+1 ) ∩ (Kn ∪ Mn+1 ) = ∅. We let Kn+1 = Kn ∪ Ln+1 ∪ φ(Mn+1 ), which is a compact subset of X containing Kn and Ln+1 . Let {θi (x)}ki=1 be a partition of unity on Mn+1 subordinate to the open cover {U (ai , 1/ln+1 )}ki=1 . Define an extension of f to φ(Mn+1 ) by setting k    f φ(x) = θi (x)bi .

(2.4)

i=1

Note that since φ|Mn+1 : Mn+1 → φ(Mn+1 ) is continuous and one-to-one, its inverse is continuous on φ(Mn+1 ), which makes f , defined by (2.4), a continuous function on φ(Mn+1 ). Finally, let fn+1 (x) = fn (x) for all x ∈ Kn , and extend f to the reset of Kn+1 continuously. Conditions (i)–(iii) are readily checked for stage n + 1, and the inductive process is complete. Next, we define f : X → E by setting f (x) = fn (x) if x ∈ Kn . Since the sequence {Ki }i1 is an exhaustion of X

JID:TOPOL 6

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.6 (1-8) M. Javaheri / Topology and its Applications ••• (••••) •••–•••

and fi (x) = fj (x) for all x ∈ Ki ∩ Kj and i, j  1, the function f is well-defined and continuous on X by Lemma 6. Let g ∈ C(X, E) be an arbitrary continuous function, K ⊆ X be an arbitrary compact set, and be an arbitrary positive number. We prove that there exists φ ∈ G such that f ◦ φ(x) − g(x) < . Without loss of generality, suppose K = Lt for some t  1 (since every compact set is a subset of some Lt , t  1, by Lemma 6). Since g is continuous on Lt and Lt is compact, there exists m > 0 such that d(x, y) < 1/m

g(x) − g(y) < /2,

⇒

(2.5)

for all x, y ∈ Lt . Let Jt,m = (a1 , . . . , as ) and for each 1  i  s, choose bi ∈ E within /2 of g(ai ). Now, there exists n such that (Lt , B, m) = (Mn , Bn , ln ). By the construction of f , there exists φ = φn ∈ G such that f ◦ φn (ai ) = bi for i = 1, . . . , k. For a given x ∈ Mn , let Ix = {i | x ∈ U (ai , 1/ln )}. For each i ∈ Ix , we have d(x, ai ) < 1/ln , and so it follows from (2.5) that bi − g(x)  bi − g(ai ) + g(ai ) − g(x) < , for all i ∈ Ix and x ∈ Mn . And so  f ◦ φ(x) − g(x) = θi (x)bi − g(x) i∈Ix

   = θ (x) b − g(x) i i i∈Ix

  max bi − g(x) θ(x) i∈Ix

 max bi − g(x)

i∈Ix

i∈Ix

< , for all x ∈ Mn , and the theorem follows. 2 3. Chaotic composition operators Let G be a semigroup acting by continuous functions on a topological space X. Let also id X denote the identity map of X i.e., id X (x) = x for all x ∈ X. We say, the action of G on X (i) is topologically transitive, if for every pair of nonempty open sets U, V ⊆ X, there exists f ∈ G such that f (U ) ∩ V = ∅; (ii) is hypercyclic, if there exists x ∈ X such that the G-orbit of x i.e., {f (x) | f ∈ G} is dense in X; (iii) has a dense fixed point set, if the set {x ∈ X | ∃f ∈ G\{id X } f (x) = x} is dense in X; (iv) is run-away on X, if for every compact set K ⊆ X, there exists f ∈ G such that f (K) ∩ K = ∅. Moreover, if G acts by continuous maps on a compact metric space (X, d), we say the action of G on X is Devaney chaotic, if the action of G is topologically transitive, has a dense fixed point set, and has sensitive dependence on initial conditions i.e., there exists δ > 0 such that for every > 0 and every x ∈ X there exist y ∈ U (x, ) and f ∈ G such that d(f (x), f (y)) > δ. The third condition of Devaney chaos (sensitive dependence on initial conditions) was included in the original definition of Chaos by Devaney (an alternative definition was proposed in [14] by Li and Yorke; see [1] for three definitions of chaos).

JID:TOPOL

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.7 (1-8) M. Javaheri / Topology and its Applications ••• (••••) •••–•••

7

For a discrete dynamical system, it turns out that sensitive dependence on initial conditions follows from topological transitivity and density of periodic points. However, in the more general setting of semigroup actions, sensitive dependence does not follow from the other two conditions. Devaney chaos for semigroup actions is preserved by conjugation on compact metric spaces only [2]. Therefore, we do not discuss chaos for semigroups of composition operators on C(X, E). It needs mentioning that C(X, E) with the compact-open topology is a metrizable space. More generally, if (Y, d) is a metrizable space and X is hemicompact, then C(X, Y ) with the compact-open topology is metrizable. A space is called hemicompact, if there exists a countable collection of compact sets such that every compact set is included in one of the compact sets in the collection. For example, if {Ci }i1 is such a collection of compact sets for the space X, then the metric ρ(f, g) =

∞ 

   min max d f (x), g(x) , 2−i , x∈Ci

i=1

gives rise to the compact-open topology on C(X, Y ). In Theorems 1 and 4, since X is assumed to be a locally compact separable metric space, it follows that X is hemicompact by Lemma 6. Which topological spaces admit run-away continuous maps? Obviously, there are no run-away maps on compact spaces. One also notices that if φ : X → X is an extension of ψ : Y → Y , and if ψ is run-away on Y , then φ is run-away on X. Recall that φ : X → X is an extension of ψ : Y → Y , if there exists a continuous onto map θ : X → Y such that the following diagram is commutative: φ

X

X

θ

Y

(3.1)

θ ψ

Y.

Since every continuous map with no fixed points on an open interval in R is run-away, every extension of such a map is run-away. The commutative diagram (3.1) induces the following commutative diagram: C(Y, E)

Cψ

C(Y, E)

θˆ

C(X, E)

(3.2)

θˆ Cφ

C(X, E)

ˆ ) = f ◦ θ for f ∈ C(Y, E). If θˆ is onto and Cψ is chaotic, then Cφ is chaotic. However, θˆ is usually where θ(f not onto (for example, when θ is not one-to-one). Nevertheless, we have Theorem 4 which is proved below. Proof of Theorem 4. The topological transitivity of Cφ follows from Theorem 1 and the fact that φ is run-away and one-to-one. It is left to show that Cφ has a dense set of periodic points. Let K be an arbitrary compact set in X and g ∈ C(X, E) be arbitrary. Then θ(K) is compact in (0, 1), and so there exist α ∈ I and n  1 so that α < min θ(K)  max θ(K) < ψ n (α). Let m = min θ(K) and M = max θ(K), and define   Ck = θ−1 ψ nk (α), ψ n(k+1) (α) ;

  Dk = θ−1 ψ nk (m), ψ nk (M ) ,

Then Ck and Dk are closed subsets of X, Dk ⊆ Ck , and φn (Ck ) = Ck+1 ,

k  0.

JID:TOPOL

AID:5077 /FLA

[m3L; v 1.120; Prn:20/12/2013; 13:44] P.8 (1-8) M. Javaheri / Topology and its Applications ••• (••••) •••–•••

8

for all k  0. Let gˆ be a continuous map on C0 such that gˆ = g on D0 and gˆ(x) = 0 for x ∈ θ−1 {α, ψ n (α)}. Such an extension of g exists by the Tietze extension theorem. Next, we define f to be the following extension of gˆ to the entire X. For any arbitrary x ∈ X, there exists l ∈ Z such that α  ψ nl (θ(x))  ψ(α). It follows that φnl (x) ∈ C0 , and so we can define f (x) = gˆ(φnl (x)). Then fˆ is a well-defined continuous map on X. In addition, we have f ◦ φn (x) = f (x) for all x ∈ X. Finally, f = gˆ = g on K, and so f could be made arbitrarily close to f in the compact-open topology by choosing K to be large enough. This completes the proof of Theorem 4. 2 An example of a space X where Theorem 4 is applicable is any metric space homeomorphic to (0, 1) × Z, where Z is a separable locally compact metric space. Then one can let φ(t, z) = (ψ(t), z) and θ(t, z) = t, where ψ : (0, 1) → (0, 1) is any run-away one-to-one map. Theorem 4 then implies that Cφ is chaotic on C(X, E). Next, we state the proof of Theorem 3. Proof of Theorem 3. Hypercyclicity of Cφ follows from Theorem 1 and Proposition 1.1 of [6], which states that a continuous self-map of an open interval is run-away if and only if it has no fixed points. It follows from Criterion 3.1 in [4] that Cφ is mixing, where one takes Y0 = X0 , the subset of continuous functions with compact support. Finally, it follows from Theorem 4 that Cφ is chaotic. 2 Acknowledgement I would like to thank the referee for making many useful suggestions and for pointing out Refs. [4,7–9]. References [1] B. Aulbach, B. Kieninger, On three definitions of chaos, Nonlinear Dyn. Syst. Theory 1 (1) (2001) 23–37. [2] J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney’s definition of chaos, Am. Math. Mon. 99 (4) (1992) 332–334. [3] F. Bayart, E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math., vol. 179, Cambridge University Press, Cambridge, 2009. [4] T. Bermúdez, A. Bonilla, J.A. Conejero, A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Stud. Math. 170 (2005) 57–75. [5] J. Bès, Dynamics of composition operators with holomorphic symbol, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107 (2) (2013) 437–449. [6] J. Bonet, P. Domański, Hypercyclic composition operators on spaces of real analytic functions, Math. Proc. Camb. Philos. Soc. 153 (03) (2012) 489–503. [7] J.A. Conejero, A. Peris, Hypercyclic translation C0 -semigroups on complex sectors, Discrete Contin. Dyn. Syst. 25 (2009) 1195–1208. [8] R. deLaubenfels, H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, Ergod. Theory Dyn. Syst. 21 (2001) 1411–1427. [9] W. Desch, W. Schappacher, G.F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergod. Theory Dyn. Syst. 17 (1997) 793–819. [10] K.-G. Grosse-Erdmann, A. Peris-Manguillot, Linear Chaos, Springer, New York, NY, 2011. [11] K.-G. Grosse-Erdmann, R. Mortini, Universal functions for composition operators with non-automorphic symbol, J. Anal. Math. 107 (2009) 355–376. [12] V. Knichal, Sur les superpositions des automorphies continues d’un intervalle fermé, Fundam. Math. 31 (1938) 79–83. [13] J.M. Lee, Introduction to Topological Manifolds, Springer, 2010. [14] T.Y. Li, J.A. Yorke, Period three implies chaos, Am. Math. Mon. 82 (10) (1975) 985–992. [15] K. Saxe, Beginning Functional Analysis, Undergrad. Texts Math., Springer-Verlag, New York, 2001. [16] J.H. Shapiro, Notes on dynamics of linear operators, http://www.mth.msu.edu/~shapiro, last visited: February 12, 2013. [17] S. Subbiah, A dense subsemigroup of S(R) generated by two elements, Fundam. Math. 117 (2) (1983) 85–90.