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Mixing, simultaneous universal and disjoint universal backward Φ-shifts
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Mixing, simultaneous universal and disjoint universal backward Φ-shifts A. Jung Fachbereich IV Mathematik, Universität Trier, D-54286 Trier, Germany
a r t i c l e
i n f o
Article history: Received 31 January 2017 Available online xxxx Submitted by R.M. Aron Keywords: Simultaneous universality Disjoint universality Backward Φ-shift Rolewicz’s theorem Julia set
a b s t r a c t Spaces of sequences which are valued in a topological space E are considered in order to study universality properties of generalized backward shifts associated to certain selfmappings of E. After providing a condition which guarantees the mixing property of backward Φ-shifts in terms of the dynamical properties of the generating selfmappings, applications to universality theory within the framework of complex dynamics are furnished. © 2017 Published by Elsevier Inc.
1. Introduction During the last years, the study of various universality properties of sequences of continuous mappings on topological spaces has turned into an active research field in analysis (see e.g. [1,9,11]). For topological spaces X, Y and continuous mappings Tn : X → Y (n ∈ N), the sequence (Tn ) is called universal if there exists an element x0 ∈ X – called universal for (Tn ) – such that {Tn x0 : n ∈ N} forms a dense set in Y . In case of a single continuous selfmap T of X, the map T is called universal whenever its sequence of iterates (T n ) is universal (where we put T n := T ◦ T n−1 for n ∈ N and T 0 := idX ), i.e. if there exists an element in X which has a dense orbit under T . Often it is assumed that the underlying topological space also has a linear structure: if X is a topological vector space and T : X → X is a linear continuous map, the words hypercyclic and universal are synonymous. In case that X is a separable complete metric space without isolated points, Birkhoff’s transitivity theorem states that T is universal if and only if T is topologically transitive, i.e. for all nonempty open sets U, V ⊂ X there exists some N ∈ N with T N (U ) ∩ V = ∅ (see e.g. [11, Theorem 1.16]). Moreover, T is called mixing if for any pair U, V of nonempty open subsets of X there exists some N ∈ N such that T n (U ) ∩ V = ∅ for all n ≥ N . If X is a complete metric space without E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2017.03.009 0022-247X/© 2017 Published by Elsevier Inc.
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isolated points, T is called chaotic (in the sense of Devaney) if T is topologically transitive and has a dense set of periodic points (x ∈ X is called a periodic point of T if there exists some N ∈ N with T N x = x; in case of N = 1, x is called a fixed point of T ). More recently, combined universality properties of finite families of continuous mappings between topological spaces have been investigated. In 2007, J. Bès and A. Peris [7] as well as independently L. Bernal-González [4] have studied the concept of disjoint hypercyclicity. In general, for topological spaces X, Y1 , . . . , YN (N ∈ N) and continuous mappings Tj,n : X → Yj (j ∈ {1, . . . , N }, n ∈ N), the sequences (T1,n ), . . . , (TN,n ) are called disjoint universal whenever there exists some x0 ∈ X such that the joint orbit {(T1,n x0 , . . . , TN,n x0 ) : n ∈ N} is dense in the product Y1 × · · · × YN . Very recently, L. Bernal-González and the author [5] introduced the weaker property of simultaneous universality: for topological spaces X, Y and continuous mappings Tj,n : X → Y (j ∈ {1, . . . , N }, n ∈ N), the sequences (T1,n ), . . . , (TN,n ) are called simultaneous universal whenever there exists some x0 ∈ X such that the closure of the joint orbit {(T1,n x0 , . . . , TN,n x0 ) : n ∈ N} contains the diagonal {(y, y, . . . , y) : y ∈ Y } of Y N . In other words, if Y1 , . . . , YN are first-countable (if Y is first-countable, resp.), the disjoint (simultaneous, resp.) universality of (Tj,n )n∈N (1 ≤ j ≤ N ) means the existence of some x0 ∈ X enjoying the property that, for each tuple (y1 , . . . , yN ) ∈ Y1 × · · · × YN (for each point y ∈ Y , resp.), there is one common sequence (nk ) in N such that Tj,nk x0 → yj (Tj,nk x0 → y, resp.) for all j = 1, . . . , N as k → ∞. A classic example of a universal map is given by the backward shift operator on sequence spaces. Denoting by X the space c0 of all complex-valued null sequences or the space q of all q-summable complex-valued sequences (1 ≤ q < ∞) and considering the map B : X → X, B((xj )j∈N ) := (xj+1 )j∈N , S. Rolewicz [14] was able to prove in 1969 that for any λ ∈ C with |λ| > 1 (and, in fact, only for these λ), the weighted backward shift operator λB is hypercyclic on X. Moreover, it is well-known that in this case λB is even mixing and chaotic (see e.g. [11, Examples 2.32 and 2.38]). In this paper, we want to study simultaneous and disjoint universality of certain finite families of generalized backward shifts. The considered shifts will be associated to selfmappings of a corresponding topological space and it will be our aim to study the above mentioned universality properties with respect to dynamical properties of the generating selfmappings. Starting point for these investigations are the papers of L. BernalGonzález [3] and A. Peris [13]. Their most important results in view of our further studies will be stated in Section 2. Subsequently, in Section 3, it will be our aim to exceed the mentioned results of Bernal’s work [3] in view of the mixing property and to combine them with a theorem of Peris’ work [13] (which follows also from Bernal’s results) in order to obtain simultaneous universality of certain finite families of backward Φ-shifts. Finally, in Section 4, we will merge proof techniques of Bernal’s work on disjoint hypercyclic operators [4] and Peris’ work [13] in order to obtain disjoint universality. 2. Universal backward Φ-shifts and complex dynamics In 2005, L. Bernal-González [3] investigated universality of so-called backward Φ-shifts in a very general setting. In order to cite his main results, we introduce the following notation coined by him (cf. [3, Section 3]): for a Hausdorff topological space E and for each a ∈ E, we denote by σ(a) the set of sequences ending with a, that is, σ(a) := J∈N σJ (a), where σJ (a) := {(xj ) ∈ E N : xj = a for all j > J}. A standard sequence space (SSS) on E is defined as a set S ⊂ E N endowed with a topology such that there is a point a ∈ E which satisfies the following properties: (S1) The space S is Baire and second-countable. (S2) The topology on S is stronger than that inherited from the product topology on E N .
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(S3) The set σ(a) is a dense subset of S. (S4) For each J ∈ N, the topology of each σJ (a) inherited from S is the product topology. If S is an SSS on E and a ∈ E is a point satisfying (S3)–(S4), then a is called a distinguished point for S. Examples of SSSs are the spaces c0 and q (0 < q < ∞), where one can simply take E = C with the usual topology. More generally, if E is a separable Banach space over R or C, the space c0 (E) of all E-valued null sequences and the space q (E) (0 < q < ∞) of all q-summable E-valued sequences are also SSSs on E (cf. [3, p. 184]). Definition 2.1. For a topological space E, an SSS S on E and a continuous selfmap T : S → S, we say that T is a backward Φ-shift on S if there exists a selfmap f : E → E such that T = Bf on S, where Bf : E N → E N , Bf ((xj )j∈N ) := (f (xj+1 ))j∈N . The following characterization of Bf itself being a backward Φ-shift on the SSSs S = c0 (E) or q (E) (0 < q < ∞) (i.e. of Bf being S-invariant and continuous) was proved in [3, Theorem 3.5]: • Bf is a backward Φ-shift on c0 (E) if and only if f is continuous and f (0) = 0, • Bf is a backward Φ-shift on q (E) (0 < q < ∞) if and only if f is continuous and lim supt→0 f (t) E / t E < ∞. It is easy yo see that f is continuous on E whenever Bf is a backward Φ-shift on S (cf. [3], Remark 3.3.(1)). In order to state some of the main results of [3], we also introduce the following conditions: (S4*) Given J ∈ N, an open set U ⊂ S and a point α ∈ σJ (a) ∩ U , there exist open sets U1 , . . . , UJ , A in ∞ (n) E such that α ∈ j=1 Aj ⊂ U for all n > J, where
(n)
Aj
⎧ ⎪ ⎨ Uj (1 ≤ j ≤ J) := A (n + 1 ≤ j ≤ n + J) ⎪ ⎩ {a} (J < j ≤ n or j > n + J).
(S5) Given an open subset A ⊂ E containing a, there exists an open set U ⊂ S with (a, a, . . . ) ∈ U such that πj (U ) ⊂ A for all j ∈ N, where πj denotes the j-projection πj : S → E, πj ((xn )n∈N ) := xj . Condition (S4*) is stronger than condition (S4) (cf. [3, Remark 3.1]), and (S4*) as well as (S5) are satisfied for S = c0 (E) and S = lq (E) (0 < q < ∞) with a = 0 in case of a separable Banach space E (cf. [3, p. 184 and p. 192]). Finally, we introduce the following local weakly mixing property of sequences of continuous selfmappings of a topological space, which was coined by L. Bernal-González (cf. [3, Definition 2.1]) and which is related to the well-known notation of weakly mixing maps: for a topological space E, continuous mappings fn : E → E (n ∈ N) and a point a ∈ E, we say that (fn ) is weakly mixing at a if for all m ∈ N and all nonempty open sets V1 , . . . , Vm , A ⊂ E with a ∈ A there exists some N ∈ N satisfying fN (A) ∩ Vj = ∅ for all
j = 1, . . . , m.
A selfmap f of E is said to be weakly mixing at a if its sequence of iterates (f n ) is weakly mixing at a.
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Main results of [3] now read as follows (cf. [3, Theorem 4.3]): Theorem 2.2 (L. Bernal-González, 2005). i) Assume that Bf : S → S is a backward Φ-shift on a SSS S satisfying (S4*) with distinguished point a, in such a way that f (a) = a and f is weakly mixing at that point. Then Bf is universal. ii) Assume that Bf : S → S is a universal backward Φ-shift on a SSS S which satisfies property (S5). Then f is weakly mixing at the distinguished point. iii) Suppose that E is a separable Banach space, that S = c0 (E) or lq (E) (0 < q < ∞) and that f : E → E is continuous. In addition, we assume f (0) = 0 if S = c0 (E) and lim supt→0 f (t) E / t E < ∞ if S = lq (E). Then the backward Φ-shift Bf is universal if and only if f is weakly mixing at the origin. In 2003, A. Peris [13] characterized universality and chaos of backward Φ-shifts of the form Bp : X → X, where X stands for the space c0 or q (1 ≤ q < ∞) and p : C → C is a polynomial with p(0) = 0. In order to formulate his main results, we need some terminology from the theory of complex dynamics that investigates the long-time behaviour of certain discrete topological dynamical systems (see e.g. [8,12,15,16]). The systems which are studied are usually given by the iteration of an entire function or a rational function. In the following, we focus on the case of entire functions, mainly polynomials. The local behaviour of the considered functions near fixed points plays an important role in the study of the long-time behaviour of the corresponding dynamical systems. Let f be an entire function (not a polynomial of degree 0 or 1) and let z0 ∈ C be a fixed point of f . Then the multiplier of f at z0 is denoted and defined by λ := f (z0 ) and the fixed point z0 is called: superattracting if λ = 0, attracting if 0 < |λ| < 1, rationally indifferent if |λ| = 1 and λ is a root of unity, irrationally indifferent if |λ| = 1 but λ is not a root of unity, repelling if |λ| > 1. The Fatou set Ff of f is defined as the set of all points z ∈ C for which there exists a neighbourhood U of z such that {f n |U : n ∈ N} is a normal family in the space of continuous functions from U to the extended complex plane C∞ . The complement of Ff is called the Julia set of f and it is denoted by Jf . It is well-known that we have Ff n = Ff and hence Jf n = Jf for all n ∈ N. Moreover, (super-)attracting fixed points of f are always contained in the Fatou set of f , whereas repelling and rationally indifferent fixed points of f always belong to the Julia set of f . Irrationally indifferent fixed points of f can be contained in Ff or in Jf . A reason for splitting the complex plane into the two disjoint subsets Ff and Jf is the following: It is well-known that the sequence of iterates (f n ) behaves quite “wildly” on Jf whereas, on the other hand, the behaviour of (f n ) on Ff is quite “nice” and well-understood (cf. the classification theorem of components of the Fatou set in the above mentioned references). The “wild” behaviour of (f n ) on Jf can be described as follows: for each point z ∈ Jf and each arbitrarily small open neighbourhood U of z, it follows directly from Montel’s theorem (see e.g. [12, Theorem 3.7]) that the set C\ n∈N f n (U ) contains at most one element, i.e. every value in the complex plane, with at most one exception, is assumed on U under the iteration of f . For z ∈ C, the grand orbit of z under f is defined as the set GO(z, f ) := {w ∈ C : there exist m, n ∈ N0 with f m (z) = f n (w)}, and z is called exceptional under f if GO(z, f ) is finite. An application of Montel’s theorem yields that the set Ef of exceptional points under f contains at most one element (see e.g. [12, Lemma 4.6]). Furthermore, in case of polynomials, the following
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sharper version of the above statement concerning the “wild” behaviour of (f n) on Jf holds (see e.g. [12, Corollary 14.2 and the subsequent considerations]): Theorem 2.3 (Expanding property of the Julia set). Let f be a polynomial of degree ≥ 2. Moreover, let K ⊂ C be compact with K ∩ Ef = ∅ and let U ⊂ C be open with U ∩ Jf = ∅. Then there exists some N ∈ N such that f n (U ) ⊃ K holds for all n ≥ N . Now, let p : C → C be a polynomial with p(0) = 0 and let us consider the backward Φ-shift Bp : X → X, where X = c0 or q (1 ≤ q < ∞). Then Peris’ main result [13, Theorem 3.2] shows that Bp is universal if and only if the origin is contained in the Julia set of p (this statement also follows from the results in Bernal’s work [3], cf. [3, Remark 5.3]). Moreover, in case of X = c0 , universality of Bp is actually equivalent to the fact that Bp is chaotic (cf. [13, Remark 3.4]), and in case of X = q (1 ≤ q < ∞), at least the stronger condition that the origin is a repelling fixed point of p guarantees that Bp is chaotic (cf. [13, Proposition 3.3]). Finally, it is conjectured that the property 0 ∈ Jp is not sufficient for Bp being chaotic on q (1 ≤ q < ∞) (cf. [13, Conjecture 3.5]). 3. Mixing and simultaneous universal backward Φ-shifts In this section, we first want to prove a “mixing-analogon” of Theorem 2.2. On the one hand, this result will be interesting in itself, and, on the other hand, it will allow us to formulate a statement concerning simultaneous universality of rotations of certain backward Φ-shifts. In order to proceed as requested, we introduce the following local mixing property (analogously to the local weakly mixing property as defined in Section 2). Definition 3.1. Let E be a topological space, fn : E → E (n ∈ N) continuous mappings and a ∈ E. We say that (fn ) is mixing at a if for all m ∈ N and all nonempty open sets V1 , . . . , Vm , A ⊂ E with a ∈ A there exists some N ∈ N such that fn (A) ∩ Vj = ∅ (j = 1, . . . , m) holds for all n ≥ N . A selfmap f of E is said to be mixing at a if its sequence of iterates (f n ) is mixing at a. Then the following statements hold: Theorem 3.2. i) Assume that Bf : S → S is a backward Φ-shift on a SSS S satisfying (S4*) with distinguished point a, in such a way that f (a) = a and f is mixing at that point. Then Bf is mixing. ii) Assume that Bf : S → S is a mixing backward Φ-shift on a SSS S which satisfies property (S5). Then f is mixing at the distinguished point. iii) Suppose that E is a separable Banach space, that S = c0 (E) or lq (E) (0 < q < ∞) and that f : E → E is continuous. In addition, we assume f (0) = 0 if S = c0 (E) and lim supt→0 f (t) E / t E < ∞ if S = lq (E). Then the backward Φ-shift Bf : S → S is mixing if and only if f is mixing at the origin.
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Proof. i) Analogously to Theorem 4.3 in [3], it suffices to show the following more general statement: for an SSS S on E satisfying (S4*) for some distinguished point a ∈ E and a sequence (fn ) of selfmappings of E which is mixing at a and fulfils fn (a) = a for all n ∈ N such that the mappings Tn : S → S, Tn ((xj )j∈N ) := (fn (xj+n ))j∈N , are well-defined and continuous, the sequence (Tn ) is mixing. We follow very closely the proof of Theorem 4.3 in [3]. To this end, let U, V ⊂ S be nonempty and open. Due to (S3), we have U ∩ σ(a) = ∅ = V ∩ σ(a) so that there exist an integer J ∈ N as well as points a1 , . . . , aJ , b1 , . . . , bJ ∈ E with α ∈ U and β ∈ V , where α := (a1 , . . . , aJ , a, a, . . . )
and β := (b1 , . . . , bJ , a, a, . . . ).
According to α ∈ σJ (a) ∩ U , (S4*) implies the existence of finitely many open sets U1 , . . . , UJ , A in E ∞ (n) such that we have α ∈ j=1 Aj ⊂ U for all n > J, where
(n)
Aj
⎧ ⎪ ⎨ Uj := A ⎪ ⎩ {a}
(1 ≤ j ≤ J) , (n + 1 ≤ j ≤ n + J) (J < j ≤ n or j > n + J)
n > J.
Moreover, there exist open sets V1 , . . . , VJ ⊂ E with β ∈ V1 × · · · × VJ × {a} × {a} × · · · ⊂ V. As (fn ) is mixing at a, we can choose some N ∈ N with N > J such that fn (A) ∩ Vj = ∅
(j = 1, . . . , J)
holds for all n ≥ N . Hence, for each n ≥ N , there exist J points t1,n , . . . , tJ,n ∈ A satisfying fn (tj,n ) ∈ Vj (j = 1, . . . , J). For each n ≥ N , we put xn := (xm,n )m∈N , where
xm,n :=
⎧ ⎪ ⎨
am
⎪ ⎩
tm−n,n a
Then, for each n ≥ N , we have xn ∈
∞ j=1
(1 ≤ m ≤ J) (n + 1 ≤ m ≤ n + J) (J < m ≤ n or m > n + J). (n)
Aj
⊂ U as well as
Tn xn = fn (xm+n,n ) m∈N = fn (t1,n ), . . . , fn (tJ,n ), fn (a), fn (a), ... = fn (t1,n ), . . . , fn (tJ,n ), a, a, ... ∈ V1 × · · · × VJ × {a} × {a} × · · · ⊂ V , and thus Tn xn ∈ Tn (U ) ∩ V = ∅. ii) Analogously to Theorem 4.5 in [3], it suffices to show the following more general statement: for an SSS S on E satisfying (S5) for some distinguished point a ∈ E and a sequence (fn ) of selfmappings of E such that the mappings Tn : S → S, Tn ((xj )j∈N ) := (fn (xj+n ))j∈N ,
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are well-defined, continuous and mixing, the sequence (fn ) is mixing at a. We follow very closely the proof of Theorem 4.5 in [3]. To this end, let V1 , . . . , Vm , A be nonempty open subsets of E with a ∈ A. According to (S5), there exists an open set U ⊂ S with (a, a, a, . . . ) ∈ U such that πj (U ) ⊂ A for all j ∈ N. Putting W := V1 × · · · × Vm × E × E × · · · , we obtain from (S2) that W is open in S. Thus, since (Tn ) is mixing, there exists some N ∈ N with Tn (U ) ∩ W = ∅ for all n ≥ N . Hence, for each n ≥ N , there exists some xn = (xj,n )j∈N ∈ U with W Tn xn = (fn (xj+n,n ))j∈N , which implies fn (xj+n,n ) ∈ Vj for all j = 1, . . . , m. As we also have xj+n,n = πj+n (xn ) ∈ πj+n (U ) ⊂ A (j ∈ N, n ≥ N ), we finally obtain that fn (xj+n,n ) ∈ fn (A) ∩ Vj = ∅ (j = 1, . . . , m) holds for all n ≥ N . iii) Under the assumptions of iii), the considerations in Section 2 show that Bf is a backward Φ-shift on S and that (S4*) and (S5) are satisfied with a = 0 for S. Hence, the assertion follows directly form parts i) and ii). 2 A first application to the framework of complex dynamics is: Theorem 3.3. Let p : C → C be a polynomial of degree ≥ 2 with p(0) = 0, X = c0 or q (0 < q < ∞) and consider the backward Φ-shift Bp : X → X. Then Bp is mixing if and only if 0 ∈ Jp . Proof. Due to the assumptions and the considerations in Section 2, we see that Bp is indeed a backward Φ-shift on X (for the q case, observe that due to p(0) = 0 the function z → p(z)/z can be extended to an entire function). If Bp is mixing, then Bp is topologically transitive so that Birkhoff’s transitivity theorem yields that Bp is universal. Hence, Peris’ result [13, Theorem 3.2] implies that the origin is contained in the Julia set of p (see also [3, Remark 5.3] in case of X = q with 0 < q < 1). Now let 0 ∈ Jp . In order to show that Bp is mixing, according to Theorem 3.2 i) it suffices to show that p is mixing at the origin. To this end, let m ∈ N and let V1 , . . . , Vm , A ⊂ C be nonempty open with 0 ∈ A. We choose points zj ∈ Vj \Ep (j = 1, . . . , m) and we put K := {z1 , . . . , zm }. As we have 0 ∈ A ∩ Jp = ∅, the expanding property of the Julia set (Theorem 2.3) yields the existence of some N ∈ N with pn (A) ⊃ K for all n ≥ N . Hence, for each n ≥ N , we have zj ∈ K ∩ Vj ⊂ pn (A) ∩ Vj = ∅ (j = 1, . . . , m), i.e. p is mixing at the origin.
2
Remark 3.4. i) Due to Peris, the equivalent properties stated in Theorem 3.3 hold if and only if Bp is universal (cf. [13, Theorem 3.2] and see also [3, Remark 5.3] in case of X = q with 0 < q < 1), and, in case of X = c0 , they are also equivalent to the fact that Bp is chaotic (cf. [13, Remark 3.4]). ii) Considering the set X = c0 (C∞ ) of all sphere-valued null sequences endowed with the metric d : X → X, d (xj )j∈N , (yj )j∈N := sup χ(xj , yj ), j∈N
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where χ denotes the chordal distance on C∞ , we obtain that X is an SSS on C∞ satisfying (S4*) for the distinguished point 0. In particular, (X, d) is a separable complete metric space without isolated points. As the Fatou set and the Julia set of rational functions can be defined analogously as in the case of entire functions and as the expanding property of the Julia set also holds for rational functions of degree ≥ 2 (see e.g. [12, Corollary 14.2 and the subsequent considerations]), the proof of Theorem 3.3 also yields the following result: Let R : C∞ → C∞ be a rational function of degree ≥ 2 with R(0) = 0, let X = c0 (C∞ ) and consider the backward Φ-shift BR : X → X. Then BR is mixing if and only if 0 ∈ JR . Observe that, indeed, BR is well-defined on X and that Peris’ result [13, Theorem 3.2] also holds for rational functions (cf. the proofs of [3, Theorem 3.5 (3)] and [13, Theorem 3.2]). Finally, we obtain the following result on simultaneous universality: Corollary 3.5. Let p : C → C be a polynomial of degree ≥ 2 with p(0) = 0 and let X = c0 or q (0 < q < ∞). Moreover, let λ1 , . . . , λN (N ∈ N) be unimodular scalars. Then the backward Φ-shifts Bp , Bλ1 p , . . . , BλN p are simultaneously universal on X if and only if 0 ∈ Jp . Proof. If Bp , Bλ1 p , . . . , BλN p are simultaneously universal, then in particular Bp is universal so that Peris’ result [13, Theorem 3.2] implies that the origin is contained in the Julia set of p (see also [3, Remark 5.3] in case of X = q with 0 < q < 1). On the other hand, if 0 ∈ Jp , Theorem 3.3 yields that Bp is mixing. Hence, in view of Birkhoff’s transitivity theorem, Bp is hereditarily universal, i.e. for each strictly increasing sequence (nk ) in N, the sequence (Bpnk )k∈N is universal. Hence, an application of Proposition 4.1 (b) of [5] implies that the rotations Bp , λ1 Bp (= Bλ1 p ), . . . , λN Bp (= BλN p ) are simultaneously universal (observe that the assumption of linearity in Proposition 4.1 (b) of [5] actually is not needed). 2 In view of Remark 3.4 ii), also the following statement holds: Let R : C∞ → C∞ be a rational function of degree ≥ 2 with R(0) = 0 and let X = c0 (C∞ ). Moreover, let λ1 , . . . , λN (N ∈ N) be unimodular scalars. Then the backward Φ-shifts BR , Bλ1 R , . . . , BλN R are simultaneously universal on X if and only if 0 ∈ JR . 4. Disjoint universal backward Φ-shifts In [4], the question was posed if for a universal map T the iterates T, T 2 are always disjoint universal (cf. [4, p. 129]). It was shown in [6] that this does not hold in general: there exists a hypercyclic linear continuous selfmap T of 2 such that T, T 2 are not disjoint hypercyclic (cf. [6, p. 855]). Moreover, in [5] it was remarked that in this case T, T 2 are even not simultaneously hypercyclic (cf. [5, Remark 5.2.2]). However, there exist examples of universal maps T such that T, T 2 are disjoint universal. For instance, Bernal has shown that for X = c0 or q (1 ≤ q < ∞), N ∈ N and a bounded sequence a = (aj )j∈N in C\{0} fulfilling the property that ⎧ ⎨ sup
min
n∈N ⎩
⎧ n ⎨
⎩
j=1
|aj | ,
2n
j=1
|aj | , . . . ,
Nn
j=1
⎫⎫ ⎬⎬ |aj | = ∞, ⎭⎭
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the weighted backward shift operators Ta , Ta2 , . . . , TaN are disjoint hypercyclic on X, where Ta : X → X, Ta ((xj )j∈N ) := (aj+1 xj+1 )j∈N (cf. [4, Proposition 5.7]). Furthermore, an example of disjoint hypercyclic weighted backward shifts with unbounded weight sequences is given by the pair D, D2 , where D denotes the differentiation operator f → f on the space H(C) of all entire functions on the complex plane, which may be viewed as the weighted backward shift (c0 , c1 , c2 , . . .) → (c1 , 2c2 , 3c3 , . . .) when considering H(C) as the space of all sequences (cj ) in C with lim sup |cj |1/j → 0 (cf. [7, Proposition 3.3], see also [2,10]). In the framework of polynomial backward Φ-shifts, the following holds: Theorem 4.1. Let p : C → C be a polynomial of degree ≥ 2 such that the origin is a repelling fixed point of p and let X = c0 or q (0 < q < ∞). Then for each N ∈ N, the backward Φ-shifts Bp , Bp2 , . . . , BpN are disjoint universal on X. Proof. The proof will follow the structure of Bernal’s proof of Proposition 5.7 in [4]. In addition, we will also mimic a technique of proof of Peris, cf. the proof of Proposition 3.3 in [13]. To simplify matters, we only prove the assumption in case of N = 2; the general case can be shown similarly. According to the disjoint blow-up/collapse criterion (see e.g. [7, Proposition 2.4]), it suffices to show that there exists a dense set D0 ⊂ X such that for all x, y, z ∈ D0 , there exist a strictly increasing sequence (nk ) in N and a sequence (xk ) in X satisfying: n (i) Bpnk x −→ 0 and Bp2 k x −→ 0 as k → ∞, n (ii) xk −→ 0, Bpnk xk −→ y and Bp2 k xk −→ z as k → ∞. Choosing D0 := (xk ) ∈ X : there exists k0 ∈ N such that xk = 0 for all k ≥ k0 and xk ∈ / Ep2 for all k ∈ N , we obtain that D0 is dense in X (observe that Ep2 is empty or a singleton, see Section 2). As each element of D0 has only finitely many nonzero entries and due to p(0) = 0, it follows that for each x = (xk ) ∈ D0 , there exists some m ∈ N such that we have Bpn x = 0 and (Bp2 )n x = 0 for all n ≥ m, i.e. condition (i) is fulfilled for each strictly increasing sequence (nk ) in N. In order to show (ii), let y, z ∈ D0 , and let M ∈ N with y = (y1 , . . . , yM , 0, 0, . . .) as well as z = (z1 , . . . , zM , 0, 0, . . .). As the origin is a repelling fixed point of p, we can fix some r > 1 with r < |p (0)|. Then there exists a δ > 0 such that we have p(U ) ⊃ rU for all open discs U centred at 0 with radius less than δ (cf. the proof of [13, Proposition 3.3]). It follows that pn (U ) ⊃ rn U
(1)
holds for all n ∈ N and all open discs U centred at 0 with radius less than δ/rn−1 . In the following, for ε > 0, we write Uε (0) := {w ∈ C : |w| < ε}. Due to 0 ∈ Jp = Jp2 and in view of the definition of D0 (observe that Ep ⊂ Ep2 ), it follows from the expansion property of the Julia set (Theorem 2.3) that there exists some m ∈ N with pm (Uδ (0)) ⊃ {y1 , . . . , yM }
and
p2m (Uδ (0)) ⊃ {z1 , . . . , zM }.
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Thus, we can find points a1,1 , . . . , aM,1 , b1,1 , . . . , bM,1 ∈ Uδ (0) with pm (al,1 ) = yl
p2m (bl,1 ) = zl ,
and
l = 1, . . . , M.
(2)
According to (1), we have pm (Uδ/rm (0)) ⊃ Uδ (0)
p2m (Uδ/rm (0)) ⊃ p2m (Uδ/r2m (0)) ⊃ Uδ (0)
and
so that there exist a1,2 , . . . , aM,2 , b1,2 , . . . , bM,2 ∈ Uδ/rm (0) with pm (al,2 ) = al,1
p2m (bl,2 ) = bl,1
and
l = 1, . . . , M.
Inductively, we can find points al,k , bl,k ∈ C with |al,k |, |bl,k | < δ/r(k−1)m as well as pm (al,k+1 ) = al,k
p2m (bl,k+1 ) = bl,k ,
and
l = 1, . . . , M,
k ∈ N.
(3)
In particular, each sequence (al,k )k∈N , (bl,k )k∈N converges to 0. Due to (2) and (3), we have pm·k (al,k ) = (pm )k (al,k ) = yl
and
p2m·k (bl,k ) = (p2m )k (bl,k ) = zl
and
p2m·2 (bl,2k ) = zl
and hence k
pm·2 (al,2k ) = yl
k
(4)
as well as k
pm·2 (bl,2k ) = p2m·2
k−1
k−1
(bl,2k ) = (p2m )2
(bl,2k ) = bl,2k−1
(5)
for all l = 1, . . . , M and all k ∈ N. Now, we define nk := m · 2k , k ∈ N, and for each k ∈ N with nk ≥ M , we put xk := 0, . . . , 0, a1,2k , . . . , aM,2k , 0, . . . , 0, b1,2k , . . . , bM,2k , 0, 0, . . . . nk −M times
nk times
Finally, we choose k0 ∈ N with nk ≥ M for all k ≥ k0 and we consider the sequence x := (xk )k≥k0 in X. As each sequence (al,k )k∈N , (bl,k )k∈N converges to 0, we obtain that xk −→ 0
as
k → ∞.
Moreover, in view of (4) and (5), we have
Bp2
nk
xk = p2nk (b1,2k ), . . . , p2nk (bM,2k ), p2nk (0), p2nk (0), . . . k k = p2m·2 (b1,2k ), . . . , p2m·2 (bM,2k ), 0, 0, . . . = (z1 , . . . , zM , 0, 0, . . .) = z
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as well as Bpnk xk =
pnk (a1,2k ), . . . , pnk (aM,2k ), pnk (0), . . . , pnk (0), pnk (b1,2k ), . . . , pnk (bM,2k ), pnk (0), pnk (0), . . . k
k
k
k
pm·2 (a1,2k ), . . . , pm·2 (aM,2k ), 0, . . . , 0, pm·2 (b1,2k ), . . . , pm·2 (bM,2k ), 0, 0, . . . = y1 , . . . , yM , 0, . . . , 0, b1,2k−1 , . . . , bM,2k−1 , 0, 0, . . . =
−→ (y1 , . . . , yM , 0, 0, . . .) = y
as
k → ∞,
as required. 2 Remark 4.2. i) In view of Remark 3.4 ii), also the following statement holds: Let R : C∞ → C∞ be a rational function of degree ≥ 2 such that the origin is a repelling fixed point of R and let X = c0 (C∞ ). Then for each 2 N N ∈ N, the backward Φ-shifts BR , BR , . . . , BR are disjoint universal on X. ii) The question arises if the statement of Theorem 4.1 is in fact an equivalence, i.e. if the fixed point of p at the origin must be repelling whenever Bp , Bp2 , . . . , BpN are disjoint universal (clearly, in this case, Bp is universal so that we obtain at least that 0 belongs to the Julia set of p). In any case, the proof of Theorem 4.1 crucially uses the fact that 0 is indeed a repelling fixed point of p – namely property (1) allows us to work with a fixed integer m ∈ N in each step of the inductive construction (this is not ensured by only applying the expanding property of the Julia set) which guarantees that the points bl,2k fulfil equation (5) that is needed to obtain the convergence Bpnk xk −→ y as k → ∞ in the last step of the proof. So we conclude with the following conjecture (cf. [13, Conjecture 3.5]): Conjecture 4.3. Let p : C → C be a polynomial of degree ≥ 2 with p(0) = 0 and let X = c0 or q (0 < q < ∞). Then the backward Φ-shifts Bp , Bp2 , . . . , BpN are disjoint universal on X for each N ∈ N if and only if 0 is a repelling fixed point of p. Acknowledgments The author thanks the referee for his/her helpful suggestions concerning the presentation of this paper. The author has been supported by DFG-Forschungsstipendium JU 3067/1-1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
F. Bayart, É. Matheron, Dynamics of Linear Operators, Cambridge University Press, Cambridge, 2009. L. Bernal-González, Universal functions for Taylor shifts, Complex Var. Theory Appl. 31 (1996) 121–129. L. Bernal-González, Backward Φ-shifts and universality, J. Math. Anal. Appl. 306 (2005) 180–196. L. Bernal-González, Disjoint hypercyclic operators, Studia Math. 182 (2007) 113–131. L. Bernal-González, A. Jung, Simultaneous universality, submitted for publication, available from https://arxiv.org/abs/ 1701.07311. J. Bès, Ö. Martin, A. Peris, Disjoint hypercyclic linear fractional composition operators, J. Math. Anal. Appl. 381 (2011) 843–856. J. Bès, A. Peris, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336 (2007) 297–315. L. Carleson, T.W. Gamelin, Complex Dynamics, Springer, New York, 1993. K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999) 345–381. K.-G. Grosse-Erdmann, Rate of growth of hypercyclic entire functions, Indag. Math. (N.S.) 11 (2000) 561–571. K.-G. Grosse-Erdmann, A. Peris, Linear Chaos, Springer, London, 2011. J. Milnor, Dynamics in One Complex Variable, 2nd edition, Vieweg, Braunschweig-Wiesbaden, 2000. A. Peris, Chaotic polynomials on Banach spaces, J. Math. Anal. Appl. 287 (2003) 487–493.
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[14] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969) 17–22. [15] D. Schleicher, Dynamics of entire functions, in: Holomorphic Dynamical Systems, in: Lecture Notes in Math., vol. 1998, Springer, Berlin, 2010, pp. 295–339. [16] N. Steinmetz, Rational Iteration, de Gruyter, Berlin, 1993.
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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Mixing, simultaneous universal and disjoint universal backward Φ-shifts A. Jung Fachbereich IV Mathematik, Universität Trier, D-54286 Trier, Germany
a r t i c l e
i n f o
Article history: Received 31 January 2017 Available online xxxx Submitted by R.M. Aron Keywords: Simultaneous universality Disjoint universality Backward Φ-shift Rolewicz’s theorem Julia set
a b s t r a c t Spaces of sequences which are valued in a topological space E are considered in order to study universality properties of generalized backward shifts associated to certain selfmappings of E. After providing a condition which guarantees the mixing property of backward Φ-shifts in terms of the dynamical properties of the generating selfmappings, applications to universality theory within the framework of complex dynamics are furnished. © 2017 Published by Elsevier Inc.
1. Introduction During the last years, the study of various universality properties of sequences of continuous mappings on topological spaces has turned into an active research field in analysis (see e.g. [1,9,11]). For topological spaces X, Y and continuous mappings Tn : X → Y (n ∈ N), the sequence (Tn ) is called universal if there exists an element x0 ∈ X – called universal for (Tn ) – such that {Tn x0 : n ∈ N} forms a dense set in Y . In case of a single continuous selfmap T of X, the map T is called universal whenever its sequence of iterates (T n ) is universal (where we put T n := T ◦ T n−1 for n ∈ N and T 0 := idX ), i.e. if there exists an element in X which has a dense orbit under T . Often it is assumed that the underlying topological space also has a linear structure: if X is a topological vector space and T : X → X is a linear continuous map, the words hypercyclic and universal are synonymous. In case that X is a separable complete metric space without isolated points, Birkhoff’s transitivity theorem states that T is universal if and only if T is topologically transitive, i.e. for all nonempty open sets U, V ⊂ X there exists some N ∈ N with T N (U ) ∩ V = ∅ (see e.g. [11, Theorem 1.16]). Moreover, T is called mixing if for any pair U, V of nonempty open subsets of X there exists some N ∈ N such that T n (U ) ∩ V = ∅ for all n ≥ N . If X is a complete metric space without E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2017.03.009 0022-247X/© 2017 Published by Elsevier Inc.
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isolated points, T is called chaotic (in the sense of Devaney) if T is topologically transitive and has a dense set of periodic points (x ∈ X is called a periodic point of T if there exists some N ∈ N with T N x = x; in case of N = 1, x is called a fixed point of T ). More recently, combined universality properties of finite families of continuous mappings between topological spaces have been investigated. In 2007, J. Bès and A. Peris [7] as well as independently L. Bernal-González [4] have studied the concept of disjoint hypercyclicity. In general, for topological spaces X, Y1 , . . . , YN (N ∈ N) and continuous mappings Tj,n : X → Yj (j ∈ {1, . . . , N }, n ∈ N), the sequences (T1,n ), . . . , (TN,n ) are called disjoint universal whenever there exists some x0 ∈ X such that the joint orbit {(T1,n x0 , . . . , TN,n x0 ) : n ∈ N} is dense in the product Y1 × · · · × YN . Very recently, L. Bernal-González and the author [5] introduced the weaker property of simultaneous universality: for topological spaces X, Y and continuous mappings Tj,n : X → Y (j ∈ {1, . . . , N }, n ∈ N), the sequences (T1,n ), . . . , (TN,n ) are called simultaneous universal whenever there exists some x0 ∈ X such that the closure of the joint orbit {(T1,n x0 , . . . , TN,n x0 ) : n ∈ N} contains the diagonal {(y, y, . . . , y) : y ∈ Y } of Y N . In other words, if Y1 , . . . , YN are first-countable (if Y is first-countable, resp.), the disjoint (simultaneous, resp.) universality of (Tj,n )n∈N (1 ≤ j ≤ N ) means the existence of some x0 ∈ X enjoying the property that, for each tuple (y1 , . . . , yN ) ∈ Y1 × · · · × YN (for each point y ∈ Y , resp.), there is one common sequence (nk ) in N such that Tj,nk x0 → yj (Tj,nk x0 → y, resp.) for all j = 1, . . . , N as k → ∞. A classic example of a universal map is given by the backward shift operator on sequence spaces. Denoting by X the space c0 of all complex-valued null sequences or the space q of all q-summable complex-valued sequences (1 ≤ q < ∞) and considering the map B : X → X, B((xj )j∈N ) := (xj+1 )j∈N , S. Rolewicz [14] was able to prove in 1969 that for any λ ∈ C with |λ| > 1 (and, in fact, only for these λ), the weighted backward shift operator λB is hypercyclic on X. Moreover, it is well-known that in this case λB is even mixing and chaotic (see e.g. [11, Examples 2.32 and 2.38]). In this paper, we want to study simultaneous and disjoint universality of certain finite families of generalized backward shifts. The considered shifts will be associated to selfmappings of a corresponding topological space and it will be our aim to study the above mentioned universality properties with respect to dynamical properties of the generating selfmappings. Starting point for these investigations are the papers of L. BernalGonzález [3] and A. Peris [13]. Their most important results in view of our further studies will be stated in Section 2. Subsequently, in Section 3, it will be our aim to exceed the mentioned results of Bernal’s work [3] in view of the mixing property and to combine them with a theorem of Peris’ work [13] (which follows also from Bernal’s results) in order to obtain simultaneous universality of certain finite families of backward Φ-shifts. Finally, in Section 4, we will merge proof techniques of Bernal’s work on disjoint hypercyclic operators [4] and Peris’ work [13] in order to obtain disjoint universality. 2. Universal backward Φ-shifts and complex dynamics In 2005, L. Bernal-González [3] investigated universality of so-called backward Φ-shifts in a very general setting. In order to cite his main results, we introduce the following notation coined by him (cf. [3, Section 3]): for a Hausdorff topological space E and for each a ∈ E, we denote by σ(a) the set of sequences ending with a, that is, σ(a) := J∈N σJ (a), where σJ (a) := {(xj ) ∈ E N : xj = a for all j > J}. A standard sequence space (SSS) on E is defined as a set S ⊂ E N endowed with a topology such that there is a point a ∈ E which satisfies the following properties: (S1) The space S is Baire and second-countable. (S2) The topology on S is stronger than that inherited from the product topology on E N .
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(S3) The set σ(a) is a dense subset of S. (S4) For each J ∈ N, the topology of each σJ (a) inherited from S is the product topology. If S is an SSS on E and a ∈ E is a point satisfying (S3)–(S4), then a is called a distinguished point for S. Examples of SSSs are the spaces c0 and q (0 < q < ∞), where one can simply take E = C with the usual topology. More generally, if E is a separable Banach space over R or C, the space c0 (E) of all E-valued null sequences and the space q (E) (0 < q < ∞) of all q-summable E-valued sequences are also SSSs on E (cf. [3, p. 184]). Definition 2.1. For a topological space E, an SSS S on E and a continuous selfmap T : S → S, we say that T is a backward Φ-shift on S if there exists a selfmap f : E → E such that T = Bf on S, where Bf : E N → E N , Bf ((xj )j∈N ) := (f (xj+1 ))j∈N . The following characterization of Bf itself being a backward Φ-shift on the SSSs S = c0 (E) or q (E) (0 < q < ∞) (i.e. of Bf being S-invariant and continuous) was proved in [3, Theorem 3.5]: • Bf is a backward Φ-shift on c0 (E) if and only if f is continuous and f (0) = 0, • Bf is a backward Φ-shift on q (E) (0 < q < ∞) if and only if f is continuous and lim supt→0 f (t) E / t E < ∞. It is easy yo see that f is continuous on E whenever Bf is a backward Φ-shift on S (cf. [3], Remark 3.3.(1)). In order to state some of the main results of [3], we also introduce the following conditions: (S4*) Given J ∈ N, an open set U ⊂ S and a point α ∈ σJ (a) ∩ U , there exist open sets U1 , . . . , UJ , A in ∞ (n) E such that α ∈ j=1 Aj ⊂ U for all n > J, where
(n)
Aj
⎧ ⎪ ⎨ Uj (1 ≤ j ≤ J) := A (n + 1 ≤ j ≤ n + J) ⎪ ⎩ {a} (J < j ≤ n or j > n + J).
(S5) Given an open subset A ⊂ E containing a, there exists an open set U ⊂ S with (a, a, . . . ) ∈ U such that πj (U ) ⊂ A for all j ∈ N, where πj denotes the j-projection πj : S → E, πj ((xn )n∈N ) := xj . Condition (S4*) is stronger than condition (S4) (cf. [3, Remark 3.1]), and (S4*) as well as (S5) are satisfied for S = c0 (E) and S = lq (E) (0 < q < ∞) with a = 0 in case of a separable Banach space E (cf. [3, p. 184 and p. 192]). Finally, we introduce the following local weakly mixing property of sequences of continuous selfmappings of a topological space, which was coined by L. Bernal-González (cf. [3, Definition 2.1]) and which is related to the well-known notation of weakly mixing maps: for a topological space E, continuous mappings fn : E → E (n ∈ N) and a point a ∈ E, we say that (fn ) is weakly mixing at a if for all m ∈ N and all nonempty open sets V1 , . . . , Vm , A ⊂ E with a ∈ A there exists some N ∈ N satisfying fN (A) ∩ Vj = ∅ for all
j = 1, . . . , m.
A selfmap f of E is said to be weakly mixing at a if its sequence of iterates (f n ) is weakly mixing at a.
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Main results of [3] now read as follows (cf. [3, Theorem 4.3]): Theorem 2.2 (L. Bernal-González, 2005). i) Assume that Bf : S → S is a backward Φ-shift on a SSS S satisfying (S4*) with distinguished point a, in such a way that f (a) = a and f is weakly mixing at that point. Then Bf is universal. ii) Assume that Bf : S → S is a universal backward Φ-shift on a SSS S which satisfies property (S5). Then f is weakly mixing at the distinguished point. iii) Suppose that E is a separable Banach space, that S = c0 (E) or lq (E) (0 < q < ∞) and that f : E → E is continuous. In addition, we assume f (0) = 0 if S = c0 (E) and lim supt→0 f (t) E / t E < ∞ if S = lq (E). Then the backward Φ-shift Bf is universal if and only if f is weakly mixing at the origin. In 2003, A. Peris [13] characterized universality and chaos of backward Φ-shifts of the form Bp : X → X, where X stands for the space c0 or q (1 ≤ q < ∞) and p : C → C is a polynomial with p(0) = 0. In order to formulate his main results, we need some terminology from the theory of complex dynamics that investigates the long-time behaviour of certain discrete topological dynamical systems (see e.g. [8,12,15,16]). The systems which are studied are usually given by the iteration of an entire function or a rational function. In the following, we focus on the case of entire functions, mainly polynomials. The local behaviour of the considered functions near fixed points plays an important role in the study of the long-time behaviour of the corresponding dynamical systems. Let f be an entire function (not a polynomial of degree 0 or 1) and let z0 ∈ C be a fixed point of f . Then the multiplier of f at z0 is denoted and defined by λ := f (z0 ) and the fixed point z0 is called: superattracting if λ = 0, attracting if 0 < |λ| < 1, rationally indifferent if |λ| = 1 and λ is a root of unity, irrationally indifferent if |λ| = 1 but λ is not a root of unity, repelling if |λ| > 1. The Fatou set Ff of f is defined as the set of all points z ∈ C for which there exists a neighbourhood U of z such that {f n |U : n ∈ N} is a normal family in the space of continuous functions from U to the extended complex plane C∞ . The complement of Ff is called the Julia set of f and it is denoted by Jf . It is well-known that we have Ff n = Ff and hence Jf n = Jf for all n ∈ N. Moreover, (super-)attracting fixed points of f are always contained in the Fatou set of f , whereas repelling and rationally indifferent fixed points of f always belong to the Julia set of f . Irrationally indifferent fixed points of f can be contained in Ff or in Jf . A reason for splitting the complex plane into the two disjoint subsets Ff and Jf is the following: It is well-known that the sequence of iterates (f n ) behaves quite “wildly” on Jf whereas, on the other hand, the behaviour of (f n ) on Ff is quite “nice” and well-understood (cf. the classification theorem of components of the Fatou set in the above mentioned references). The “wild” behaviour of (f n ) on Jf can be described as follows: for each point z ∈ Jf and each arbitrarily small open neighbourhood U of z, it follows directly from Montel’s theorem (see e.g. [12, Theorem 3.7]) that the set C\ n∈N f n (U ) contains at most one element, i.e. every value in the complex plane, with at most one exception, is assumed on U under the iteration of f . For z ∈ C, the grand orbit of z under f is defined as the set GO(z, f ) := {w ∈ C : there exist m, n ∈ N0 with f m (z) = f n (w)}, and z is called exceptional under f if GO(z, f ) is finite. An application of Montel’s theorem yields that the set Ef of exceptional points under f contains at most one element (see e.g. [12, Lemma 4.6]). Furthermore, in case of polynomials, the following
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sharper version of the above statement concerning the “wild” behaviour of (f n) on Jf holds (see e.g. [12, Corollary 14.2 and the subsequent considerations]): Theorem 2.3 (Expanding property of the Julia set). Let f be a polynomial of degree ≥ 2. Moreover, let K ⊂ C be compact with K ∩ Ef = ∅ and let U ⊂ C be open with U ∩ Jf = ∅. Then there exists some N ∈ N such that f n (U ) ⊃ K holds for all n ≥ N . Now, let p : C → C be a polynomial with p(0) = 0 and let us consider the backward Φ-shift Bp : X → X, where X = c0 or q (1 ≤ q < ∞). Then Peris’ main result [13, Theorem 3.2] shows that Bp is universal if and only if the origin is contained in the Julia set of p (this statement also follows from the results in Bernal’s work [3], cf. [3, Remark 5.3]). Moreover, in case of X = c0 , universality of Bp is actually equivalent to the fact that Bp is chaotic (cf. [13, Remark 3.4]), and in case of X = q (1 ≤ q < ∞), at least the stronger condition that the origin is a repelling fixed point of p guarantees that Bp is chaotic (cf. [13, Proposition 3.3]). Finally, it is conjectured that the property 0 ∈ Jp is not sufficient for Bp being chaotic on q (1 ≤ q < ∞) (cf. [13, Conjecture 3.5]). 3. Mixing and simultaneous universal backward Φ-shifts In this section, we first want to prove a “mixing-analogon” of Theorem 2.2. On the one hand, this result will be interesting in itself, and, on the other hand, it will allow us to formulate a statement concerning simultaneous universality of rotations of certain backward Φ-shifts. In order to proceed as requested, we introduce the following local mixing property (analogously to the local weakly mixing property as defined in Section 2). Definition 3.1. Let E be a topological space, fn : E → E (n ∈ N) continuous mappings and a ∈ E. We say that (fn ) is mixing at a if for all m ∈ N and all nonempty open sets V1 , . . . , Vm , A ⊂ E with a ∈ A there exists some N ∈ N such that fn (A) ∩ Vj = ∅ (j = 1, . . . , m) holds for all n ≥ N . A selfmap f of E is said to be mixing at a if its sequence of iterates (f n ) is mixing at a. Then the following statements hold: Theorem 3.2. i) Assume that Bf : S → S is a backward Φ-shift on a SSS S satisfying (S4*) with distinguished point a, in such a way that f (a) = a and f is mixing at that point. Then Bf is mixing. ii) Assume that Bf : S → S is a mixing backward Φ-shift on a SSS S which satisfies property (S5). Then f is mixing at the distinguished point. iii) Suppose that E is a separable Banach space, that S = c0 (E) or lq (E) (0 < q < ∞) and that f : E → E is continuous. In addition, we assume f (0) = 0 if S = c0 (E) and lim supt→0 f (t) E / t E < ∞ if S = lq (E). Then the backward Φ-shift Bf : S → S is mixing if and only if f is mixing at the origin.
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Proof. i) Analogously to Theorem 4.3 in [3], it suffices to show the following more general statement: for an SSS S on E satisfying (S4*) for some distinguished point a ∈ E and a sequence (fn ) of selfmappings of E which is mixing at a and fulfils fn (a) = a for all n ∈ N such that the mappings Tn : S → S, Tn ((xj )j∈N ) := (fn (xj+n ))j∈N , are well-defined and continuous, the sequence (Tn ) is mixing. We follow very closely the proof of Theorem 4.3 in [3]. To this end, let U, V ⊂ S be nonempty and open. Due to (S3), we have U ∩ σ(a) = ∅ = V ∩ σ(a) so that there exist an integer J ∈ N as well as points a1 , . . . , aJ , b1 , . . . , bJ ∈ E with α ∈ U and β ∈ V , where α := (a1 , . . . , aJ , a, a, . . . )
and β := (b1 , . . . , bJ , a, a, . . . ).
According to α ∈ σJ (a) ∩ U , (S4*) implies the existence of finitely many open sets U1 , . . . , UJ , A in E ∞ (n) such that we have α ∈ j=1 Aj ⊂ U for all n > J, where
(n)
Aj
⎧ ⎪ ⎨ Uj := A ⎪ ⎩ {a}
(1 ≤ j ≤ J) , (n + 1 ≤ j ≤ n + J) (J < j ≤ n or j > n + J)
n > J.
Moreover, there exist open sets V1 , . . . , VJ ⊂ E with β ∈ V1 × · · · × VJ × {a} × {a} × · · · ⊂ V. As (fn ) is mixing at a, we can choose some N ∈ N with N > J such that fn (A) ∩ Vj = ∅
(j = 1, . . . , J)
holds for all n ≥ N . Hence, for each n ≥ N , there exist J points t1,n , . . . , tJ,n ∈ A satisfying fn (tj,n ) ∈ Vj (j = 1, . . . , J). For each n ≥ N , we put xn := (xm,n )m∈N , where
xm,n :=
⎧ ⎪ ⎨
am
⎪ ⎩
tm−n,n a
Then, for each n ≥ N , we have xn ∈
∞ j=1
(1 ≤ m ≤ J) (n + 1 ≤ m ≤ n + J) (J < m ≤ n or m > n + J). (n)
Aj
⊂ U as well as
Tn xn = fn (xm+n,n ) m∈N = fn (t1,n ), . . . , fn (tJ,n ), fn (a), fn (a), ... = fn (t1,n ), . . . , fn (tJ,n ), a, a, ... ∈ V1 × · · · × VJ × {a} × {a} × · · · ⊂ V , and thus Tn xn ∈ Tn (U ) ∩ V = ∅. ii) Analogously to Theorem 4.5 in [3], it suffices to show the following more general statement: for an SSS S on E satisfying (S5) for some distinguished point a ∈ E and a sequence (fn ) of selfmappings of E such that the mappings Tn : S → S, Tn ((xj )j∈N ) := (fn (xj+n ))j∈N ,
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are well-defined, continuous and mixing, the sequence (fn ) is mixing at a. We follow very closely the proof of Theorem 4.5 in [3]. To this end, let V1 , . . . , Vm , A be nonempty open subsets of E with a ∈ A. According to (S5), there exists an open set U ⊂ S with (a, a, a, . . . ) ∈ U such that πj (U ) ⊂ A for all j ∈ N. Putting W := V1 × · · · × Vm × E × E × · · · , we obtain from (S2) that W is open in S. Thus, since (Tn ) is mixing, there exists some N ∈ N with Tn (U ) ∩ W = ∅ for all n ≥ N . Hence, for each n ≥ N , there exists some xn = (xj,n )j∈N ∈ U with W Tn xn = (fn (xj+n,n ))j∈N , which implies fn (xj+n,n ) ∈ Vj for all j = 1, . . . , m. As we also have xj+n,n = πj+n (xn ) ∈ πj+n (U ) ⊂ A (j ∈ N, n ≥ N ), we finally obtain that fn (xj+n,n ) ∈ fn (A) ∩ Vj = ∅ (j = 1, . . . , m) holds for all n ≥ N . iii) Under the assumptions of iii), the considerations in Section 2 show that Bf is a backward Φ-shift on S and that (S4*) and (S5) are satisfied with a = 0 for S. Hence, the assertion follows directly form parts i) and ii). 2 A first application to the framework of complex dynamics is: Theorem 3.3. Let p : C → C be a polynomial of degree ≥ 2 with p(0) = 0, X = c0 or q (0 < q < ∞) and consider the backward Φ-shift Bp : X → X. Then Bp is mixing if and only if 0 ∈ Jp . Proof. Due to the assumptions and the considerations in Section 2, we see that Bp is indeed a backward Φ-shift on X (for the q case, observe that due to p(0) = 0 the function z → p(z)/z can be extended to an entire function). If Bp is mixing, then Bp is topologically transitive so that Birkhoff’s transitivity theorem yields that Bp is universal. Hence, Peris’ result [13, Theorem 3.2] implies that the origin is contained in the Julia set of p (see also [3, Remark 5.3] in case of X = q with 0 < q < 1). Now let 0 ∈ Jp . In order to show that Bp is mixing, according to Theorem 3.2 i) it suffices to show that p is mixing at the origin. To this end, let m ∈ N and let V1 , . . . , Vm , A ⊂ C be nonempty open with 0 ∈ A. We choose points zj ∈ Vj \Ep (j = 1, . . . , m) and we put K := {z1 , . . . , zm }. As we have 0 ∈ A ∩ Jp = ∅, the expanding property of the Julia set (Theorem 2.3) yields the existence of some N ∈ N with pn (A) ⊃ K for all n ≥ N . Hence, for each n ≥ N , we have zj ∈ K ∩ Vj ⊂ pn (A) ∩ Vj = ∅ (j = 1, . . . , m), i.e. p is mixing at the origin.
2
Remark 3.4. i) Due to Peris, the equivalent properties stated in Theorem 3.3 hold if and only if Bp is universal (cf. [13, Theorem 3.2] and see also [3, Remark 5.3] in case of X = q with 0 < q < 1), and, in case of X = c0 , they are also equivalent to the fact that Bp is chaotic (cf. [13, Remark 3.4]). ii) Considering the set X = c0 (C∞ ) of all sphere-valued null sequences endowed with the metric d : X → X, d (xj )j∈N , (yj )j∈N := sup χ(xj , yj ), j∈N
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where χ denotes the chordal distance on C∞ , we obtain that X is an SSS on C∞ satisfying (S4*) for the distinguished point 0. In particular, (X, d) is a separable complete metric space without isolated points. As the Fatou set and the Julia set of rational functions can be defined analogously as in the case of entire functions and as the expanding property of the Julia set also holds for rational functions of degree ≥ 2 (see e.g. [12, Corollary 14.2 and the subsequent considerations]), the proof of Theorem 3.3 also yields the following result: Let R : C∞ → C∞ be a rational function of degree ≥ 2 with R(0) = 0, let X = c0 (C∞ ) and consider the backward Φ-shift BR : X → X. Then BR is mixing if and only if 0 ∈ JR . Observe that, indeed, BR is well-defined on X and that Peris’ result [13, Theorem 3.2] also holds for rational functions (cf. the proofs of [3, Theorem 3.5 (3)] and [13, Theorem 3.2]). Finally, we obtain the following result on simultaneous universality: Corollary 3.5. Let p : C → C be a polynomial of degree ≥ 2 with p(0) = 0 and let X = c0 or q (0 < q < ∞). Moreover, let λ1 , . . . , λN (N ∈ N) be unimodular scalars. Then the backward Φ-shifts Bp , Bλ1 p , . . . , BλN p are simultaneously universal on X if and only if 0 ∈ Jp . Proof. If Bp , Bλ1 p , . . . , BλN p are simultaneously universal, then in particular Bp is universal so that Peris’ result [13, Theorem 3.2] implies that the origin is contained in the Julia set of p (see also [3, Remark 5.3] in case of X = q with 0 < q < 1). On the other hand, if 0 ∈ Jp , Theorem 3.3 yields that Bp is mixing. Hence, in view of Birkhoff’s transitivity theorem, Bp is hereditarily universal, i.e. for each strictly increasing sequence (nk ) in N, the sequence (Bpnk )k∈N is universal. Hence, an application of Proposition 4.1 (b) of [5] implies that the rotations Bp , λ1 Bp (= Bλ1 p ), . . . , λN Bp (= BλN p ) are simultaneously universal (observe that the assumption of linearity in Proposition 4.1 (b) of [5] actually is not needed). 2 In view of Remark 3.4 ii), also the following statement holds: Let R : C∞ → C∞ be a rational function of degree ≥ 2 with R(0) = 0 and let X = c0 (C∞ ). Moreover, let λ1 , . . . , λN (N ∈ N) be unimodular scalars. Then the backward Φ-shifts BR , Bλ1 R , . . . , BλN R are simultaneously universal on X if and only if 0 ∈ JR . 4. Disjoint universal backward Φ-shifts In [4], the question was posed if for a universal map T the iterates T, T 2 are always disjoint universal (cf. [4, p. 129]). It was shown in [6] that this does not hold in general: there exists a hypercyclic linear continuous selfmap T of 2 such that T, T 2 are not disjoint hypercyclic (cf. [6, p. 855]). Moreover, in [5] it was remarked that in this case T, T 2 are even not simultaneously hypercyclic (cf. [5, Remark 5.2.2]). However, there exist examples of universal maps T such that T, T 2 are disjoint universal. For instance, Bernal has shown that for X = c0 or q (1 ≤ q < ∞), N ∈ N and a bounded sequence a = (aj )j∈N in C\{0} fulfilling the property that ⎧ ⎨ sup
min
n∈N ⎩
⎧ n ⎨
⎩
j=1
|aj | ,
2n
j=1
|aj | , . . . ,
Nn
j=1
⎫⎫ ⎬⎬ |aj | = ∞, ⎭⎭
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the weighted backward shift operators Ta , Ta2 , . . . , TaN are disjoint hypercyclic on X, where Ta : X → X, Ta ((xj )j∈N ) := (aj+1 xj+1 )j∈N (cf. [4, Proposition 5.7]). Furthermore, an example of disjoint hypercyclic weighted backward shifts with unbounded weight sequences is given by the pair D, D2 , where D denotes the differentiation operator f → f on the space H(C) of all entire functions on the complex plane, which may be viewed as the weighted backward shift (c0 , c1 , c2 , . . .) → (c1 , 2c2 , 3c3 , . . .) when considering H(C) as the space of all sequences (cj ) in C with lim sup |cj |1/j → 0 (cf. [7, Proposition 3.3], see also [2,10]). In the framework of polynomial backward Φ-shifts, the following holds: Theorem 4.1. Let p : C → C be a polynomial of degree ≥ 2 such that the origin is a repelling fixed point of p and let X = c0 or q (0 < q < ∞). Then for each N ∈ N, the backward Φ-shifts Bp , Bp2 , . . . , BpN are disjoint universal on X. Proof. The proof will follow the structure of Bernal’s proof of Proposition 5.7 in [4]. In addition, we will also mimic a technique of proof of Peris, cf. the proof of Proposition 3.3 in [13]. To simplify matters, we only prove the assumption in case of N = 2; the general case can be shown similarly. According to the disjoint blow-up/collapse criterion (see e.g. [7, Proposition 2.4]), it suffices to show that there exists a dense set D0 ⊂ X such that for all x, y, z ∈ D0 , there exist a strictly increasing sequence (nk ) in N and a sequence (xk ) in X satisfying: n (i) Bpnk x −→ 0 and Bp2 k x −→ 0 as k → ∞, n (ii) xk −→ 0, Bpnk xk −→ y and Bp2 k xk −→ z as k → ∞. Choosing D0 := (xk ) ∈ X : there exists k0 ∈ N such that xk = 0 for all k ≥ k0 and xk ∈ / Ep2 for all k ∈ N , we obtain that D0 is dense in X (observe that Ep2 is empty or a singleton, see Section 2). As each element of D0 has only finitely many nonzero entries and due to p(0) = 0, it follows that for each x = (xk ) ∈ D0 , there exists some m ∈ N such that we have Bpn x = 0 and (Bp2 )n x = 0 for all n ≥ m, i.e. condition (i) is fulfilled for each strictly increasing sequence (nk ) in N. In order to show (ii), let y, z ∈ D0 , and let M ∈ N with y = (y1 , . . . , yM , 0, 0, . . .) as well as z = (z1 , . . . , zM , 0, 0, . . .). As the origin is a repelling fixed point of p, we can fix some r > 1 with r < |p (0)|. Then there exists a δ > 0 such that we have p(U ) ⊃ rU for all open discs U centred at 0 with radius less than δ (cf. the proof of [13, Proposition 3.3]). It follows that pn (U ) ⊃ rn U
(1)
holds for all n ∈ N and all open discs U centred at 0 with radius less than δ/rn−1 . In the following, for ε > 0, we write Uε (0) := {w ∈ C : |w| < ε}. Due to 0 ∈ Jp = Jp2 and in view of the definition of D0 (observe that Ep ⊂ Ep2 ), it follows from the expansion property of the Julia set (Theorem 2.3) that there exists some m ∈ N with pm (Uδ (0)) ⊃ {y1 , . . . , yM }
and
p2m (Uδ (0)) ⊃ {z1 , . . . , zM }.
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Thus, we can find points a1,1 , . . . , aM,1 , b1,1 , . . . , bM,1 ∈ Uδ (0) with pm (al,1 ) = yl
p2m (bl,1 ) = zl ,
and
l = 1, . . . , M.
(2)
According to (1), we have pm (Uδ/rm (0)) ⊃ Uδ (0)
p2m (Uδ/rm (0)) ⊃ p2m (Uδ/r2m (0)) ⊃ Uδ (0)
and
so that there exist a1,2 , . . . , aM,2 , b1,2 , . . . , bM,2 ∈ Uδ/rm (0) with pm (al,2 ) = al,1
p2m (bl,2 ) = bl,1
and
l = 1, . . . , M.
Inductively, we can find points al,k , bl,k ∈ C with |al,k |, |bl,k | < δ/r(k−1)m as well as pm (al,k+1 ) = al,k
p2m (bl,k+1 ) = bl,k ,
and
l = 1, . . . , M,
k ∈ N.
(3)
In particular, each sequence (al,k )k∈N , (bl,k )k∈N converges to 0. Due to (2) and (3), we have pm·k (al,k ) = (pm )k (al,k ) = yl
and
p2m·k (bl,k ) = (p2m )k (bl,k ) = zl
and
p2m·2 (bl,2k ) = zl
and hence k
pm·2 (al,2k ) = yl
k
(4)
as well as k
pm·2 (bl,2k ) = p2m·2
k−1
k−1
(bl,2k ) = (p2m )2
(bl,2k ) = bl,2k−1
(5)
for all l = 1, . . . , M and all k ∈ N. Now, we define nk := m · 2k , k ∈ N, and for each k ∈ N with nk ≥ M , we put xk := 0, . . . , 0, a1,2k , . . . , aM,2k , 0, . . . , 0, b1,2k , . . . , bM,2k , 0, 0, . . . . nk −M times
nk times
Finally, we choose k0 ∈ N with nk ≥ M for all k ≥ k0 and we consider the sequence x := (xk )k≥k0 in X. As each sequence (al,k )k∈N , (bl,k )k∈N converges to 0, we obtain that xk −→ 0
as
k → ∞.
Moreover, in view of (4) and (5), we have
Bp2
nk
xk = p2nk (b1,2k ), . . . , p2nk (bM,2k ), p2nk (0), p2nk (0), . . . k k = p2m·2 (b1,2k ), . . . , p2m·2 (bM,2k ), 0, 0, . . . = (z1 , . . . , zM , 0, 0, . . .) = z
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as well as Bpnk xk =
pnk (a1,2k ), . . . , pnk (aM,2k ), pnk (0), . . . , pnk (0), pnk (b1,2k ), . . . , pnk (bM,2k ), pnk (0), pnk (0), . . . k
k
k
k
pm·2 (a1,2k ), . . . , pm·2 (aM,2k ), 0, . . . , 0, pm·2 (b1,2k ), . . . , pm·2 (bM,2k ), 0, 0, . . . = y1 , . . . , yM , 0, . . . , 0, b1,2k−1 , . . . , bM,2k−1 , 0, 0, . . . =
−→ (y1 , . . . , yM , 0, 0, . . .) = y
as
k → ∞,
as required. 2 Remark 4.2. i) In view of Remark 3.4 ii), also the following statement holds: Let R : C∞ → C∞ be a rational function of degree ≥ 2 such that the origin is a repelling fixed point of R and let X = c0 (C∞ ). Then for each 2 N N ∈ N, the backward Φ-shifts BR , BR , . . . , BR are disjoint universal on X. ii) The question arises if the statement of Theorem 4.1 is in fact an equivalence, i.e. if the fixed point of p at the origin must be repelling whenever Bp , Bp2 , . . . , BpN are disjoint universal (clearly, in this case, Bp is universal so that we obtain at least that 0 belongs to the Julia set of p). In any case, the proof of Theorem 4.1 crucially uses the fact that 0 is indeed a repelling fixed point of p – namely property (1) allows us to work with a fixed integer m ∈ N in each step of the inductive construction (this is not ensured by only applying the expanding property of the Julia set) which guarantees that the points bl,2k fulfil equation (5) that is needed to obtain the convergence Bpnk xk −→ y as k → ∞ in the last step of the proof. So we conclude with the following conjecture (cf. [13, Conjecture 3.5]): Conjecture 4.3. Let p : C → C be a polynomial of degree ≥ 2 with p(0) = 0 and let X = c0 or q (0 < q < ∞). Then the backward Φ-shifts Bp , Bp2 , . . . , BpN are disjoint universal on X for each N ∈ N if and only if 0 is a repelling fixed point of p. Acknowledgments The author thanks the referee for his/her helpful suggestions concerning the presentation of this paper. The author has been supported by DFG-Forschungsstipendium JU 3067/1-1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
F. Bayart, É. Matheron, Dynamics of Linear Operators, Cambridge University Press, Cambridge, 2009. L. Bernal-González, Universal functions for Taylor shifts, Complex Var. Theory Appl. 31 (1996) 121–129. L. Bernal-González, Backward Φ-shifts and universality, J. Math. Anal. Appl. 306 (2005) 180–196. L. Bernal-González, Disjoint hypercyclic operators, Studia Math. 182 (2007) 113–131. L. Bernal-González, A. Jung, Simultaneous universality, submitted for publication, available from https://arxiv.org/abs/ 1701.07311. J. Bès, Ö. Martin, A. Peris, Disjoint hypercyclic linear fractional composition operators, J. Math. Anal. Appl. 381 (2011) 843–856. J. Bès, A. Peris, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336 (2007) 297–315. L. Carleson, T.W. Gamelin, Complex Dynamics, Springer, New York, 1993. K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999) 345–381. K.-G. Grosse-Erdmann, Rate of growth of hypercyclic entire functions, Indag. Math. (N.S.) 11 (2000) 561–571. K.-G. Grosse-Erdmann, A. Peris, Linear Chaos, Springer, London, 2011. J. Milnor, Dynamics in One Complex Variable, 2nd edition, Vieweg, Braunschweig-Wiesbaden, 2000. A. Peris, Chaotic polynomials on Banach spaces, J. Math. Anal. Appl. 287 (2003) 487–493.
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[14] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969) 17–22. [15] D. Schleicher, Dynamics of entire functions, in: Holomorphic Dynamical Systems, in: Lecture Notes in Math., vol. 1998, Springer, Berlin, 2010, pp. 295–339. [16] N. Steinmetz, Rational Iteration, de Gruyter, Berlin, 1993.