局部緊緻群上的權重平移算子

國立臺灣大學理學院數學系 碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

局部緊緻群上的權重平移算子

Weighted translation operators on locally compact groups

陳奎佑 Kui-Yo Chen

指導教授：陳怡全 博士, 陳俊全 博士

Advisor: Yi-Chiuan Chen, Ph.D., Chiun-Chuan Chen, Ph.D.

中華民國 106 年 5 月

May 2017

誌謝

首先我要感謝陳怡全與陳俊全兩位指導老師給我不少的建議，無論是研究上或寫作上，

這篇論文的很多靈感也都是來自他們的啟發。特別是陳怡全老師建議我考慮已知的非週期性 元素的超循環權重平移算子的共通性質，而他的建議引導出我論文的第二章。也很感謝陳中 川老師引領我進入算子理論和推薦我不少對我影響重大的書籍。也感謝陳其誠老師在 p-adic 方面與撰寫論文技巧方面給我一些幫助。

另外，我還想感謝我的很多同學們：李俊緯、郭子模、胡亦行、黃子豪、楊大緯、吳宗 堂。他們在多次的談話中給了我不少靈感。還要特別是李俊緯同學在 latex 與寫作方面也給了 我很多的幫助。

最後要感謝我爸媽，感謝他們的支持，包含經濟上的支持和興趣上的支持，讓我有機會 在數學的環境裡自由發展。

中文摘要

我們主要探討的問題是「局部緊緻群上是否存在超循環權重平移算子？」。首先我們刻 劃了非週期性元素的一些等價條件，並給出非週期性元素的超循環權重平移算子存在性。另 一方面，我們也給出一些週期性元素的權重平移算子是超循環算子的必要條件。

關鍵詞：非週期性、週期性、超循環算子、權重平移算子、局部緊緻群、 空間。

Abstract

Our main question is ``Does there exist a hypercyclic weighted translation operator on locally compact groups?''. In this article, we give several equivalent characterization of aperiodicity of an element on locally compact group. If is an aperiodic element, we show that there exists a frequently hypercyclic weighted translation . If is a periodic element, we give necessary conditions of hypercyclic weighted translation .

Keywords: Aperiodicity, periodicity, hypercyclicity, weighted translation operator, locally compact group, -space.

目 錄

口試委員會審定書 ………... i

誌謝 ……….…….. ii

中文摘要 ………..…...….. iii

Abstract ………..…..…….. iv

1 Introduction ……….… 1

2 Weighted translation operators with aperiodic elements ……….... 3

2.1 Notations ……….. 3

2.2 Equivalent statements of aperiodicity ……….…… 4

2.3 Existence of hypercyclic weighted translations …………...…………... 12

3 Weighted translation operators with periodic elements ……….… 16

3.1 Notations ………..…..…. 16

3.1.1 Notation for p-adic analysis ………..………. 17

3.2 Homogeneous equidistribution on compact groups ………... 18

3.3 A necessary condition of hypercyclic weighted translation operators ... 22

3.4 Some properties of hypercyclic weighted translations on p-adic ……... 25

Bibliography ……….. 27

Chapter 1 Introduction

Let G be a locally compact group and let a be an element of G. The main aim of this article is to investigate the existence of hypercyclic weighted translation operators associated to a and try to construct the hypercyclic weighted translation operators if such operators are known to exist. The goal has been achieved in the case where a is aperiodic. The details in the aspect makes up the content of Part I (Chapter2) of this article. In Part II (Chapter 3) we prove several necessary conditions for the existence of hypercyclic weighted translation operators associated to a periodic a.

This article is inspired by [5], and also [3, 4], which characterize the chaoticity and hypercyclicity of a weighted translation operator on the L^p space of a locally compact group.

An operator T on a Banach space X is called hypercyclic if there exists a vector x ∈ X such that its orbit

orb(T, x) := {Tⁿx|n ∈ N}

is dense in the whole space X. In this article, we focus on weighted translation operators.

The weighted translation operator T_a,w is a bounded linear self-map on the Banach space L^p(G) (by using the right Haar measure on G), for some p ∈ [1, ∞), defined by

T_a,w(f )(x) := w(x)f (xa⁻¹),

where the weight w is a bounded continuous function from G to (0, ∞). We denote T_a,1 by T_aso that T_aw is the function w translation by a, while T_a,w is a weighted translation operator.

We call an element a of G torsion if it has finite order. An element a is periodic if the closed subgroup G(a) generated by a (i.e. G(a) = < a >) is compact in G. An element a is aperiodic if it is not periodic. Such classification is meaningful. For example, we have the following important result.

Lemma. [5, Lemma 1.1.] Let G be a locally compact group and let a ∈ G be a torsion element. Then any weighted translation T_a,w : L^p(G) → L^p(G) is not hypercyclic, for 1 ≤ p < ∞.

[5, Lemma 1.1] gives the non-existence of hypercyclic weighted translations when the element a is torsion. So the question of the existence of hypercyclic weighted translations will be focused on the case in which a is non-torsion, this is, a is either non-torsion periodic or a is aperiodic.

All examples of hypercyclic weighted translation operators in previously known lit- erature are all associated to aperiodic a. One of the most important concrete examples is the weighted backward shift operator on `^p(Z) [7, Example 4.15. p.102] and there are also some classical analogous examples relevant to semigroups [12]; in fact, it can correspond to our case for several admissible weights by conjugation. In Chapter2, we prove the following main theorem of this article, which is also labeled as Theorem2.3.2.

Theorem 1 (Theorem 2.3.2). Let G be a second countable locally compact group and let a be an aperiodic element in G, then there exists a weighted translation operator T_a,w which is mixing, chaotic and frequently hypercyclic on L^p(G) for all p ∈ [1, ∞), simultaneously.

The main theorem is proved by showing that for every aperiodic a, there always exists an weighted translation associated to a, which satisfies Frequent Hypercyclicity Criterion (see Section 2.1). In general, Frequent Hypercyclicity Criterion is a strong condition that implies the hypercyclicity (see Section2.1).

The existence of hypercyclic weighted translation operators associated to non-torsion periodic element is still an open problem, even no single non-torsion periodic a is known to be have a hypercyclic weighted translation operator.

Note that in Chapter 2 we don’t need G to be Hausdorff, while in Chapter 3 we assume G is.

In Section 3.3, we find a necessary condition of hypercyclic weighted translation operators for non-torsion periodic element, which is also labeled as Theorem 3.3.1.

Theorem 2 (Theorem3.3.1). Let G be a compact Hausdorff group, and a be an element in G such that G = < a >. If T_a,w is hypercyclic on L^p(G) for some p ∈ [1, ∞), then

Z

G

ln w = 0.

In Section 3.4, we focus on the groups Zp and Qp (note that any non-zero element in these group are non-torsion periodic). And we also find a necessary condition of hypercyclic weighted translation operators, which is also labeled as Corollary3.4.2and Corollary3.4.3.

Theorem 3 (Corollary 3.4.2and Corollary 3.4.3). Let G = Zp or Qp. If w is a locally constant weight for the weighted translation operator T_a,w, then T_a,w is not hypercyclic on L^p0(G) for any prime number p, any p⁰ ∈ [1, ∞) and any a ∈ G.

Chapter 2

Weighted translation operators of aperiodic elements

2.1 Notations and Preliminary

An operator T is called weakly mixing if T ⊕ T defined on X × X is hypercyclic. An operator T is called mixing if for any nonempty open subsets U, V in X, there exists N ∈ N such that TⁿU ∩ V 6= ∅ for all n > N . An operator T is called chaotic if it is hypercyclic and the set of periodic points is dense. An operator T is called frequently hypercyclic if there is some x ∈ X such that for any nonempty open subset U of X, nk = O(k), where nk is a strictly increasing sequence of integers such that T^nkx is k-th element lying in U (by [7, Proposition 9.3.], this is an equivalent statement of [7, Definition 9.2.]).

The graph below displays some relations between these dynamical properties which have been discussed, see [7]:

Frequently Hypercyclicity

Chaos Frequent

Hypercyclicity Criterion

Weakly Mixing

Mixing

Hypercyclicity

The graph above shows the three conditions in the second column which are implied by Frequent Hypercyclicity Criterion (in left) follows from a useful theorem.

Theorem. (Frequent Hypercyclicity Criterion, [7, Theorem 9.9 and Proposition 9.11.]).

Let T be an operator on a separable Fr´echet space X. If there is a dense subset X₀ of X and a map S : X₀ → X₀ such that, for any x ∈ X₀,

(1)

∞

P

n=0

Tⁿx converges unconditionally,

(2)

∞

P

n=0

Sⁿx converges unconditionally, (3) T Sx = x,

then T is frequently hypercyclic. Moreover, T is also chaotic and mixing. In particular, it is also weakly mixing and hypercyclic.

In the paragraphs above, only general conclusions are described; in the following discussions, we will focus on the weighted translation operators.

To analyze the weighted translation operators, we would like to consider more topo- logical properties of the elements of the locally compact group in next section.

Lemma. [5, Lemma 2.1.] An element a in a second countable group G is aperiodic if, and only if, for each compact subset K ⊆ G, there exists N ∈ N such that K ∩Kaⁿ = ∅ for n > N .

In [5, Lemma 2.1], they gave an equivalence statement of aperiodicity when G is a second countable locally compact Hausdorff group. We give another equivalence statement when G is second countable, which we define in Definition2.2.8 and named as terminal pair. And in section2.3, we will use it to prove the existence of hypercyclic weighted translation operators.

2.2 Equivalent statements of aperiodicity

Proposition 2.2.1. Let G, G⁰ be locally compact groups, and φ(a) be an aperiodic element of G⁰, where φ : G → G⁰ is a continuous homomorphism. Then a is also an aperiodic element of G. In other words, continuous homomorphisms pull back the aperiodicity.

Proof. If such a is periodic, then < a > is a compact group, and so is φ(< a >); but φ(a) ∈ φ(< a >) derives a contradiction, since there is no aperiodic element in compact group.

Let G be a topological group. we define the Hausdorffication of G by the natural continuous quotient map π : G → eG, where eG := G/{e} and e denotes the identity element of G. (This may be related to the “Hausdorffication” which is defined as a left adjoint of forget functor in general topology). The reason that we consider the Hausdorffication is lots of statements in this paper without the assumption that G is Hausdorff. Actually, we first assume G is Hausdorff in proving. Next, we prove the non-Hausdorff case by considering its Hausdorffication and show it can preserve or pull back some conclusion we want.

The claim form (1) to (4) in next Lemma are basic arguments for topological groups.

But it is not so easy to find the reference for the non-Hausdorff group. For the reader’s convenience, we will state it below with simple proof for each claim.

Lemma 2.2.2. Let G be a topological group, and let π : G → eG be its Hausdorffication.

Then we have the following statements:

(1) Each open or closed subset of G is a union of the cosets of {e}.

(2) There’s a one to one correspondence between the set of open (closed) of G and eG’s.

In particular, G is first (second) countable if and only if eG also, and eG is Hausdorff.

(3) π is an open, closed and proper mapping. In particular, G is locally compact iff eG also. On the other hand, the one to one correspondence in (2), not only just closed sets but also closed compact sets (since π is proper).

(4) Let Y be an arbitrary Hausdorff topology space, then Hom(G, Y ) ∼= Hom( eG, Y ).

(i.e. There’s a natural one to one correspondence between the set of continuous functions from G to Y and eG’s. Equivalently, one can say that any continuous function from G to Y factor through eG. In other words, this is the universal property of Hausdorffication.)

(5) a is an aperiodic in G iff π(a) is an aperiodic in eG.

proof of (1). Since Sx = Sx for any S ⊆ G and x ∈ G. Choose S = {e} and x ∈ {e}, we have {x} = {e}x = {e}x = {e} (last equality follows by x ∈ {e} and {e} is a subgroup of G). This shows that {e} is indiscrete topology space and so is all cosets of {e}. This result implies the statement we are looking for.

proof of (2). This correspondence is given by (U 7→ π(U )) and ( eU 7→ π⁻¹( eU )) for U is open in G and eU is open in eG. To verify that these two maps compose to identity for two sides, we only need to say π⁻¹π(U ) ⊆ U for any U is open in G, since the others are relatively obvious. Assume there exists x ∈ π⁻¹π(U ) \ U , that is, π(x) ∈ π(U ). This means that there is some y ∈ U such that π(x) = π(y). Hence xy⁻¹ ∈ Ker π = {e} and x ∈ {e}y. But by (1), {e}y ⊆ U , since y ∈ U . A contradiction occurs as this implies x ∈ U ∩ π⁻¹π(U ) \ U = ∅.

proof of (3). The openness and closedness follow by (1) and (2) immediately. For the properness, let eK be compact in eG. We will check that π⁻¹( eK) is also compact. Let {Uα} be a open covering of π⁻¹( eK). By correspondence, {π(Uα)} be a open covering of K. Hence it induce a finite subcovering {π(Ue _i)}. Then we can check the corresponding finite collection {U_i} is actually a covering of π⁻¹( eK).

proof of (4). Let w ∈ Hom(G, Y ). We define w(e x) := w(x) wheree ex = {e}x, then we is a well-defined continuous function on eG. On the other hand, let w ∈ Hom( ee G, Y ).

We have w := w ◦ π, it is also a well-defined continuous function on G. It is easy toe check that these two mappings between Hom(G, Y ) and Hom( eG, Y ) are inverse to each others.

proof of (5). If π(a) is a periodic, then < π(a) > is compact and so is π⁻¹(< π(a) >), since π is proper. However, a ∈ π⁻¹(< π(a) >) which displays a contradiction. The other side follows from Proposition2.2.1 immediately.

The idea of the proofs of Proposition2.2.3, Lemmas2.2.4and 2.2.5and Proposition 2.2.6 follows from [5, Lemma 2.1].

Proposition 2.2.3. Let G be a locally compact group, and let a ∈ G satisfy the following properties: for any compact subset K of G, there exists N ∈ N such that K ∩ Kaⁿ = ∅ for n > N . Then a is an aperiodic element in G.

Proof. Suppose a is a periodic element. Then the closed subgroup G(a) generated by a is compact. Now we set K = G(a), we have K ∩ Kaⁿ = G(a) 6= ∅ for all n ∈ Z, which is the negation of sufficient condition.

Lemma 2.2.4. Let G be a first countable locally compact Hausdorff group with an aperiodic element a in G. Then G(a) is a second countable compactly generated abelian group.

Proof. Since a is an aperiodic element, G(a) is a non-compact closed abelian subgroup of G (the commutativity follows from net arguement and Hausdorffness of G(a)). More- over, by [8, Theorem 5.14], there exists a compactly generated subgroup G⁰ of G which contains G(a), since {e, a} is a compact subset of G.

Firstly, we will check that it is second countable. Since G(a) is a first countable locally compact Hausdorff group, it is metrizable by [2, Birkhoff-Kakutani metrizable theorem]. On the other hand, since the set {a^j}_j∈Z is dense in G(a), G(a) is a separable metrizable space, as a result, it is second countable.

In the last step, we will check that it is compactly generated. Note that not every subgroup of a compactly generated group is compactly generated (even a closed sub- group), but in our case it holds. Let G⁰ be generated by the compact set K₀ where K₀ := V₀ for some open V₀ containing {e, a} (the existence follows from the proof in [8, Theorem 5.14]). We will claim that G(a) is generated by the compact set K₁ where K₁ := V₁ and V₁ := V₀ ∩ G(a).

Since the set {a^j}_j∈Z is dense in G(a), this implies that {V₁a^j}_j∈Z is a covering of G(a) (we will explain it in the Remark below). So G(a) ⊆ ∪_j∈ZV₁a^j ⊆ ∪_j∈ZK₁a^j ⊆ G(a) (last step follows from K1 ⊆ G(a)). That means that any x ∈ G(a) can be written as the form ka^j for some k ∈ K₁ and some j ∈ Z (that is, G(a) is generated by K1, since a is also in K₁).

Remark. To prove that {V₁a^j}_j∈Z is a covering of G(a), it is sufficient to say that every x ∈ G(a) which is a limit of a subsequence {a^nk} has been covered by {V₁a^nk} (we

view V₁ as the relative open neighborhood of e in G(a) here). Now choose V₂ the symmetric open neighborhood of e in V₁, then a^nk ∈ V₂x for k large enough, hence x ∈ V₂⁻¹a^nk = V₂a^nk ⊆ V₁a^nk, so the proof is completed.

Lemma 2.2.5. Let G be a first countable locally compact Hausdorff group with a ∈ G being an aperiodic element. Then G(a) is topologically isomorphic to Z.

Proof. According to Lemma2.2.4 and [8, Theorem 9.8], G(a) ∼= Rⁿ× Z^m× F for some n, m ∈ N and F being a compact group. The element a is identified as an element in Rⁿ× Z^m× F \ ({0} × {0} × F). Hence the cyclic subgroup < a > has no accumulation point, which implies G(a) being a discrete group, that is, G(a) =< a >, which is also isomorphic to Z.

Proposition 2.2.6. Let G be a first countable locally compact group with an aperiodic element a in G. Then, for any compact subset K of G, there exists N ∈ N such that K ∩ Kaⁿ = ∅ for n > N .

Proof. We first consider the case that G is Hausdorff.

If we assume there exists a compact set K such that K ∩ Kaⁿ 6= ∅ for infinitely many n’s, then for those n’s, aⁿ ∈ K⁻¹K; but it is impossible since there must admit a convergent subsequence in the compact set K⁻¹K. This contradicts with the Lemma 2.2.5, so the case of Hausdorff is verified.

Let G be a first countable and not necessary Hausdorff locally compact group and π : G → eG be its Hausdorffication. For any compact set K in G, π(K) is compact in G. So there exists N such thate

π(K) ∩ π(K)π(a)ⁿ= ∅ for n > N, then

K ∩ Kaⁿ ⊆ (π⁻¹◦ π)(K) ∩ (π⁻¹◦ π)(K)aⁿ

= π⁻¹(π(K)) ∩ π⁻¹(π(K)π(a)ⁿ)

= π⁻¹(π(K) ∩ π(K)π(a)ⁿ)

= π⁻¹(∅)

= ∅ for n > N.

Combining Proposition2.2.3and Proposition2.2.6, we have the following Theorem.

Theorem 2.2.7. Let G be a first countable locally compact group, then the following are equivalent:

(1) a ∈ G is an aperiodic element.

(2) For any compact subset K of G, there exists N ∈ N such that K ∩ Kaⁿ = ∅ for n > N .

Definition 2.2.8 (terminal pair). Let G be a locally compact group and a ∈ G. We say G has a terminal pair (A, B) w.r.t. a if there exists a pair of disjoint closed subsets (A, B) of G such that for any given compact subset K in G, we have

Kaⁿ⊆ A and

Ka⁻ⁿ ⊆ B for n large enough.

More intuitively, a shifts any compact subset positively (resp. negatively) into A (resp. B).

Example 2.2.9. One of the simplest cases is G = Z or R and a = 1, the terminal pair w.r.t. 1 can be given by (A, B) = ([100, ∞), (−∞, −100]).

Example 2.2.10. Let G be a general linear group GL(n, C), a ∈ G with some eigen- value λ such that |λ| 6= 1, then G admits a terminal pair w.r.t. a.

Proof. Without loss of generality, we can assume a itself is a Jordan form,

a =











λ ∗ · · · 0 ... . .. ... ...

0 · · · λ⁰ ∗ 0 · · · 0 λ⁰⁰









 ,

since we can act a conjugate automorphism on G.

Notation: Let any x ∈ G, we write x = [x₁|x₂|...|x_n], where x_i ∈ Cⁿ are the column of x. (Note that x_i will never be zero vector since x is invertible.)

Consider the map f : G → R, f (x) = ln kx1k. By the calculation, we have f (xaⁿ) = f (x) + n ∗ ln |λ|.

Set (A, B) = (f⁻¹([1, ∞)), f⁻¹((−∞, −1])), so for any compact subset K in G,

x∈Kinf f (xaⁿ) = ( inf

x∈Kf (x)) + n ∗ ln|λ|

and

sup

x∈K

f (xaⁿ) = (sup

x∈K

f (x)) + n ∗ ln|λ|, which imply (A, B) is a terminal pair w.r.t. a.

Proposition 2.2.11. Let G be a locally compact group and admit a terminal pair w.r.t.

a. Then a is an aperiodic element.

Proof. Suppose a is a periodic element but G also admits a terminal pair w.r.t. a, then G(a)aⁿ= G(a) for all n ∈ Z, which implies that A and B are not disjoint.

Proposition 2.2.12. Let G, G⁰ be locally compact groups, and let G⁰ admit a terminal pair w.r.t. φ(a), where φ : G → G⁰ is a continuous homomorphism. Then G also admits a terminal pair w.r.t. a. In other words, continuous homomorphisms pull back the terminal pair.

Proof. Let (A_G⁰, B_G⁰) be a terminal pair w.r.t. φ(a). Equivalently, for any given com- pact subset K⁰ in G⁰, we have

K⁰φ(a)ⁿ⊆ A_G⁰ and

K⁰φ(a)⁻ⁿ⊆ B_G⁰ for n large enough.

Set (A_G, B_G) = (φ⁻¹(A_G⁰), φ⁻¹(B_G⁰)). By continuity of φ, (A_G, B_G) is also a pair of disjoint closed sets in G. For any given compact subset K in G, φ(K) is also compact, we have

Kaⁿ⊆ φ⁻¹(φ(Kaⁿ)) = φ⁻¹(φ(K)φ(a)ⁿ) ⊆ φ⁻¹(A_G⁰) = A_G and

Ka⁻ⁿ⊆ φ⁻¹(φ(Ka⁻ⁿ)) = φ⁻¹(φ(K)φ(a)⁻ⁿ) ⊆ φ⁻¹(B_G⁰) = B_G for n large enough.

Example 2.2.13. Let G = S¹× R and a = (0, 1) ∈ S¹× R, where S¹ denotes the circle group. Then the natural quotient map φ : G → R sends a to 1, so terminal pair w.r.t.

a can be given by (A, B) = (φ⁻¹((−∞, −100]), φ⁻¹([100, ∞))).

Proposition 2.2.14. Let G be locally compact group and π : G → eG be the Hausdorf- fication of G. Then (A, B) is a terminal pair w.r.t. a iff ( eA, eB) = (π(A), π(B)) is a terminal pair w.r.t. π(a) with closed subsets A, B of G.

Proof. Suppose (A, B) is a terminal pair w.r.t. a. Then eA and eB are also disjoint closed since π is closed and they are union of cosets of {e}. Moreover, ( eA, eB) is also a terminal pair w.r.t. π(a). This follows by the one to one correspondence between the sets of closed compact subsets of G and eG (i.e. eK = π(π⁻¹( eK))). The other side follows from Proposition2.2.12.

Recall that any second countable locally compact Hausdorff groups are compatible with a proper “right” invariant metric d [11]. In this metric space the Heine-Borel property holds. Therefore, BR(x) := {y ∈ G|d(x, y) < R} is a precompact open ball, and B_R(xa) = B_R(x)a by right invariant.

Note that [11] said that there’s a proper “left” invariant metric say d_L, but it’s easy to induce a proper “right” invariant metric d by set d(x, y) := dL(x⁻¹, y⁻¹).

Theorem 2.2.15. Let G be a second countable locally compact Hausdorff group. Then the following are equivalent:

(1) a ∈ G is an aperiodic element.

(2) For any compact subsets K and K⁰ in G, d(Ka<sup>`</sup>, K<sup>0</sup>) → ∞ as ` → ∞.

Proof. (2⇒1). Choose K = K⁰ = {e}, then d(e, a<sup>`</sup>) → ∞ as ` → ∞. But if a is periodic, then G(a) is compact, hence it is bounded. This implies that aⁿare uniformly bounded, contradicting to d(e, a<sup>`</sup>) → ∞ as ` → ∞.

(1⇒2). Suppose d(Ka^{`</sup>, K<sup>0</sup>) is uniformly bounded by C > 0, then for any ` ∈ N, there exists k`∈ K, k<sup>0</sup><sub>`</sub> ∈ K⁰ such that d(k`a<sup>`}, k_{`</sub><sup>0</sup>) = d(k`a<sup>`</sup>k<sub>`}⁰⁻¹, e) < C. Since the closed ball centered at e is compact, so there’s a subsequence {`⁰} , k⁻¹ ∈ K⁻¹, k⁰ ∈ K⁰ and an element b in the ball such that

k_{`</sub><sup>−1</sup>0 → k<sup>−1</sup>, k<sub>`}⁰0 → k⁰,

k<sub>`</sub><sup>0</sup>a<sup>`0</sup>k⁰⁻¹<sub>`</sub>0 → b as `⁰ → ∞.

Hence a<sup>`0</sup> → k<sup>−1</sup>bk<sup>0</sup> as `⁰ → ∞. This contradicts the Lemma 2.2.5, since a^`0 never converges in G(a) ∼= Z.

Theorem 2.2.16. Let G be a second countable locally compact group. Then the fol- lowing are equivalent:

(1) a ∈ G is an aperiodic element.

(2) For any compact subset K of G, there exists N ∈ N such that K ∩ Kaⁿ = ∅ for n > N .

(3) G has a terminal pair w.r.t. a.

Proof. According to discussion above, it is sufficient to prove the case (1⇒3).

We first consider the case that G is Hausdorff.

Set

J :=n

x ∈ G | d(x, e) ≤ d(xaⁿ⁰, e) for all n⁰ ∈ Zo and

N_x := min {N ∈ N | 2d(x, e) + 2 < d(xaⁿ, e) for all n ∈ Z with |n| > N} .

The reason for the definition of J is that we want to simulate the special case G = R², a = (1, 0). In this special case, the terminal pair w.r.t. a can be setting as ({(x, y)|x ≥ 100}, {(x, y)|x ≤ −100}). To deduce this, we need some kind of sense like y-axis which is orthogonal to a, thus, we define J to be the simulation of y-axis.

Note that they are well-defined, since Theorem 2.2.15 implies that both d(xaⁿ, e) and d(xa⁻ⁿ, e) → ∞ as n → ∞. And for each x ∈ G, there exists xaⁿ ∈ J (i.e.

G = {xaⁿ|x ∈ J, n ∈ Z}), and this also follows from Theorem 2.2.15.

Note that B :=n B¹

4(xaⁿ)o

x∈J,n∈Z

is a covering of G.

Claim 1: For any x, y ∈ J , n > N and m > N , we have d(xaⁿ, ya^−m) > 1.

Proof of Claim 1: Suppose 1 ≥ d(xaⁿ, ya^−m) = d(x, ya^−n−m) = d(xa^n+m, y), then 2 + 2d(x, e) < d(xa^n+m, e)

≤ d(xa^n+m, y) + d(y, e)

≤ 1 + d(y, e)

≤ 1 + d(ya^−n−m, e)

≤ 1 + d(ya^−n−m, x) + d(x, e)

≤ 2 + d(x, e), a contradiction.

Claim 2: For any x, y ∈ J , n > N_x and m > N_y, we have d(B¹

4(xaⁿ), B¹

4(ya^−m)) > 1 4. Proof of Claim 2: Suppose d(B¹

4(xaⁿ), B¹

4(ya^−m)) ≤ ¹₄ < ¹₃, then there exist x⁰ ∈ B¹

4(xaⁿ) and y⁰ ∈ B¹

4(ya^−m) such that d(x⁰, y⁰) < ¹₃, so d(xaⁿ, ya^−m) < ¹₄ +¹₄ +¹₃ < 1, an absurdity.

So far we are ready to set our terminal pair. Set (A, B) := (A⁰, B⁰), where A⁰ := [

x∈J,n>Nx

B¹

4(xaⁿ) and

B⁰ := [

x∈J,n>Nx

B¹

4(xa⁻ⁿ).

By Claim 2, d(A⁰, B⁰) ≥ ¹₄ > 0, so A and B are disjoint closed set.

Now we are going to show that for each compact set K shifts into A and B. By compactness of K, there exists finitely many balls in B, n

B¹

4(x_iaⁿⁱ)o`

i=1 which covers K. Set

N := 2 max1≤i≤`{|ni− Nxi|, |ni+ Nxi|}, so ni+ n > Nxi and ni− n < −Nxi for n > N . Then

Kaⁿ⊆ ∪^`_i=1B¹

4(xiaⁿⁱ)aⁿ = ∪^`_i=1B¹

4(xiaⁿⁱ⁺ⁿ) ⊆ A and

Ka⁻ⁿ⊆ ∪^`_i=1B¹

4(xiaⁿⁱ)a⁻ⁿ= ∪^`_i=1B¹

4(xiaⁿⁱ⁻ⁿ) ⊆ B for n > N , so the case of Hausdorff is verified.

Now we are going to consider the general case. Let G be a second countable locally compact group and π : G → eG be its Hausdorffication, so π(a) is also aperiodic, hence G has a terminal pair w.r.t. π(a). The proof is completed by Propositione 2.2.14.

2.3 Existence of hypercyclic weighted translations

In the following section, we would like to discuss how the existence of terminal pair affects the existence of hypercyclic weighted translation operators.

Lemma 2.3.1. Let G be a second countable locally compact group and admit a terminal pair w.r.t. a. Then there exists a weighted translation operator T_a,w which satisfies the frequent hypercyclicity criterion on L^p(G) for all p ∈ [1, ∞), simultaneously.

Proof. We are going to construct a weight w by Urysohn’s lemma and verify the frequent hypercyclicity criterion [7, Theorem 9.9 and Proposition 9.11.] directly.

Let π : G → eG be the Hausdorffication of G.

Recall that any locally compact Hausdorff group is normal [8, Theorem 8.13]. So eG is normal. Using the notation in Proposition2.2.14, we setw|e_Ae= 2⁻¹ andw|e_Be = 2. By Urysohn’s lemmaw is a well-defined continuous function on ee G with w(G) ⊂ [2e ⁻¹, 2] ⊂ (0, ∞). Thus we can induce w :=w ◦ π with w|e _A= 2⁻¹, w|_B = 2 and w(G) ⊂ [2⁻¹, 2] ⊂ (0, ∞).

To verify such T_a,w satisfying frequent hypercyclicity criterion, we set X₀ := {bounded compact support functions on G}

which is a dense subspace in L^p(G) for all p ∈ [1, ∞), and also set T = T_a,w and S = T_a,w⁻¹ = T_a⁻¹_,w⁰, where w⁰ := (T_a⁻¹w)⁻¹.

Now let ϕ ∈ X₀ and K := supp(ϕ) (note that supp(ϕ(· aⁱ)) = Ka⁻ⁱ).

Claim: kTⁿϕk_∞ and kSⁿϕk_∞ decay to zero by exponential type.

That is,

kTⁿϕk_∞≤ Cγ⁻ⁿ and

kSⁿϕk_∞≤ Cγ⁻ⁿ for some C > 0, γ > 1 and n large enough.

Now we only need to prove the first case, since the second case is symmetric with the first case by replacing A to B, B to A, a to a⁻¹ , w to w⁰ and T to S. Since K is compact, there is a large number N such that

Kaⁿ⊆ A and

Ka⁻ⁿ ⊆ B for n ≥ N , then

kTⁿϕk_∞ =

n−1

Y

i=0

w(xa⁻ⁱ)ϕ(xa⁻ⁿ) _∞

=

n−1

Y

i=0

w(xaⁿ⁻ⁱ)ϕ(x) ∞

=

n

Y

j=1

w(xa^j)ϕ(x) ∞

(j = n − i)

≤

n

Y

j=1

w(xa^j)|K

∞

kϕk∞

≤

N

Y

j=1

w(xa^j) ∞

kϕk_∞

n

Y

j=N +1

w(xa^j)|_{K ∞}

=

N

Y

j=1

w(xa^j) ∞

kϕk_∞

n

Y

j=N +1

kw(x)|_Ka^jk_∞

≤

N

Y

j=1

w(xa^j) ∞

kϕk_∞

n

Y

j=N +1

2⁻¹ (since Ka^j ⊆ A).

So the claim is verified.

Finally, we will show that

∞

X

n=0

Tⁿϕ converges unconditionaly

and ∞

X

n=0

Sⁿϕ converges unconditionaly.

Again, we only need to verify the first case, and we will prove that they are actually convergent absolutely. By aperiodicity of a, there is a large number N₀ such that

K ∩ Ka⁻ⁿ = ∅ for n ≥ N0,

then we can easily check C<sub>`</sub> := {Ka<sup>−(N0j+`)</sup>}_j∈Z is a mutually disjoint collection for each

` = 0, 1, ..., N0− 1. So

∞

X

n=0

Tⁿϕ p

≤

N0−1

X

`=0

∞

X

j=0

T^N0j+`ϕ  p

≤

N0−1

X

`=0

∞

X

j=0

T^N0j+`ϕ  p

.

For any given ` = 0, 1, ..., N₀− 1,

∞

X

j=0

T^N0j+`ϕ 

p

p

= Z 

∞

X

j=0

T^N0j+`ϕ     

p

=

Z ^∞

X

j=0

T^N0j+`ϕ 

p (C_` are mutually disjoint)

=

∞

X

j=0

Z

T^N0j+`ϕ 

p

≤

∞

X

n=0

kTⁿϕk^p_p

=

∞

X

n=0

Z

Kaⁿ

|Tⁿϕ|^p (supp(Tⁿϕ) ⊆ Kaⁿ)

≤

∞

X

n=0

|Kaⁿ| kTⁿϕk^p_∞ (|Kaⁿ| = |K|)

= |K|

∞

X

n=0

kTⁿϕk^p_∞< ∞.

Notice that the last inequality follows from kTⁿϕk_∞decaying to zero by exponential type.

Therefore, the condition of frequent hypercyclicity criterion holds.

Remark. The existence of weighted translation operator T_a,w is not unique. In fact, there are uncountable many weighted translations satisfying this lemma by setting w|A ≡ α and w|_B ≡ β for α ∈ (0, 1) and β ∈ (1, ∞) whatever you like in the proof.

Theorem 2.3.2. Let G be a second countable locally compact group and let a be an aperiodic element in G. Then there exists a weighted translation operator T_a,w which is mixing, chaotic and frequently hypercyclic on L^p(G) for all p ∈ [1, ∞), simultaneously.

Proof. By Theorem2.2.16 and Lemma 2.3.1.

Example 2.3.3. Let G be an arbitrary Lie group and let a be an aperiodic element in G. Then there exists a weighted translation operator T_a,w which is mixing, chaotic and frequently hypercyclic on L^p(G) for all p ∈ [1, ∞), simultaneously.

Example 2.3.4. Let G be a general linear group GL(n, C). Then a is a periodic element of G iff a is diagonalizable with each eigenvalue has norm 1 (i.e. if λ is a eigenvalue of a, then |λ| = 1). Without using Theorem 2.2.16, it would be hard to verify G has a terminal pair w.r.t. a when a is aperiodic, for instance, a =−1 1

0 −1

 .

Whatever, if a ∈ GL(n, C) is not diagonalizable or has an eigenvalue that doesn’t has norm 1, then there exist a positive continuous function w such that T_a,w is hypercyclic on L^p(GL(n, C)).

Remark. To explain why a is periodic if and only if a is diagonalizable with all eigenvalue has norm 1. We only need to prove the case that if a is nondiagonalizable then a is aperiodic. Since other cases follow by Example 2.2.10 and Proposition 2.2.11. The same as the Example2.2.10, we can assume a is itself a Jordan form:

a =











λ 1 · · · 0 0 λ · · · ... ... · · · . .. ∗ 0 · · · 0 λ⁰









 .

Notation: Let any x ∈ G, we write x = [x₁|x₂|...|x_n], where x_i ∈ Cⁿ are the column of x. (Note that x_i will never be zero vector since x is invertible.)

Consider the map f : G → R, f (x) = |ln kx2k|, by the calculation, we have f (aⁿ) = 

(n − 1) ln |λ| + ln |n| +¹₂ln |1 + ^|λ|_n2²| 

 → ∞ as n → ∞ whatever λ might be (λ never be zero since a ∈ GL(n, C)). Suppose a is periodic. Then f (G(a)) is compact in R, but {f (aⁿ)} is unbounded in R, an absurdity.

Chapter 3

Weighted translation operators of periodic elements

3.1 Notations and Preliminary

In this chapter, we only consider that G is a compact Hausdorff group except few examples.

The same as Chapter2, section3.2 is prepared for the section3.3, which argues that the sequence {a⁻ⁿ}_n∈N is homogeneous equidistribution (Definition 3.2.2). And section 3.3 will use it to prove a necessary condition for some hypercyclic weighted translation operators with non-torsion periodic element. For the section 3.4, we focus on p-adic groups. And we also give a necessary condition for hypercyclic weighted translation for it.

We denote eU to be an arbitrary borel set with U being an open set and U ⊆ eU ⊆ U . (Obviously, this definition is not well-defined for a given open set U , but we denote those sets by the same symbol). (See also [9, Chapter 3], they call eU a µ-continuity set when |∂U | = 0).

Example. Let G = S¹ = [0, 1) and U = (0,¹₂), where S¹ denotes the circle group.

There are 4 possibilities of the expression eU , (0,¹₂), [0,¹₂), (0,¹₂] and [0, ¹₂].

We denote A to be an algebra (closed under finite union and complement) generated by the sets {every eU |∂U has measure 0}.

Example. In the case of G = S¹ = [0, 1), these sets (0,¹₂), [0,¹₂), (0,¹₂] and [0,¹₂] belong to A.

There is a simple observation:

Proposition 3.1.1. The elements in A are one of the forms below:

(1) eU , where ∂U has measure 0.

(2) S = eU \ U ⊆ ∂U , where eU has the form (1).

(3) eU ∪ ∪^k_i=1S_i, where eU has the form (1), and S_i have the form (2).

Note that this classification is not disjoint.

Proof.

1. We first consider the union and complement of form (1). It is not hard to see that the union of form (1) is also form (1). But for complement of form (1) there are two possibilities, form (1) or form (2), determined by whether it is dense in G.

2. For the form (2), let S ⊆ ∂U be an element of the form (2) and K be form (1) or (2) or (3), where eU has the form (1). Then S ∪ K has the form (3) and S^c has the form (1), this follows by the facts that ∂S^c = ∂S ⊆ S ⊆ ∂U is measure 0 and S^c = int(S^c) ∪ (S^c\ int(S^c)).

3. Finally, we consider the form (3).

It is not hard to see that the union of the elements of the form (3) are also the form (3).

Let eU ∪ ∪^k_i=1S_i be an element of the form (3). Then its complement is eU^c∩ (∩^k_i=1S_i^c), which is the finite intersection of the form (1). And the finite intersection of the form (1) is either the form (1) or the form (2).

In conclusion, the sets generated by the form (1) are also of the three forms.

Recall the representation theory of compact group [6, Chapter 5]. We denote ˆG to be the set of unitary equivalence classes of irreducible unitary representations of the compact Hausdorff G. Let [π] ∈ ˆG, π : G → U (H_π) ⊂ M (H_π) be a group homomor- phism, where H_π is denoted to be the finite dimension complex Hilbert space w.r.t. π [6, Theorem 5.2]. M (H_π) is denoted to be the collection of linear operators on H_π and U (H_π) is its subcollection of unitary matries. We denote π_i,j(x) :=< π(x)e_i, e_j > to be the matrix elements, where {e_i} is the standard basis of H_π.

3.1.1 Notation for p-adic analysis

This subsection is prepared for the discussions which refer to the p-adic groups.

Let Qp and Zp denote the p-adic field and its ring of integer for some prime number p. Since they are both DVR (discrete valuation ring). We denote their valuation by v_p, (ex: v5(100) = v5(5²) + v5(4) = 2 + 0 = 2). Hence the p-adic norm | · |p := p^vp(·) defines their topology. (See [10]).

We denote M_a as a multiplication operator which is defined by Ma(f )(x) := f (ax).

Remark. In the discussion of the p-adic groups the weighted translation operator will become T_a,w(f )(x) := w(x)f (x − a), since we view p-adic groups as additive groups.

Both weighted translation operators and multiplication operators act on the L^p0(G), where G = Qp or Zp and p⁰ ∈ [1, ∞). (Note that the notation for “p” is different from other places in this article.)

3.2 Homogeneous equidistribution on compact groups

Let C(G) denote the continuous complex value functions on G.

Lemma 3.2.1. Let G be a compact Hausdorff group, and let a be an element in G such that π(a) do not have nontrivial fixed points (equivalently, I_Hπ− π(a) is invertible) for all [π] ∈ ˆG \ {[1]}. Then for any f ∈ C(G), define gf,N(x) := _N¹

N −1

P

n=1

f (xa⁻ⁿ). We have

g_f,N converges uniformly to the constant Z

G

f.

Proof. Let S be the collection consisting all f which satisfies this Lemma. It is non- empty, since all constant functions belong to S.

Recall that span n

all matrix elements πi,j of π|[π] ∈ ˆG o

is dense in C(G) [6, Theo- rem 5.11]. In particular, the only closed subspace of C(G) containing all matrix elements is C(G) itself. So it is sufficient to show S is a closed subspace containing all matrix elements.

Note that kg_h,Nk_∞≤ khk_∞ for any h ∈ C(G).

Claim: S is a closed subspace in C(G) (w.r.t. k · k^∞).

S is a subspace, since for any f, f⁰ ∈ S and A, B ∈ C we have g^{Af +Bf0,N} = Agf,N + Bg_f⁰_,N. So g_{Af +Bf}⁰_,N also converges uniformly to the constantR

GAf + Bf⁰.

Let f_i ∈ S which converges uniformly to f . That means that for any ε > 0, we have kf − fik_∞< ε/3 for some i and there is a number N⁰ such that

gfi,N −R

Gfi

_∞< ε/3 for all N > N⁰. Then

g_f,N − Z

G

f _∞

≤

g_fi,N − Z

G

f_{i ∞}

+ kg_fi,N − g_f,Nk_∞+ Z

G

f_i− Z

G

f _∞

≤

g_fi,N − Z

G

f_{i ∞}

+

g_(fi−f ),N

_∞+ kf_i− f k_∞

< ε/3 + ε/3 + ε/3 = ε.

Hence S is a closed subspace.

Claim: If [π] ∈ ˆG \ {[1]}, then π_i,j ∈ S.

Firstly, we will show that R

Gπ is zero matrix. Hence R

Gπ_i,j = 0.

By the property of Haar measure on compact groups that it is two-sided invariant, we have

( Z

G

π) = ( Z

G

π(x)dx)

= ( Z

G

π(xy)dx) = ( Z

G

π(x)dx)π(y) = ( Z

G

π)π(y)

and

( Z

G

π) = ( Z

G

π(x)dx)

= ( Z

G

π(yx)dx) = π(y)(

Z

G

π(x)dx) = π(y)(

Z

G

π) for any y ∈ G. So R

Gπ commutes with all π(G), by Schur’s Lemma [6, Lemma 3.5], R

Gπ = cI_Hπ for some c ∈ C. If c 6= 0, then (R

Gπ) = π(y)(R

Gπ) implies I_Hπ = π(y) for all y ∈ G, which contradicts to the assumption that [π] 6= [1]. ThusR

Gπ is zero matrix.

Secondly, let k·k_M denote the operator norm. We will show that

kgπ,Nk_M

_∞ → 0 as N → ∞. In particular,

g_πi,j,N −R

Gπ_{i,j ∞} =

g_πi,j,N

_∞ → 0 as N → ∞. Hence π_i,j ∈ S.

kg_π,Nk_M

_∞ = 1 N

N −1

X

n=1

π(xa⁻ⁿ) M

∞

≤ 1 N

kπ(x)k_M

N −1

X

n=1

π(a)⁻ⁿ M

∞

≤ 1 N

N −1

X

n=1

π(a)⁻ⁿ M

(π(x) and π(a) are unitary.)

≤ 1 N

(I_Hπ− π(a))⁻¹ M

I_Hπ− π(a)^N M

≤ 2 N

(I_Hπ− π(a))⁻¹

M → 0 as N → ∞.

Definition 3.2.2. Let G be a compact Hausdorff group. We say a sequence {x_k}_k∈N is a homogeneous equidistribution w.r.t. A, if A is a collection of subsets of G and

sup

x∈G

Dens_N,K,{xk}_k∈N(x) − |K|

→ 0 as N → ∞ for any K ∈ A, where Dens_N,K,{xk}_k∈N(x) := dens_N(x⁻¹K, {x_k}_k∈N) and

densN(K, {xk}_k∈N) := #{xk ∈ K|k < N }

N = 1

N

N −1

X

k=1

χK(xk).

Lemma 3.2.3. Let G be a compact Hausdorff group, and a be an element in G such that π(a) do not have nontrivial fixed points for all [π] ∈ ˆG \ {[1]}. Then {a⁻ⁿ}_n∈N is a homogeneous equidistribution w.r.t. A.

Proof. We seperate the proof into two parts. For the first part, we will prove that {a⁻ⁿ}_n∈N is a homogeneous equidistribution w.r.t. the form (1) in A. (We have defined A in Section 3.1.)

For any ε > 0 and U open in G with ∂U being measure 0, define f_U,ε⁺ and f_U,ε⁻ to be some continuous functions which satisfy the following:

(1) Both image of f_U,ε⁺ and f_U,ε⁻ lie in [0, 1].

(2) f_U,ε⁺ |_U ≡ 1 and f_U,ε⁺ |_V^c ≡ 0, where V is an open neighborhood of U such that

|V \ U | = |V \ U | < ε/2.

(3) f_U,ε⁻ |_K ≡ 1 and f_U,ε⁻ |_U^c ≡ 0, where K is a compact subset of U such that |U \ K| =

|U \ K| < ε/2.

The existence of f_U,ε⁺ and f_U,ε⁻ follow by the Uryshon’s lemma (It works, since G is compact Hausdorff, hence normal.) and the regularity of Haar measure. From the construction, we immediatly have

χ_K ≤ f_U,ε⁻ ≤ χ

Ue ≤ f_U,ε⁺ ≤ χ_V, hence (use the notation in the proof of Lemma3.2.1)

g_f⁻

U,ε,N ≤ dens_N(x⁻¹U , {ae ⁻ⁿ}_n∈N) ≤ g_f⁺

U,ε,N for any N . By Lemma 3.2.1, there exists N⁰ independent of x such that

| eU | − ε = Z

G

χ_Ue − ε

≤ Z

G

f_U,ε⁻ − ε/2

< g_f⁻

U,ε,N

≤ dens_N(x⁻¹U , {ae ⁻ⁿ}_n∈N) = Dens_{N, eU ,{a}−n}_n∈N(x)

< g_f⁺

U,ε,N

≤ Z

G

f_U,ε⁺ + ε/2

≤ Z

G

χ_Ue + ε

= | eU | + ε for all N > N⁰. Then we have

sup

x∈G

Dens_{N, eU ,{a}−n}_n∈N(x) − | eU |   < ε.

For the sencond part, we will consider the additivity of the function Dens.

Let X, Y be disjoint subsets of G. If any two of the followings hold, then so does the third:

(1) {a⁻ⁿ}_n∈N is a homogeneous equidistribution w.r.t. {X}.

(2) {a⁻ⁿ}_n∈N is a homogeneous equidistribution w.r.t. {Y }.

(3) {a⁻ⁿ}_n∈N is a homogeneous equidistribution w.r.t. {X ∪ Y }.

This follows by the equations

Dens_N,X∪Y,{a⁻ⁿ_}n∈N = Dens_N,X,{a⁻ⁿ_}n∈N+ Dens_N,Y,{a⁻ⁿ_}n∈N and

|X ∪ Y | = |X| + |Y |.

For the form (2) in A. Let S = eU \ U , where eU has the form (1). We can write U = S ∪ U as a disjoint union. Hence {ae ⁻ⁿ}_n∈N is a homogeneous equidistribution w.r.t. {S} (since U is itself another form (1).)

Similar to the form (3) in A. We can write it as the finite disjoint union of the form (1) and (2). So {a⁻ⁿ}_n∈N is a homogeneous equidistribution w.r.t. the form (3) in A.

Hence the lemma has been proved.

Corollary 3.2.4. Let G be a compact Hausdorff group and a ∈ G. Then the following are equivalent:

(1) G = < a >.

(2) For any π ∈ ˆG\{[1]}, π(a) does not have nontrivial fixed points for all [π] ∈ ˆG\{[1]}.

Proof. (1⇒2) Since G = < a > is abelian, ˆG is its daul group. And we know that dim Hπ = 1 for all π ∈ ˆG. So for any π ∈ ˆG \ {1}, 1 − π(a) 6= 0 is invertible, since if π(a) = 1, we have π(a)ⁿ= 1 for all n ∈ Z, a contradiction as π ≡ 1.

(2⇒1) By Lemma3.2.3, {a⁻ⁿ}_n∈Nis a homogeneous equidistribution w.r.t. A. This implies that {a⁻ⁿ}_n∈N is dense in G (it’s sufficient to say any that non-empty open set in G contains an element of form (1) in A, see [9, Chapter 3, example 1.2.]), hence G = < a >.

Corollary 3.2.5. Let G be a finite group. Then the following are equivalent:

(1) G is non-cyclic.

(2) For any a ∈ G, there exists π ∈ ˆG \ {[1]} such that π(a) has an eigenvalue 1.

3.3 A necessary condition of hypercyclic weighted translation operators

Theorem 3.3.1. Let G be a compact Hausdorff group, and let a be an element in G such that G = < a >. If T_a,w is hypercyclic on L^p(G) for some p ∈ [1, ∞), then

Z

G

ln w = 0.

Proof. First, we will observe some facts for the step functions:

Let φ > 0 be a strictly positive A-step function (i.e φ can write as

k

P

i=1

α_iχ_Ei, where α_i > 0 and E_i ∈ A are disjoint). Set m_i := |E_i|. By Lemma 3.2.3, for any ε > 0, there are N⁰ such that

m_i− ε < 1

N#{xa⁻ⁿ∈ E_i|n < N } < m_i+ ε for any x ∈ G and N > N⁰ and i = 1 · · · k. Then

k

Y

i=1

α^m_i ^i−ε <

k

Y

i=1

α_i^{#{xa−n∈Ei|n<N }}

!_N¹

<

k

Y

i=1

α^m_i ^i+ε.

Hence

k

Y

i=1

α^m_i ^i−ε < (φ_N)^N1 <

k

Y

i=1

α^m_i ^i+ε,

where φN(x) :=

N −1

Q

n=0

φ(xa⁻ⁿ), since

N −1

Q

n=0

φ(xa⁻ⁿ) =

k

Q

i=1

α^#{xa_i ^{−n∈Ei|n<N }}.

Now we are going to claim two things: Let φ be a strictly positive A-step function, then

(1) w ≥ φ implies R

Gln φ ≤ 0.

(2) w ≤ φ implies R

Gln φ ≥ 0.

Since R

Gln φ =

k

P

i=1

m_iln α_i = ln

 _k Q

i=1

α^m_i ⁱ

 , so R

Gln φ ≤ 0 ⇔

k

Q

i=1

α^m_i ⁱ ≤ 1 and R

Gln φ ≥ 0 ⇔

k

Q

i=1

α^m_i ⁱ ≥ 1.

For (1). Suppose

k

Q

i=1

α^m_i ⁱ > 1 but w ≥ φ. We can choose ε small enough such that 1 < (φ_N)^N1. Then 1 < φ_N ≤ w_N, where w_N(x) :=

N −1

Q

n=0

w(xa⁻ⁿ). But T_a,w^N = T_aN,wN

is also hypercyclic [1], contradict to the fact that a weighted translation operator with the wieght greater then 1 will never be a hypercyclic operator.