Strong topological shift equivalence for dynamical systems

Topology and its Applications 265 (2019) 106830

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

Strong topological shift equivalence for dynamical systems Bingzhe Hou a,∗ , Geng Tian b a b

School of Mathematics, Jilin University, 130012, Changchun, PR China Department of Mathematics, Liaoning University, 110036, Shenyang, PR China

a r t i c l e

i n f o

Article history: Received 11 February 2019 Received in revised form 11 June 2019 Accepted 27 July 2019 Available online 1 August 2019 MSC: 54H20 37B99

a b s t r a c t In this paper, we introduce some new equivalence relations for topological dynamical systems named strong topological shift equivalence and topological shift equivalence, which are similar to the strong shift equivalence and shift equivalence for subshifts of finite type. We study the relations between the new equivalences and other equivalences such as topological conjugacy, mutually topological semi-conjugacy and canonical homeomorphism extensions being topologically conjugate. Some properties and examples are shown. In particular, mean topological dimension is an invariant for topological shift equivalence but not for mutually topologically semi-conjugate equivalence. In this topic, linear operators are also considered. © 2019 Elsevier B.V. All rights reserved.

Keywords: Strong topological shift equivalence Mutually topological semi-conjugacy Canonical homeomorphism extension Mean topological dimension Linear operators

1. Introduction and preliminaries In this paper, a dynamical system (X, f ) means a complete metric spaces X and a continuous map f : X → X. Furthermore, if X is compact, (X, f ) is said to be a compact dynamical system; if f is surjective, (X, f ) is said to be a surjective dynamical system. Definition 1.1. Let (X, f ) and (Y, g) be two dynamical systems. f is said to be topologically semi-conjugate to g, if there exists a continuous surjective map h : X → Y such that h ◦ f = g ◦ h. Furthermore, if the map h is a homeomorphism, we say that h is a topological conjugacy from f to g, denoted by f ∼ =h g or f ∼ = g. Moreover, if f is topologically semi-conjugate to g and g is topologically semi-conjugate to f , we say that f and g are mutually topologically semi-conjugate; if there exists positive integer N such that for all n > N , f n is topologically conjugate to g n , we say that f and g are eventually topologically conjugate. * Corresponding author. E-mail addresses: [email protected] (B. Hou), [email protected] (G. Tian). https://doi.org/10.1016/j.topol.2019.106830 0166-8641/© 2019 Elsevier B.V. All rights reserved.

B. Hou, G. Tian / Topology and its Applications 265 (2019) 106830

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Remark 1.2. In fact, the above definitions and related conclusions in the following except in section 3 for linear operators, will work if the base spaces are compact Hausdorff spaces instead of complete metric spaces. A major task in dynamical systems is to give the classifications in the sense of topologically conjugate equivalence. There are also some other equivalence relations for topological systems. For instance, mutually topologically semi-conjugate equivalence is a weaker equivalence relation than topologically conjugate equivalence, which was studied in [8] for rotations on the unit circle and associated crossed product C ∗ -algebras. In the present paper, inspired by the concept of shift equivalence in symbolic dynamical systems, we introduce some new equivalences in topological dynamical systems. Definition 1.3. Two dynamical systems (X, f ) and (Y, g) (or two continuous maps f and g) are said to be elementarily equivalent, if there exist continuous maps α : X → Y and β : Y → X such that f = β ◦ α and g = α ◦ β. In this case we write (α, β) : f  g. If there exist dynamical systems (Xi , fi ), i = 0, 1, . . . , l, such that (X0 , f0 ) = (X, f ), (Xl , fl ) = (Y, g), and (Xi , fi ) is elementarily equivalent to (Xi+1 , fi+1 ) for each i = 0, 1, . . . , l − 1, then (X, f ) and (Y, g) are said to be strong topological shift equivalent of lag l. In this case we write f ≈ g (lag l). Moreover, (X, f ) and (Y, g) are said to be strong topological shift equivalent (write f ≈ g), if (X, f ) and (Y, g) are strong topological shift equivalent of some lag. Definition 1.4. Two dynamical systems (X, f ) and (Y, g) (or two continuous maps f and g) are said to be topological shift equivalent of lag l, if there exist two continuous maps α : X → Y and β : Y → X such that α ◦ f = g ◦ α, f ◦ β = β ◦ g, f l = β ◦ α and g l = α ◦ β. In this case we write (α, β) : f ∼ g (lag l). Moreover, (X, f ) and (Y, g) are said to be topological shift equivalent (write f ∼ g), if (X, f ) and (Y, g) are strong topological shift equivalent of some lag. Similar to the (strong) shift equivalence of subshifts of finite type (see [13]), we have the following conclusions immediately. Proposition 1.5. Strong topological shift equivalence is an equivalence relation on dynamical systems. Proposition 1.6. Let (X, f ), (Y, g) and (Z, h) be three dynamical systems. (1) (2) (3) (4)

A topological shift equivalence with lag l = 1 is the same as an elementary equivalence. If f ∼ g (lag l), then f ∼ g (lag l ) for all l ≥ l. If f ∼ g (lag l) and g ∼ h (lag l ), then f ∼ h (lag l + l ). Topological shift equivalence is an equivalence relation.

Proposition 1.7. Strong topological shift equivalence implies topological shift equivalence. More precisely, if f ≈ g (lag l), then f ∼ g (lag l).

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In the next section, we show some basic properties of (strong) topological shift equivalence, some examples are also given. In section 3, canonical homeomorphism extension and mean topological dimension are considered. For a noninvertible dynamical system, it is natural to study its canonical homeomorphism extension. The notion of mean topological dimension was first introduced by Gromov [7] to study dynamical properties of certain spaces of homomorphic maps and complex varieties and in further, was used by Lindenstrauss and Weiss [9] to answer in the negative a question raised by Auslander. Obviously, mean topological dimension is an invariant under topological conjugacy. We show that it is also an invariant under the weaker equivalencetopological shift equivalence. For more about mean topological dimension, we refer to Coornaert’s book [2] which is a good introduction. We also consider some C ∗ -algebras from dynamical systems. Note that one can associate to a selfhomeomorphism h on X, a C ∗ -dynamical system (C(X), h∗ ) and a crossed product C ∗ -algebra C(X) h Z. Hence we could study the topologically dynamical systems by studying the corresponding C ∗ -algebra dynamical systems and crossed product C ∗ -algebras. In this aspect of dynamical systems and operator algebras, we refer to a recent book [5], in which one can see the related definitions and developments. In particular, the remarkable work of Giordano, Putnam and Skau [6] showed that the transformation group C ∗ -algebras of two minimal homeomorphisms of the Cantor set are isomorphic if and only if the homeomorphisms are strong orbit equivalent. Furthermore, J. Tomiyama [14] proved that two transitive systems (X, α) and (X, β) are flip conjugate if and only if their crossed product C ∗ -algebras are isomorphic preserving C(X). If the continuous map h is noninvertible, one could consider the semicrossed product C ∗ -algebras (we refer to [3]), which can be seemed as a generalization of crossed product. Our interest is in dealing with noninvertible dynamics, so we study the associate semicrossed products of (strong) topological shift equivalent systems. In the final section, we study (strong) topological shift equivalence for linear operators. 2. Basic properties and examples In this section, we will show some basic properties and examples of (strong) topological shift equivalence, and compare it to some other relations of the equivalences including topological conjugacy, eventually topological conjugacy and mutually topological semi-conjugacy. Proposition 2.1. Let (X, f ) and (Y, g) be two dynamical systems. If (X, f ) and (Y, g) are topologically conjugate, then they are elementarily equivalent. Proof. Suppose that h : X → Y is a topological conjugacy from f to g. Let α = h ◦ f and β = h−1 . Then f =β◦α

and g = h ◦ f ◦ h−1 = α ◦ β,

and hence (α, β) : f  g. 2 On the other hand, strong topological shift equivalence does not imply (eventually) topological conjugacy. The following is a simple counterexample. Example 2.2. Let X be a complete metric spaces including at least two points, x0 be a point in X and f : X → X be a constant map defined by f (x) = x0 for all x ∈ X. Let Y = {y0 } be a single point space and g : Y → Y be the identity map. Obviously, (X, f ) and (Y, g) are not topologically conjugate. However, (X, f ) and (Y, g) are strong topological shift equivalent. In fact, let α : X → Y defined by α(x) = y0

for any x ∈ X,

4

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and β : Y → X defined by β(y0 ) = x0 , then f =β◦α

and g = α ◦ β.

In general, we pay more attentions on surjective dynamical systems. In this case, there are also counterexamples. Example 2.3. Let f : I → I be the standard tent map defined by f (x) = 1 − |2x − 1|,

for x ∈ I,

where I is the unit closed interval [0, 1]. Let g : T → T be the quadratic map defined by g(z) = z 2 ,

for z ∈ T ,

where T is the unit circle. Obviously, f and g are not topologically conjugate. However, they are elementarily equivalent. In fact, consider α : I → T defined by α(t) = e2πit

for any t ∈ I,

and β : T → I defined by β(z) =

1 arccos(Rez) π

for any z ∈ T .

Then, one can see both of α and β are continuous surjective maps, and f =β◦α

and g = α ◦ β.

Proposition 2.4. Let (X, f ) and (Y, g) be two surjective dynamical systems. If (X, f ) and (Y, g) are topological shift equivalent, then (X, f ) and (Y, g) are mutually topologically semi-conjugate. Proof. Suppose that (α, β) : f ∼ g (lag l). Since f l = β ◦ α and g l = α ◦ β are surjective, α and β are surjective. Furthermore, by α ◦ f = g ◦ α and f ◦ β = β ◦ g, α is a topological semi-conjugacy from f to g (while β is a topological semi-conjugacy from g to f ). Therefore, (X, f ) and (Y, g) are mutually topologically semi-conjugate. 2 Theorem 2.5. Let X and Y be two complete metric spaces, f : X → X and g : Y → Y be two homeomorphisms. Then the following are equivalent. (1) (X, f ) and (Y, g) are topologically conjugate. (2) (X, f ) and (Y, g) are strong topological shift equivalent. (3) (X, f ) and (Y, g) are topological shift equivalent.

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Proof. It is obvious of “(1) ⇒ (2) ⇒ (3)”. Now it suffices to prove (3) ⇒ (1). Suppose that (α, β) : f ∼ g (lag l). Since f l = β ◦ α and g l = α ◦ β are homeomorphisms, α and β are homeomorphisms. Furthermore, by α ◦ f = g ◦ α and f ◦ β = β ◦ g, α is a topological conjugacy from f to g (while β is a topological conjugacy from g to f ). 2 From Proposition 2.4, one can see that (strong) topological shift equivalence implies mutually topologically semi-conjugate equivalence. Next, we will give an example to show that the converse if false. Example 2.6. Let f, g : T → T be two rational rotations on unit circle defined by f (z) = e2πi·1/p z and g(z) = e2πi·q/p z,

for z ∈ T ,

where p and q are relatively prime positive integers, i.e. (p, q) = 1. One can see that f and g are mutually topologically semi-conjugate but not topological conjugate, since they have different rotation numbers and for hq , hm : T → T defined by hp (z) = z p

and hm (z) = z m ,

where mq = 1 mod p, we have hp ◦ f = g ◦ hp

and f ◦ hm = hm ◦ g.

Furthermore, it follows from Theorem 2.5 that f and g are not (strong) topological shift equivalent. Following the counterexamples of Williams’ conjecture given by K. Kim and F. Roush in [10] or [11], there exist two matrices A and B with nonnegative entries being shift equivalent but not strong shift equivalent. Then the induced subshifts of finite type (X, σA ) and (Y, σB ) are eventually topologically conjugate but not topologically conjugate. Notice that σA : X → X and σB : Y → Y are two homeomorphisms. Then, by Theorem 2.5, (X, σA ) and (Y, σB ) are not topological shift equivalent. Hence, together with Example 2.3, it is independent of eventually topological conjugacy and topological shift equivalence. 3. Canonical homeomorphism extension and mean topological dimension Following from previous discussions, we should pay attention on non-invertible continuous surjective maps to study (strong) topological shift equivalence. In the present section, we consider the canonical homeomorphism extensions of non-invertible maps, and in further, mean topological dimension is shown to be an invariant for topological shift equivalence. Firstly, let us review the concept of canonical homeomorphism extension. For a surjective dynamical  f), where system (X, f ), its canonical homeomorphism extension means a dynamical system (X,  = {(. . . x−1 , x0 , x1 . . .) ∈ X Z ; xn ∈ X and xn+1 = f (xn ) for any n ∈ Z}, X  i.e. and f is the backward shift on X, f(. . . x−1 , x0 , x1 , x2 . . .) = (. . . f (x−1 ), f (x0 ), f (x1 ), f (x2 ) . . .) = (. . . x−1 , x0 , x1 , x2 . . .).

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 f) is the minimal homeomorphism extension of (X, f ). Notice that f is a homeomorphism and (X, Theorem 3.1. Let (X, f ) and (Y, g) be two surjective dynamical systems. If (X, f ) and (Y, g) are topolog f) and (Y , g) are topologically ical shift equivalent, then their canonical homeomorphism extensions (X, conjugate. Proof. Suppose that (α, β) : f ∼ g (lag l).  → Y by Define α :X α (. . . x−1 , x0 , x1 , x2 . . .) = (. . . α(x−1 ), α(x0 ), α(x1 ), α(x2 ) . . .),  by and define β : Y → X  . . x−1 , x0 , x1 , x2 . . .) = (. . . β(x−1 ), β(x0 ), β(x1 ), β(x2 ) . . .). β(. Since α ◦ f = g ◦ α, f ◦ β = β ◦ g, f l = β ◦ α

and g l = α ◦ β,

α  and β are well-defined continuous maps satisfying α  ◦ f = g ◦ α  and f ◦ β = β ◦ g,   and gl = α  ◦ β. fl = β ◦ α  : f ∼ g (lag l), and consequently, by Theorem 2.5, (X,  f) and (Y , g) are topologically It implies ( α, β) conjugate. 2 However, the converse of the conclusion in Theorem 3.1 is not true. Proposition 3.2. There exist two surjective compact dynamical systems (X, f ) and (Y, g) such that they  f) and (Y , g) are are not topological shift equivalent, but their canonical homeomorphism extensions (X, topologically conjugate. Proof. Let f : X → X be a continuous surjective map but not a homeomorphism. Obviously, the canonical  f) are the same (topologically conjugate) dynamical system homeomorphism extensions of (X, f ) and (X,  f). Similar to the proof of Theorem 2.5, one can see that if the surjective map f and the homeomorphism (X,  f are topological shift equivalent, then they are topological conjugate and f is a homeomorphism. Thus,  f) can not be topological shift equivalent. 2 (X, f ) and (X, Remark 3.3. Now one can see that for continuous surjective maps, topologically conjugate equivalence is strictly stronger than strong topological shift equivalence, topological shift equivalence is strictly stronger than mutually topologically semi-conjugate equivalence, topological shift equivalence is strictly stronger than canonical homeomorphism extensions being topologically conjugate, and it is independent of mutually topologically semi-conjugate equivalence, eventually topologically conjugate equivalence and canonical homeomorphism extensions being topologically conjugate. So the remaining question is, “Does topological shift equivalence imply strong topological shift equivalence?” This could be seen as an analogue of Williams’ conjecture for continuous maps.

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Applying Theorem 3.1, one can obtain a result concerned with related C ∗ -algebras. B. Duncan and J. Peters in [4] defined the semicrossed product algebra and the left regular algebra for a surjective compact dynamical systems (X, f ), and they proved that the C ∗ -envelope of the left regular algebra is isomorphic to  f) (Davidson, Fuller and Kakariadis [3] obtained another proof). Then we have the crossed product of (X, the following result immediately. Theorem 3.4. Let (X, f ) and (Y, g) be two surjective compact dynamical systems. If (X, f ) and (Y, g) are topological shift equivalent, then the C ∗ -envelopes of the left regular algebras induced by (X, f ) and (Y, g) are isomorphic. Remark 3.5. The standard tent map on the unit closed interval and the quadratic map on the unit circle are not topologically conjugate, but their canonical homeomorphism extensions are topologically conjugate by Example 2.3 and Theorem 3.1. Furthermore, the C ∗ -envelopes of the left regular algebras induced by them are isomorphic. Now, let us consider mean topological dimension. In the present paper, we study it following the way in Coornaert’s book [2]. In particular, it is useful of a metric approach to mean topological dimension. Let (X, d) be a compact metric space and f : X → X be a continuous map. Given n ∈ N. For any x, y ∈ X, define dn (x, y) by dn (x, y) =

max

0≤k≤n−1

d(f k (x), f k (y)).

Clearly dn is a metric on X. For every ε > 0, define dimε (X, dn ) by dimε (X, dn ) = inf dim(K) K

where K runs over all compact metrizable spaces for which there exists an ε-injective continuous map T : (X, dn ) → K. Here dim(K) means the topological dimension of K and the ε-injectivity of T means T (x) = T (y) ⇒ dn (x, y) ≤ ε, for all x, y ∈ X. Furthermore, define mdimε (X, f ) by mdimε (X, f ) = lim

n→+∞

dimε (X, dn ) . n

Then the mean topological dimension of (X, f ), denoted by mdim(X, f ), is defined by mdim(X, f ) = lim mdimε (X, f ). ε→0

As well-known, mean topological dimension is an invariant for topologically conjugate equivalence. Now we can see that it is also an invariant for topological shift equivalence, but not for mutually topologically semi-conjugate equivalence. Theorem 3.6. Let ((X, d), f ) and ((Y, ρ), g) be two surjective compact dynamical systems. If (X, f ) and (Y, g) are topological shift equivalent, then mdim(X, f ) = mdim(Y, g).

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1 mdim(X, f ) for any l ∈ N. So it suffices to prove that mean topological l dimension is an invariant for elementary equivalence. Now suppose that (X, f ) and (Y, g) are elementary equivalence, i.e., there exist continuous surjective maps α : X → Y and β : Y → X such that Proof. Notice that mdim(X, f l ) =

f =β◦α

and g = α ◦ β.

For any ε > 0, there exists δ > 0 such that for any x, x ∈ X with d(x, x ) < δ, we have ρ(α(x), α(x )) < ε. In addition, by the compactness of Y , there exists an ε-injective continuous map T0 : (Y, ρ) → Y0 , where Y0 is a compact metrizable space with finite dimension n0 . For any n ∈ N, let T : (X, dn ) → K an ε-injective continuous map. Then, define T : Y → Y0 × K by T(y) = (T0 (y), T (β(y))),

for any y ∈ Y.

Obviously, T is continuous. Now consider two points y and y  in Y with T(y) = T(y  ). It is easy to see ρ(y, y  ) < ε by the ε-injectivity of T0 . For any 0 ≤ k ≤ n − 1, it follows from the ε-injectivity of T that d(f k (β(y)), f k (β(y  ))) = d((β ◦ α)k (β(y)), (β ◦ α)k (β(y  ))) < δ. Furthermore, ρ(α ◦ (β ◦ α)k (β(y)), α ◦ (β ◦ α)k (β(y  ))) = ρ(g k+1 (β(y)), g k+1 (β(y  ))) < ε. This implies that T is an ε-injective map. Therefore, dimε (Y, ρn ) ε→0 n→+∞ n n0 + dimδ (X, dn ) ≤ lim lim δ→0 n→+∞ n

mdim(Y, g) = lim lim

= mdim(X, f ). Similarly we also have mdim(X, f ) ≤ mdim(Y, g). This finishes the proof. 2 Proposition 3.7. There exist two surjective compact dynamical systems (X, f ) and (Y, g), such that they are mutually topologically semi-conjugate but have different mean topological dimensions. Proof. Let X = [0, 1]Z , Y = ([0, 1] × [0, 1])Z , f and g be the shifts on X and Y respectively. Then (X, f ) and (Y, g) are two surjective compact dynamical systems. Let h : [0, 1] → [0, 1] × [0, 1] be the Peano curve and π : [0, 1] × [0, 1] → [0, 1] be the natural projection to the first [0, 1]. Define H : X → Y by H((xn )n∈Z ) = (h(xn ))n∈Z for any (xn )n∈Z ∈ X, and define Π : Y → X by H((yn )n∈Z ) = (π(yn ))n∈Z for any (yn )n∈Z ∈ Y . Therefore, it is easy to see that H is a topological semi-conjugacy from (X, f ) to (Y, g) and Π is a topological semi-conjugacy from (Y, g) to (X, f ). However, mdim(X, f ) = 1 = 2 = mdim(Y, g). 2 4. Linear operators In this section, we will study (strong) topological shift equivalence for linear operators. Theorem 4.1. Let V be an k-dimensional complex vector space, and A, B : V → V be two linear operators. Then the following are equivalent.

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(1) (V , A) and (V , B) are eventually topologically conjugate. (2) (V , A) and (V , B) are strong topological shift equivalent. (3) (V , A) and (V , B) are topological shift equivalent. Proof. For any n ∈ N, let ⎡ ⎢ ⎢ Jn = ⎢ ⎢ ⎣

0

1 0

⎤ 1 .. .

..

.

0

⎥ ⎥ ⎥ ⎥ 1⎦ 0

.

n×n

By making Jordan standard forms, A and B are similar to the following matrices respectively

J 0

0 T





V1 and V2

0 V1 , T  V2

J 0

where V = V1 ⊕ V2 and V = V1 ⊕ V2 , J and J  are direct sums of some Jn blocks, T and T  are invertible. Without loss of generality, we may assume A=

J 0

0 T



and B =

J 0

0  , T

(1) ⇒ (2). Suppose that (V , A) and (V , B) are eventually topologically conjugate. Then there exists positive integer L, such that for l > L,



0 0 0 0 l ∼ l = A . = B = 0 Tl 0 (T  )l

Consequently, T l is topological conjugate to (T  )l for l > L. According to the topological conjugate classification of finite dimensional linear operators (see [12] or [1]), one can see that T is topological conjugate to T  . For any n ∈ N, let ⎡ ⎢ ⎢ Pn = ⎢ ⎢ ⎣

⎤

1

⎥ ⎥ ⎥ ⎥ ⎦

1 ..

. 1 0

.

n×n

Then Pn Jn = Jn

and Jn Pn =

Jn−1 0

0 . 0

So each Jn is strong topological shift equivalent to the zero homomorphism. Thus,

0 A≈ 0

0 T



0 0  ∼ ≈ B. T =T  0 T

By the way, for n ≥ 3, Jn is not elementarily equivalent to the zero homomorphism. (2) ⇒ (3). It is obvious.

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(3) ⇒ (1). Now, suppose that A and B are topological shift equivalent. Then

0 T  0

0 T



0 0  T . ≈A∼B≈ 0 T

Since T and T  are invertible matrices (homeomorphisms), it follows from Theorem 2.5 that T and T  are topologically conjugate. Let h be a topological conjugacy from T to T  , and let  h : V → V defined by  h(v1 , v2 ) = (v1 , h(v2 )), for any (v1 , v2 ) ∈ V1 ⊕ V2 = V . Then

0 0 0 T



0 ∼ =h 0

0  . T

Notice that there exists positive integer L, such that for l > L Al =

0 0

0 ∼ 0 =  h 0 Tl

0 = Bl. (T  )l

Therefore, (V , A) and (V , B) are eventually topologically conjugate. 2 Corollary 4.2. Let V be an k-dimensional complex vector space, and A, B : V → V be two linear operators. Then (V , AB) and (V , BA) are eventually topologically conjugate. Similarly, the above two conclusions are also true for linear operators on n-dimensional real vector space. However, they are false for bounded linear operators on infinite dimensional vector spaces. Example 4.3. Let R and S be two bounded linear operators on a separable complex Hilbert space with infinite dimensional matrix representations under an fixed orthogonal basis ⎡

0 1 0 ⎢ R=⎢ ⎣

⎤ 1 0

1 .. .

..

⎡

0 ⎢1 ⎥ ⎥ and S = ⎢ ⎣ ⎦

⎤ 0 1

0 .. .

.

..

⎥ ⎥. ⎦ .

Furtherly, let ⎡ ⎢ A = SR = ⎢ ⎣

⎤

0

⎥ ⎢ ⎥ and B = RS = ⎢ ⎦ ⎣

1 1 ..

⎡

.

⎤

1

⎥ ⎥. ⎦

1 1 ..

.

It is not difficult to see that A and B are elementarily equivalent but not eventually topologically conjugate. References [1] T.V. Budnitska, Topological classification of affine operators on unitary and Euclidean spaces, Linear Algebra Appl. 434 (2011) 582–592. [2] M. Coornaert, Topological Dimension and Dynamical Systems, Springer, 2015. [3] K. Davidson, A. Fuller, E. Kakariadis, Semicrossed products of operator algebras by semigroups, Mem. Am. Math. Soc. 247 (1168) (2014) 3108–3124.

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[4] B.L. Duncan, J.R. Peters, Operator algebras and representations from commuting semigroup actions, J. Oper. Theory 74 (1) (2015) 23–43. [5] T. Giordano, D. Kerr, N.C. Phillips, A. Toms, Crossed Products of C ∗ –Algebras, Topological Dynamics, and Classification, Advanced Courses in Mathematics-CRM Barcelona, Birkhäuser Basel, Springer Nature Switzerland AG part of Springer Nature, 2018. [6] T. Giordano, I. Putnam, C. Skau, Topological orbit equivalence and C ∗ -crossed products, J. Reine Angew. Math. 469 (1995) 51–111. [7] M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I, Math. Phys. Anal. Geom. 2 (1999) 323–415. [8] Bingzhe Hou, Hongzhi Liu, Xiaotian Pan, Mutual embeddability equivalence relation for rotation algebras, J. Math. Anal. Appl. 452 (1) (2017) 495–504. [9] E. Lindenstrauss, B. Weiss, Mean topological dimension, Isr. J. Math. 115 (2000) 1–24. [10] K.H. Kim, F.W. Roush, Williams’ conjecture is false for reducible subshifts, J. Am. Math. Soc. 5 (1992) 213–215. [11] K.H. Kim, F.W. Roush, Williams’ conjecture is false for irreducible subshifts, Ann. Math. 149 (1999) 545–558. [12] N.H. Kuiper, J.W. Robbin, Topological classification of linear endomorphisms, Invent. Math. 19 (2) (1973) 83–106. [13] D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. [14] J. Tomiyama, Topological full groups and structure of normalizers in transformation group C ∗ -algebras, Pac. J. Math. 173 (1996) 571–583.