The Retentivity of Chaos under Topological Conjugation

The definitions of Devaney chaos (DevC), exact Devaney chaos (EDevC), mixing Devaney chaos (MDevC), and weak mixing Devaney chaos (WMDevC) are extended to topological spaces. This paper proves that these chaotic properties are all preserved under topological conjugation. Besides, an example is given to show that the Li-Yorke chaos is not preserved under topological conjugation if the domain is extended to a general metric space.


Introduction and Preliminaries
The existence of chaotic behavior in deterministic systems has attracted researchers for many years.In engineering applications such as biological engineering, imaging, encryption, and chaos control, chaoticity of a system is an important subject for investigation.The complexity of a topological dynamical system is intensively discussed since the term chaos is introduced by Li and Yorke [1] in 1975.However, the definition of chaos in the sense of Li-Yorke is inconveniently in research of engineering applications.In 1989, Devaney [2] stated another known definition of chaos called "Devaney chaos" today.A map  is said to be chaotic in the sense of Devaney (or DevC for short) on a close set  if  is transitive on , the set of periodic points of  is dense in , and  has sensitive dependence on initial conditions.
In 1992, Banks et al. [3] proved that if  : (, ) → (, ) is transitive and has dense periodic points, then  has sensitive dependence on initial conditions, where (, ) is a metric space which has no isolated point.This causes that Devaney chaos is preserved under topological conjugation on general infinite metric spaces.If sensitivity is replaced by some other dynamical properties, such as weakly mixing (WMix), mixing (Mix), or exact, then we obtain stronger notions of chaos comparing with the original one (see [4][5][6][7]), which are called Weak Mixing Devaney chaos (WMDevC), Mixing Devaney chaos (MDevC), and Exact Devaney chaos (EDevC), respectively.
From [7], on metric spaces, implications EDevC ⇒ MDevC ⇒ WMDevC ⇒ DevC (1) immediately follow from their definitions.Since an invertible map may not be topologically exact, then MDevC does not imply EDevC.In [8], Carnnrl gave a dynamical system which is WMix but not Mix on a compact metric space.And an example in [9] is constructed to show that there exists a DevC map that is not WMDevC.Hence, the inverse of (1) is not proper.This paper extends the definitions of DevC, WMDevC, MDevC, and EDevC to topological spaces and proves that topological conjugation preserves these chaotic properties.Moreover, we use an example to show that Li-Yorke chaos is not preserved under topological conjugation on metric spaces.
Throughout this paper, let The open neighborhood system of a point  ∈  is denoted by ().The set of periods and periodic points of a continuous self-map  are denoted by () and Per(), respectively.
Definition 2 (see [10]).Let  and  be two topological spaces and let  :  →  and  :  →  be two continuous maps.Maps  and  are said to be (topologically) conjugate if there exists a homeomorphism ℎ :  →  such that ℎ ∘  =  ∘ ℎ, where "∘" denotes composition of two maps.For short, we call  and  ℎ-conjugate maps.
Some more results related to topological conjugation may be found in [10][11][12].

Devaney Chaoticity
Throughout this section,  :  →  and  :  →  are two ℎ-conjugate maps defined on two topological spaces  and , where ℎ :  →  is a homeomorphism.Definition 3. The map is exact if, for every nonempty open set  ⊂ , there exists some  ∈ N such that   () = .The map  is (topological) transitive (resp., mixing) if for any two nonempty open sets ,  ⊂ , there exists some  ∈ N such that   () ∩  ̸ =  (resp.,   () ∩  ̸ =  for all  ≥ ).The map  is weak mixing if  ×  is transitive on  × .

Definition 4. (1)
The map  :  →  is chaotic in the sense of Devaney if  is transitive on  and the set of periodic points of  is dense in .
(2) The map  exhibits exact Devaney chaos (resp., mixing Devaney chaos and weakly mixing Devaney chaos) if  is exact (resp., mixing and weakly mixing) and chaotic in the sense of Devaney.Devaney chaos, exact Devaney chaos, mixing Devaney chaos, and weak mixing Devaney chaos are briefly denoted by DevC, EDevC, MDevC, and WMDevC, respectively.
Similarly, from these definitions, we have these implications that (1) is right and its inverse is not proper on topological spaces.

Proposition 5. The map 𝑓 is exact if and only if 𝑔 is exact.
Proof.Necessity.Given any nonempty open set  in , take  = ℎ −1 ().Clearly,  is a nonempty open set in .Since  is exact, there exists an  ∈ N such that   () = .Noting that  and  are ℎ-conjugate maps and that ℎ is a homeomorphism, it follows that This implies that  is exact.
Similarly to the proof of necessity, the sufficiency follows immediately.
This implies that Thus,  is mixing.Sufficiency can be proved similarly.
This implies that Thus,  is weak mixing.Sufficiency can be proved similarly.
Theorem 8 (see [3]).The map  is DevC on  if and only if  is DevC on .
Proof.Necessity.Similarly to the proof of Proposition 6, it can be verified that  is transitive on , as  is transitive.
According to the definition of periodic points, it is not difficult to check that Per () ⊃ ℎ (Per ()) .
This implies that Per() = .Summing up the above discussion, it follows that  is chaotic in the sense of Devaney.
Sufficiency can be proved similarly.Remark 12.According to the results obtained in this section, it is easy to see that DevC, EDevC, MDevC, and WMDevC are all preserved under topological conjugation on metric spaces.The next section will show that this does not hold for the Li-Yorke chaos.

Li-Yorke Chaoticity
First, we give the definition of Li-Yorke chaos.
Since both  1 and  2 are discrete metrics on , -map and -map are both continuous.Proposition 14. -map and -map are topologically conjugate.
(i) For any ,  ∈  0 with  ̸ = , we have (iii) For any  ∈  0 and any  ∈ Per() = Z + , it is easy to see that there exists a  ∈ N such that   () ≤  and that   () =  +  ≥  + 1 for any  >  + 1.Then, for any We thus conclude that -map is chaotic in the sense of Li-Yorke.
The following theorem holds obviously by Propositions 14, 15, and 16.
Theorem 17.For a continuous self-map which is defined on a metric space, Li-Yorke's chaoticity is not preserved under topological conjugation.Remark 18. Section 2 discusses some notions of "chaos" which can be defined on topological spaces and Section 3 studies Li-Yorke chaos on metric spaces.There are some other problems for further research; For example, could Li-Yorke chaos be extended to topological spaces?According to Section 3, we can conclude that even if Li-Yorke chaos may be extended to topological spaces, topological conjugation does not preserve it.

Proposition 7 .
The map  is weakly mixing if and only if  is weakly mixing.Proof.Necessity.For any two nonempty open sets ,  ⊂  × , according to the construction of product topology, it follows that there exist nonempty open sets  1 ,  2 ,  1 ,  2 ⊂  such that  1 ×  2 ⊂  and  1 ×  2 ⊂ .Since  is weakly mixing, there exists  ∈ N such that Applying Propositions 5, 6, and 7 and Theorem 8, the following results are straightforward.The map  is EDevC if and only if  is EDevC.The map  is MDevC if and only if  is MDevC.The map  is WMDevC if and only if  is WMDevC.