Shift spaces and distributional chaos
Introduction
There are many notions of chaos. The fist one was the chaos in the sense of Li and Yorke which was introduced in [4]. Distributional chaos was introduced in [7]. In the case of continuous functions mapping a compact real interval into itself, there is equivalence between distributional chaos and positivity of topological entropy (see [7]), moreover these notions of chaos imply chaos in the sense of Li and Yorke, but not conversely. But in more general spaces this statement does not hold true. There exists system which exhibit distributional chaos and has zero topological entropy (Example 18, Example 20) and exists system with positive topological entropy which is not distributionally chaotic (cf. [2], [5]).
The main aim of this paper is to give sufficient conditions for the existence of distributional chaos (Theorem 15) and the existence of chaos in the sense of Li and Yorke (Theorem 16) in the case of shift spaces.
Cocyclic shifts appear in natural way in the computations of cohomological Conley index for maps (see [8]). Cocyclic shifts, as introduced in [3], are the supports of locally constant cocycles into the semigroup S of all linear endomorphisms of a finite-dimensional vector space on the full shift over a finite alphabet. They generalize [10], where sofic shifts are introduced by taking for S any finite semigroup. In general shift spaces, as mentioned in Section 5 (Example 18, Example 20, Example 21), different notions of chaos are not equivalent. This cannot happen among cocyclic shifts. In this spaces concepts of distributional and Li and Yorke chaoses coincide (Theorem 23, Theorem 25). Moreover they are equivalent to the positivity of entropy. It occurs that all chaotic behaviour is carried on the sequence of horseshoes. Existence of a horseshoe is also the only possibility the cocyclic shift to be uncountable (Theorem 22). It leads in the case of sofic shifts to effective algorithms detecting distributional chaos (cf. Remark 26).
The paper is organized as follows. In the next section we recall some definitions of different types of chaos. In the third one we give definitions of sofic and cocyclic shifts. In Section 4 we give some conditions for the general shift to be distributionally chaotic and chaotic in the sense of Li and Yorke. We use them in the following section where we give examples of chaotic shifts. The last section is devoted to cocyclic and sofic shifts.
Section snippets
Preliminaries
Let be a finite set (an alphabet). The (full) -shift is the product space with the shift map and metric d on , given by ρ((xi), (yi)) = 2−j, where is minimal number such that xj ≠ yj. Any non-empty closed subset X of invariant under σ (i.e. σ(X) ⊂ X) is called a shift. If X and Y are shifts and Y ⊂ X then Y is said to be subshift of X.
Elements of the set are called n-words (n-blocks) over . A word (block) over is any element of the set .
Sofic shifts and cocyclic shifts
In this section we will recall some definitions.
We call G = (V, E, i, t) a graph if V and E are finite sets, V ≠ ∅ and i and t are maps from E to V.
A vertex v ∈ V is stranded if the set {e∣i(e) = v} is empty. Graph is essential if it has not stranded vertices.
The graph G is irreducible (strongly connected) if for any two vertices I, J ∈ VG exist paths π = e1⋯em, η = f1⋯fk on graph G such that I = i(e1), J = t(em) and J = i(f1), I = t(fk).
A labelled graph is a pair , where G is a graph and assigns to each
Distributional chaos in general shift spaces
In this section we are going to present a sufficient conditions useful for proving distributional chaos and chaos in the sense of Li and Yorke among class of shift spaces. Definition 12 Let for some . We will use following notion: Theorem 13 Let X be a shift and let sequences and {un},{vn} ⊂ B(X) fulfill the following conditions: ∣un∣ = ∣vn∣ = qn, and cn → ∞, there exists constant such
Applications
In this section we will use toolbox given by Theorem 15, Theorem 16. We will show two examples of a shift which are DS chaotic, but have zero entropy. We will also show LY chaotic shift which do not contain any pair of DS chaotic points (and has zero entropy). This situation is only possible to occur outside the class of sofic (and more generally cocyclic) shifts, which will be explained in the next section. Example 18 Let . Let us take following sequence of integers:
Chaos on cocyclic shifts and sofic shifts
Now we express positivity of entropy in terms of cardinality of cocyclic shift. Theorem 22 Let X be a cocyclic shift. Then X has positive entropy iff X is uncountable. Proof It has been shown in [3] that X has positive entropy iff there exists such that σn : X → X has an embedded full 2-shift; namely there exist a1, a2, a1 ≠ a2 blocks of length n such that . Full 2-shift is uncountable, therefore X is uncountable. Now let suppose that X has zero entropy. As stated in [3], X has zero entropy
Two sided shift spaces
It should be noted that all presented results are also true when two sided (an thus invertible) shift spaces are considered. Proofs in this case are the same. It is the same for presented examples, as it is enough to put letter c at all negative positions of given sequences and sum all images of σn over . Sets of bi-infinite sequences obtained in this way are closed and shift invariant. But this construction ensures that properties of one sided representatives (like zero of entropy) hold.
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