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In this article, we study the relations between the dynamical behavior of the shift (Ω, σ) and its induced counterpart (K(Ω), σ). We consequently show that many of the chaotic properties of (K(Ω), σ) can be easily exhibited by the sequences in Ω.

2. THEORETICAL PRELIMINARIES We now introduce some basics from dynamical systems, hyperspace topologies and .

2.1. Dynamical Systems. Let (X, τ) (resp. (X, d)) be a topological (resp. metric) space and let f : X → X be a . The pair (X,f) is referred as a . We state some dynamical properties here, though we refer to [1, 4, 5, 6, 17] for more details. A point x ∈ X is called periodic if f n(x) = x for some positive integer n, where f n = f ◦ f ◦ f ◦ ... ◦ f (n times). The least such n is called the period of the point x. If there exists a δ > 0 such that for every x ∈ X and for each ǫ > 0 there exists y ∈ X and a positive integer n such that d(x, y) < ǫ and d(f n(x),f n(y)) > δ, then f is said to be sensitive (δ-sensitive). The constant δ is called the sensitivity constant for f. f is said to be cofinitely sensitive, if there exists δ > 0 such that the set of instances Nf (U,δ)= {n ∈ N : there exist y,z ∈ U with d(f n(y),f n(z)) > δ} is cofinite. f is called syndetically sensitive if there exists δ > 0 with the property that for every ǫ-neighborhood U of x, Nf (U,δ) is syndetic. In general, cofinitely sensitive ⇒ syndetically sensitive ⇒ sensitive f is Li-Yorke sensitive if there exists δ > 0 such that for each x ∈ X and for each ǫ> 0, there exists y ∈ X with d(x, y) < ǫ such that lim inf d(f n(x),f n(y)) = 0 but lim sup d(f n(x),f n(y)) > δ. A very strong form of n→∞ n→∞ sensitivity is expansivity. f is called expansive (δ-expansive) if for any pair of distinct elements x, y ∈ X, there exists k ∈ N such that d(f k(x),f k(y)) >δ. f is called transitive if for any pair of non-empty open sets U, V in X, there exist a positive integer n such that f n(U) V 6= φ, and is called totally transitive if f n is transitive for each n ∈ N. f is called weakly mixing if f × f is transitive.T f is called mixing or topologically mixing if for each pair of non-empty open sets U, V in X, there exists a positive integer k such that f n(U) V 6= φ for all n ≥ k. f is called locally eventually onto (leo) if for each non-empty open set U, there exists aT positive integer k ∈ N such that f k(U)= X. Among the above topological properties, the following relation holds, leo ⇒ topological mixing ⇒ weakly mixing ⇒ totally transitivity ⇒ transitivity

2.2. Hyperspace Topologies. For a Hausdorff space (X, τ), a hyperspace (K(X), ∆) comprises of all nonempty compact subsets of X endowed with the topology ∆, where the topology ∆ is defined using the topology τ of X. The topology ∆, that we consider here will be either the Vietoris topology or the Hausdorff Metric topology (when X is a ). We briefly describe these topologies. k Define, hU1,U2,...,Uki = {E ∈ K(X): E ⊆ Ui and E Ui 6= φ ∀i}. The topology generated by the iS=1 T collection of all such sets, where k varies over all possible natural numbers and Ui varies over all possible open subsets of X, is known as the Vietoris topology. For a metric space (X, d) and for any two non-empty compact subsets A1, A2 of X, define, dH (A1, A2) = inf{ǫ> 0 : A ⊆ Sǫ(B) and B ⊆ Sǫ(A)} where Sǫ(A)= S(x, ǫ) and S(x, ǫ)= {y ∈ X : d(x, y) <ǫ}. dH is a ∈ xSA metric on K(X) and is known as the Hausdorff metric, which generates the Hausdorff metric topology on K(X). It is known that K(X) is compact if and only if X is compact and in this case, the Hausdorff metric topology is equivalent to the Vietoris topology. Also, it is known that the collection of finite sets is dense in K(X). We can talk of these topologies for any subspace Ψ of K(X). See [3, 12] for details.

2.3. Symbolic Dynamics. We study the sequence space generated by a symbol set A, where |A| may be finite or infinite. In general, we study the sequence space AN or AZ. N Let A be a discrete alphabet set (|A| may be finite or infinite). Let ΣA = A be the space of all infinite + sequences over A. Define D1 :ΣA × ΣA → R as, ∞ δ(xi,yi) D1(¯x, y¯)= 2i iP=0 where,x ¯ = (xi), y¯ = (yi) and δ is the discrete metric. ∞ N δ(xi,yi) Then, D1 defines a metric which generates the on A . Similarly, D1(¯x, y¯)= 2|i| −∞ i=P defines a metric which generates the product topology on AZ, the space of all bifinite sequences over A. It can be seen that the set [i0i1 ...ik]= {(xn): xr = ir, 0 ≤ r ≤ k} is a clopen set in ΣA, and is referred to as a . Any open set, in AN, is a countable union of such sets. Consequently, the cylinder sets form a basis for the product topology on ΣA. The shift (left shift) operator, defined as σ(x0x1 ...) = x1x2x3 ... is N known to be continuous when the space is equipped with the metric D1. We refer the system (A , σ) as the full shift (shift) space. DYNAMICS OF THE INDUCED SHIFT MAP 3

N Z When A is a finite set, A (resp. A ) is a compact metrizable space. Let Σ ⊆ ΣA be a closed σ-invariant subset of ΣA. If there exists a finite collection of words (finite strings) that are forbidden in any sequence in Σ, then the subsystem (Σ, σ) is called a subs