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function. 

## 3 Unpredictable sequences 

The definition of an unpredictable sequence is as follows. 

Definition 3.1 A bounded sequence { κ$_{i}$ } in a metric space ( X, d ) is called unpredictable if there exist a positive number ϵ$_{0}$ and sequences { j$_{n}$ } , { k$_{n}$ } of positive integers both of which diverge to infinity such that d ( κ$_{i}$$_{+}$$_{j}$ n , κ$_{i}$ ) → 0 as n → ∞ for each i in bounded intervals of integers and d ( κ$_{j}$ n + k n , κ$_{k}$ n ) ≥ ϵ$_{0}$ for each n ∈ N . 

Let us consider the space Σ$_{2}$ = { s = ( s$_{0}$s$_{1}$s$_{2}$ . . . ) | s$_{j}$ = 0 or 1 } of infinite sequences of 0 ’s and 1 ’s with the metric 

<img src='content_image/299872.jpg'>

where s = ( s$_{0}$s$_{1}$s$_{2}$ . . . ) , t = ( t$_{0}$t$_{1}$t$_{2}$ . . . ) ∈ Σ$_{2}$ . The Bernoulli shift σ : Σ$_{2}$ → Σ$_{2}$ is defined as σ ( s$_{0}$s$_{1}$s$_{2}$ . . . ) = ( s$_{1}$s$_{2}$s$_{3}$ . . . ) . The map σ is continuous and Σ$_{2}$ is a compact metric space [7, 17]. 

Through the proof of Lemma 3.1. [6], we constructed an element s ∗∗ = ( s ∗∗ 0 s ∗∗ 1 s ∗∗ 2 . . . ) of Σ$_{2}$ which is unpredictable in the sense of Definition 3.1 by placing all blocks of 0 ’s and 1 ’s in a specific order without any repetitions and extending it to the left hand side by appropriately choosing the terms s ∗∗ i for negative values of i. 

Now, we take into account the logistic map 

<img src='content_image/299873.jpg'>

where i ∈ Z and F$_{µ}$ ( s ) = µs (1 − s ) . The interval [0 , 1] is invariant under the iterations of (3.3) for µ ∈ (0 , 4] [9]. 

It was proved by Shi and Yu [16] that for each µ ∈ [3 + (2 / 3) 1 / $^{2}$, 4] , there exist a natural number h$_{0}$ > 4 and a Cantor set Λ ⊂ [0 , 1] such that the map F h 0 µ on Λ is topologically conjugate to the Bernoulli shift σ on Σ$_{2}$ . This gives immediately the proof of the following theorem. 

Theorem 3.1 For each µ ∈ [3 + (2 / 3) 1 / $^{2}$, 4] , the logistic map (3.3) possesses an unpredictable solution. 

The next section is devoted to examples. 

