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## 4 An unpredictable solution of the logistic map 

In this section, we will demonstrate the presence of an unpredictable solution of the equation 

where F$_{µ}$ ( s ) = µs (1 − s ) is the logistic map. 

The result is provided in the next theorem. 

<img src='content_image/316696.jpg'>

Theorem 4.1 For each µ ∈ [3 + (2 / 3) 1 / $^{2}$, 4] and sequence of positive numbers δ$_{n}$ → 0 , there exists a solution { η$_{n}$ } , n ∈ $^{Z}$, of equation (4.4) such that 

(i) | η$_{i}$$_{n}$$_{+}$$_{r}$ − η$_{r}$ | < δ$_{n}$, r = − h$_{0}$n, − h$_{0}$n + 1 , . . . , h$_{0}$n, 

(ii) | η$_{i}$$_{n}$$_{+}$$_{j}$$_{n}$ − η$_{j}$$_{n}$ | ≥ ϵ$_{0}$ for each n ∈ $^{N}$, 

where ϵ$_{0}$ is a positive number, h$_{0}$ > 4 is a natural number, and { i$_{n}$ } , { j$_{n}$ } are integer valued sequences both of which diverge to infinity. 

Proof. Fix µ ∈ [3 + (2 / 3) 1 / $^{2}$, 4] and a sequence { δ$_{n}$ } of positive real numbers with δ$_{n}$ → 0 as n → ∞ . Take a neighborhood U ⊂ [0 , 1] of the point 1 − 1 /µ. According to Theorem 6 of paper [20], there exist a natural number h$_{0}$ > 4 and a Cantor set Λ ⊂ U such that the map F h$_{0}$ µ on Λ is topologically conjugate to the Bernoulli shift σ on Σ$_{2}$ . Therefore, there exists a homeomorphism S : Σ$_{2}$ → Λ such that S ◦ σ = F h$_{0}$ µ ◦ S. Since S is uniformly continuous on Σ$_{2}$ , for each n ∈ $^{N}$, there exists a number δ$_{n}$ > 0 such that for each s $^{1}$, s2 ∈ Σ$_{2}$ with d ( s $^{1}$, s$^{2}$) < δ$_{n}$, we have ∣ η 1 − η 2 ∣ < δ$_{n}$/µ h$_{0}$ − $^{1}$, where η 1 = S ( s $^{1}$) , η 2 = S ( s $^{2}$) . 

Let { m$_{n}$ } be an increasing sequence of natural numbers such that 2 − m$_{n}$ < δ$_{n}$ for each n ∈ $^{N}$. Accord- ing to Lemma 3.1, there exist a sequence s ∗∗ ∈ Σ$_{2}$ and sequences { α$_{n}$ } , { β$_{n}$ } both of which diverge to in- finity such that d ( σ α$_{n}$ + $^{r}$( s $^{∗∗}$) , σ $^{r}$( s $^{∗∗}$)) ≤ 2 − $^{m$_{n}$}$, r = − n, − n +1 , . . . , n, and d ( σ α$_{n}$ + $^{β$_{n}$}$( s $^{∗∗}$) , σ $^{β$_{n}$}$( s $^{∗∗}$)) ≥ 1 for each n ∈ $^{N}$. 

Now, let { η$_{n}$ } , n ∈ $^{Z}$, be the solution of (4.4) with η$_{h}$$_{0}$$_{k}$ = S ( σ $^{k}$( s $^{∗∗}$)) , k ∈ $^{Z}$. Since the inequality | F$_{µ}$ ( u$_{1}$ ) − F$_{µ}$ ( u$_{2}$ ) | ≤ µ | u$_{1}$ − u$_{2}$ | is valid for every u$_{1}$, u$_{2}$ ∈ [0 , 1] , we have for each n ∈ $^{N}$that | η$_{i}$$_{n}$$_{+}$$_{r}$ − η$_{r}$ | < δ$_{n}$, r = − h$_{0}$n, − h$_{0}$n + 1 , . . . , h$_{0}$n, where i$_{n}$ = h$_{0}$α$_{n}$. Besides, using the arguments presented in [21], one can verify the existence of a positive number ϵ$_{0}$ such that | η$_{i}$$_{n}$$_{+}$$_{j}$$_{n}$ − η$_{j}$$_{n}$ | ≥ ϵ$_{0}$, n ∈ $^{N}$, where j$_{n}$ = h$_{0}$β$_{n}$. 

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By the topological equivalence and results on Σ$_{2}$ of the last section, one can make several observations from the proved theorem. Any number η$_{n}$, n ∈ $^{Z}$, is an unpredictable point of the logistic map dynamics, and the Cantor set Λ mentioned in the proof of Theorem 4.1 is a quasi-minimal set. Moreover, the sequence { η$_{n}$ } is unpredictable. By Theorem 3.1 mentioned in [2], the dynamics on the quasi-minimal 

