Tsallis entropy of dynamical systems

Abstract

In the papers by Riecan et al. (Fuzzy Sets Syst. 96, 1998) and Ebrahimzadeh et al. (Mathematics, 5, 2017) some fuzzy modifications of Shannon and logical entropy have been studied and the general scheme involving the presented models has been introduced. The present paper deals with providing analogies of these results for the case of Tsallis entropy. In particular, chain rules and concavity property for Tsallis entropy of partitions on the appropriate algebraic structure are established. Using the concept of Tsallis entropy of partitions, we define Tsallis entropy of dynamical systems and prove concavity property of Tsallis entropy of dynamical systems. Finally, we prove the version of Kolmogorov-Sinai Theorem for Tsallis entropy.

Keywords

Tsallis entropy M-preserving transformation dynamical system generator

1 Introduction

The study of concept of entropy is very important in current sciences. The entropy has been applied in information theory, physics, computer science, statistics, chemistry, biology, sociology, general systems theory and other many fields. The classical approach in information theory is based on Shannon entropy [10, 14]. In connection with the problem of the isomorphism of dynamical systems, Kolmogorov and Sinai [12 , 16] using Shannon entropy defined the entropy of dynamical systems. They received a tool for distinction of non-isomorphic dynamical systems by means of which proved the existence of non-isomorphic Bernoulli shifts. Some investigations concerning entropy of dynamical systems and related notions were carried in [3–5]. In [13], a general algebraic theory was introduced.

The entropy known as Tsallis entropy in statistical physics was proposed by Havrda and Charvat in [11] and Tsallis discussed this entropy in [17]. In [1 , 18], the authors deal with studying the concept of Tsallis entropy. If $P = (p_{1}, . . ., p_{n}) \in ℝ^{n}$ is a probability distribution, then Tsallis entropy of P is defined as the formula: $S_{q} (P) = - \sum_{i = 1}^{n} p_{i}^{q} {ln}_{q} p_{i}, (q \neq 1)$ (1) with one parameter q as an extension of Shannon entropy and logical entropy, where q-logarithm function is defined by ${ln}_{q} (x) \equiv \frac{x^{1 - q} - 1}{1 - q}$ for any nonnegative real number x and q [8]. So we obtain $S_{q} (P) = - \sum_{i = 1}^{n} p_{i}^{q} \frac{p_{i}^{1 - q} - 1}{1 - q} = \frac{1}{q - 1} (1 - \sum_{i = 1}^{n} p_{i}^{q}) .$ (2) Tsallis entropy plays an essential role in nonextensive statistics [8]. Furuichi in [8], studied the notions of Tsallis entropy, Tsallis conditional entropy, Tsallis joint entropy, Tsallis relative entropy and Tsallis mutual entropy of probability distributions. In the cited paper, Furuichi also studied difference between Tsallis entropy and Shannon entropy and proved chain rules for Tsallis entropy and Tsallis relative entropy and some of their properties. In the papers [6, 13], the concepts of Shannon entropy and logical entropy of partitions and dynamical systems on an appropriate algebraic structure, have been defined and studied (See also [7]). Since the notion of Tsallis entropy of S_q (P) is a generalization of the notion of logical entropy of S₂ (P) and we studied the notion of logical entropy on the appropriate algebraic structure, this research is important. The relation of between Tsallis entropy and Shannon entropy is presentedin [8].

In this paper, we provide analogies of the results in [6, 13] for the case of Tsallis entropy. We extend the definitions of Tsallis entropy and Tsallis conditional entropy to the appropriate algebraic structure. In fact, we introduce a general algebraic theory for the case of Tsallis entropy. We show that the basic properties of Shannon entropy and logical entropy of partitions in the appropriate algebraic structure, valid for the case of Tsallis entropy, in the case of q > 1. So the suggested measure (for q > 1) can be used (in addition to the Shannon entropy and logical entropy of partitions and dynamical systems) as a measure the amount of uncertainty in random events.

The paper is organized as follows. In the next section, we give the basic definitions. In Section 3, Tsallis entropy of finite partitions on the appropriate algebraic structure is defined. We prove the subadditivity (for q > 1) and concavity properties for the notion of Tsallis entropy of partitions. In Section 4, we define the notion of Tsallis conditional entropy of finite partitions and prove the chain rules for Tsallis conditional entropy of partitions. We also study the notions of Tsallis entropy and Tsallis conditional entropy of independent partitions. In Section 5, we define Tsallis entropy of a dynamical system using the suggested concept of Tsallis entropy of finite partitions. We prove the concavity property for Tsallis entropy of dynamical systems. Finally, an analogy of the Kolmogorov-Sinai Theorem on generators of dynamical systems for the case of Tsallis entropy is proved.

2 Basic definitions

As in [13] we shall consider the algebraic structure (F, ⊕, ⊗, 1_F) where F is a non-empty partially ordered set, ⊕ is a partial binary operation on F, ⊗ is a binary operation on F, 1_F is a fixed element of F, and two mappings m : F ⟶ [0, 1] and u : F ⟶ F, where the following conditions are satisfied:

(F₁) m (f ⊗ g) = m (g ⊗ f), for any f, g ∈ F, and if f ⊕ g exists, then g ⊕ f exists, too, and m (f ⊕ g) = m (g ⊕ f); (F₂) m (f ⊗ (g ⊗ h)) = m ((f ⊗ g) ⊗ h) for any f, g, h ∈ F, and if (f ⊕ g) ⊕ h exists, then f ⊕ (g ⊕ h) exists, too, and m (f ⊕ (g ⊕ h)) = m ((f ⊕ g) ⊕ h); (F₃) for any f, g, h ∈ F, if (f ⊕ g) ⊗ (f ⊕ h) exists, then f ⊕ (g ⊗ h) exists and m (f ⊕ (g ⊗ h)) = m ((f ⊕ g) ⊗ (f ⊕ h)); (F₄) f ⊗ g ≤ f = 1_F ⊗ f, for every f, g∈ F ; (F₅) if $\oplus_{i = 1}^{n} f_{i}$ exists, then $m (\oplus_{i = 1}^{n} f_{i}) = \sum_{i = 1}^{n} m (f_{i})$ ; (F₆) if f, g ∈ F, f ≤ g then m (f) ≤ m (g); (F₇) if f ∈ F such that m (f) = m (1_F), then m (f ⊗ g) = m (g), for every g ∈ F; (F₈) for any f, g ∈ F, if f ⊕ g exists, then u (f) ⊕ u (g) exists, too, and m (u (f ⊕ g)) = m (u (f) ⊕ u (g)); (F₉) u : F ⟶ F is an m- preserving transformation, i.e., m (u (f)) = m (f) for every f ∈ F. Some examples about this algebraic structure were presented in [6].

3 Tsallis Entropy of partitions in F

In this section, we define the notion of Tsallis entropy of finite partitions on the algebraic structure defined in Section 2. We prove the concavity and subadditivity properties for Tsallis entropy of partitions and study Tsallis entropy of partitions in F, under the relation of refinement.

Definition 3.1.[13] By a partition (in F) we mean a finite collection $A = {f_{1}, . . ., f_{n}} \subset F$ such that $\oplus_{i = 1}^{n} f_{i}$ exists, and: $m (1_{F}) = m (\oplus_{i = 1}^{n} f_{i}) = \sum_{i = 1}^{n} m (f_{i}) .$

Definition 3.2.[13] If $A = {f_{1}, . . ., f_{n}}$ and $B = {g_{1}, . . ., g_{p}}$ are two partitions in F, then the common refinement of these partitions is defined as $A \lor B = {f_{i} \otimes g_{j} : f_{i} \in A, g_{j} \in B}$ if $A \neq B,$ and $A \lor A = A .$ We say that $B$ is a refinement of $A$ , denoted by $A ≺ B$ , if there exists a partition I (1), …, I (n) of the set {1, …, p} such that m (f_i) = ∑_j∈I(i)m (g_j), for every i = 1, …, n. [6]

Note that $A \lor B$ is a partition in F, too. [6]

We shall now define the notion of Tsallis entropy of finite partitions. Let $A = {f_{1}, . . ., f_{n}}$ be a partition in F. Tsallis entropy of $A$ is defined by: $S_{q} (A) = - \sum_{i = 1}^{n} (m (f_{i}))^{q} {ln}_{q} m (f_{i}),$ (3) where q > 0, q ≠ 1. Then by $\ln_{q} (x) \equiv \frac{x^{1 - q} - 1}{1 - q},$ we obtain

$\begin{matrix} S_{q} (A) & = & - \sum_{i = 1}^{n} (m (f_{i}))^{q} \frac{(m (f_{i}))^{1 - q} - 1}{1 - q} \\ = & \frac{1}{q - 1} (\sum_{i = 1}^{n} m (f_{i}) - \sum_{i = 1}^{n} (m (f_{i}))^{q}) \\ = & \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} (m (f_{i}))^{q}) . \end{matrix}$

Therefore, $S_{q} (A)$ can be defined as follows:

Definition 3.3. Let $A = {f_{1}, . . ., f_{n}}$ be a partition in F. Tsallis entropy of $A$ defined as the number $S_{q} (A) = \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} (m (f_{i}))^{q})$ (4) where q > 0, q ≠ 1.

In the following theorem, we prove the concavity property for the notion of Tsallis entropy.

Theorem 3.4. Let $A$ be a finite partition in F corresponding to m₁ and m₂. Then for every r ∈ [0, 1], $(i) A$ is a partition in F corresponding to rm₁ + (1 - r) m₂, too; $(ii) r S_{q} (A, m_{1}) + (1 - r) S_{q} (A, m_{2}) \leq S_{q} (A, r m_{1} + (1 - r) m_{2}) .$

Proof. (i) It is easy to see that the conditions (F₁), …, (F₉) hold for m₁ + (1 - r) m₂, too. Assume that $A = {f_{1}, . . ., f_{n}} .$ Since $A$ is a partition in F corresponding to m₁ and m₂, from Definition 3.1, we obtain

$\begin{matrix} (r m_{1} + (1 - r) m_{2}) (\oplus_{i = 1}^{n} f_{i}) \\ = r m_{1} (\oplus_{i = 1}^{n} f_{i}) + (1 - r) m_{2} (\oplus_{i = 1}^{n} f_{i}) \\ = r \sum_{i = 1}^{n} m_{1} (f_{i}) + (1 - r) \sum_{i = 1}^{n} m_{2} (f_{i}) \\ = \sum_{i = 1}^{n} (r m_{1} + (1 - r) m_{2}) (f_{i}) . \end{matrix}$

Also

$\begin{matrix} (r m_{1} + (1 - r) m_{2}) (\oplus_{i = 1}^{n} f_{i}) \\ = r m_{1} (\oplus_{i = 1}^{n} f_{i}) + (1 - r) m_{2} (\oplus_{i = 1}^{n} f_{i}) \\ = r m_{1} (1_{F}) + (1 - r) m_{2} (1_{F}) \\ = (r m_{1} + (1 - r) m_{2}) (1_{F}) . \end{matrix}$ (ii) Since the function f : R → R defined, for every x ∈ R, by f (x) = x^q is convex when q > 1, we have for any x₁, x₂ ∈ R and everyr ∈ [0, 1],

$(r x_{1} + (1 - r) x_{2})^{q} \leq {rx}_{1}^{q} + (1 - r) x_{2}^{q} .$ Now if we put for an arbitrary i ∈ {1, …, n}, x₁ = m₁ (f_i) and x₂ = m₂ (f_i), then we get (rm₁ (f_i) + (1 - r) m₂ (f_i)) ^q ≤ r (m₁ (f_i)) ^q + (1 - r) (m₂ (f_i)) ^q. Therefore $\sum_{i = 1}^{n} ({rm}_{1} (f_{i}) + (1 - r) m_{2} (f_{i}))^{q} \leq r \sum_{i = 1}^{n} (m_{1} (f_{i}))^{q} + (1 - r) \sum_{i = 1}^{n} (m_{2} (f_{i}))^{q} .$ Thus $r S_{q} (A, m_{1}) + (1 - r) S_{q} (A, m_{2}) = \frac{1}{q - 1} r (m_{1} (1_{F}) - \sum_{i = 1}^{n} (m_{1} (f_{i}))^{q}) + \frac{1}{q - 1} (1 - r) (m_{2} (1_{F}) - \sum_{i = 1}^{n} (m_{2} (f_{i}))^{q}) = \frac{1}{q - 1} (({rm}_{1} + (1 - r) m_{2}) (1_{F}) - r \sum_{i = 1}^{n} (m_{1} (f_{i}))^{q} - (1 - r) \sum_{i = 1}^{n} (m_{2} (f_{i}))^{q}) \leq \frac{1}{q - 1} (({rm}_{1} + (1 - r) m_{2}) (1_{F}) - \sum_{i = 1}^{n} (r m_{1} (f_{i}) + (1 - r) m_{2} (f_{i}))^{q}) = S_{q} (A, r m_{1} + (1 - r) m_{2}) .$

In the case of q < 1, since f (x) = x^q is concave we get for any f_i, (rm₁ (f_i) + (1 - r) m₂ (f_i)) ^q ≥ r (m₁ (f_i)) ^q + (1 - r) (m₂ (f_i)) ^q. Summing these inequalities over i = 1, …, n, we have $\sum_{i = 1}^{n} ({rm}_{1} (f_{i}) + (1 - r) m_{2} (f_{i}))^{q} \geq r \sum_{i = 1}^{n} (m_{1} (f_{i}))^{q} + (1 - r) \sum_{i = 1}^{n} (m_{2} (f_{i}))^{q} .$ Hence similar to the above since $\frac{1}{q - 1} < 0,$ we obtain the assertion for this case, too. Thus the proof of the theorem is completed. □

Lemma 3.5. Let $A = {f_{1}, . . ., f_{n}}$ and $B = {g_{1}, . . ., g_{p}}$ be two partitions in F, then we have: $(i) \sum_{i = 1}^{n} m (f_{i} \otimes g_{j}) = m (g_{j});$ $(ii) \sum_{j = 1}^{p} m (f_{i} \otimes g_{j}) = m (f_{i}) .$

Proof. (i) According to (F7), (F3) and (F5) we get, for each j = 1,.., p, $m (g_{j}) = m ((\oplus_{i = 1}^{n} f_{i}) \otimes g_{j}) = m (\oplus_{i = 1}^{n} (f_{i} \otimes g_{j})) = \sum_{i = 1}^{n} m (f_{i} \otimes g_{j}) .$ (ii) It is similar to the part (i) of this lemma. □

Now we prove the subadditivity of Tsallis entropy in F.

Theorem 3.6. Let $A$ and $B$ be finite partitions in F and let m (1_F) =1. If q > 1, then $S_{q} (A \lor B) \leq S_{q} (A) + S_{q} (B) .$ Proof. Let $A = {f_{1}, . . ., f_{n}}$ and $B = {g_{1}, . . ., g_{p}} .$ Since m (1_F) =1, we get $\sum_{j = 1}^{p} m (g_{j}) = 1,$ and from Lemma 3.5, we obtain $\sum_{j = 1}^{p} m (f_{i} \otimes g_{j}) = m (f_{i}), \sum_{i = 1}^{n} \frac{m (f_{i} \otimes g_{j})}{m (g_{j})} = 1,$ and $\sum_{i = 1}^{n} \sum_{j = 1}^{p} m (f_{i} \otimes g_{j}) = 1 .$ Therefore from (2), we have $S_{q} (\sum_{j = 1}^{p} m (f_{1} \otimes g_{j}), . . ., \sum_{j = 1}^{p} m (f_{n} \otimes g_{j})) = \frac{1}{q - 1} (1 - \sum_{i = 1}^{n} (\sum_{j = 1}^{p} m (f_{i} \otimes g_{j}))^{q}) = \frac{1}{q - 1} (1 - \sum_{i = 1}^{n} (m (f_{i}))^{q}) = \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} (m (f_{i}))^{q}) = S_{q} (A) .$ Let $(p_{1}, . . ., p_{m}) \in ℝ^{n}$ and $(v_{1_{j}}, . . ., v_{n_{j}}) \in ℝ^{n}$ be probability distributions. From Theorem 7 in [2], since q > 1, we have

$\sum_{j = 1}^{p} {p_{j}}^{q} S_{q} (v_{1_{j}}, . . ., v_{n_{j}}) \leq S_{q} (\sum_{j = 1}^{p} p_{j} v_{1_{j}}, . . ., \sum_{j = 1}^{r} p_{j} v_{n_{j}}) .$ (5) Put p_j = m (g_j) and $v_{i_{j}} = \frac{m (f_{i} \otimes g_{j})}{m (g_{j})}$ for every i = 1, …, n and j = 1, …, p. Thus from (4) and Lemma 3.5 (ii), we obtain $S_{q} (A \lor B) = \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}) = \frac{1}{q - 1} (m (1_{F}) - \sum_{j = 1}^{p} (m (g_{j}))^{q} \sum_{i = 1}^{n} (\frac{m (f_{i} \otimes g_{j})}{m (g_{j})})^{q}) = \frac{1}{q - 1} (1 - \sum_{j = 1}^{p} (m (g_{j}))^{q} + \sum_{j = 1}^{p} (m (g_{j}))^{q} - \sum_{j = 1}^{p} (m (g_{j}))^{q} \sum_{i = 1}^{n} (\frac{m (f_{i} \otimes g_{j})}{m (g_{j})})^{q}) = \frac{1}{q - 1} (1 - \sum_{j = 1}^{p} (m (g_{j}))^{q}) + \sum_{j = 1}^{p} (m (g_{j}))^{q} \frac{1}{q - 1} (1 - \sum_{i = 1}^{n} (\frac{m (f_{i} \otimes g_{j})}{m (g_{j})})^{q}) = S_{q} (B) + \sum_{j = 1}^{p} (m (g_{j}))^{q} S_{q} (\frac{m (f_{1} \otimes g_{j})}{m (g_{j})}, . . ., \frac{m (f_{n} \otimes g_{j})}{m (g_{j})}) \leq S_{q} (B) + S_{q} (\sum_{j = 1}^{p} m (f_{1} \otimes g_{j}), . . ., \sum_{j = 1}^{p} m (f_{n} \otimes g_{j})) = S_{q} (A) + S_{q} (B) .$ Thus the proof of this theorem is completed. □

Now Tsallis entropy of finite partitions in F, under the relation of refinement will be studied.

Theorem 3.7. Let $A = {f_{1}, . . ., f_{n}}, B = {g_{1}, . . ., g_{p}}$ be partitions in F. Then for every $q > 0, q \neq 1; A ≺ B$ implies that $S_{q} (A) \leq S_{q} (B) .$

Proof. Since $A ≺ B,$ there exists a partition I (1), …, I (n) of the set {1, …, p} such that m (f_i) = ∑_j∈I(i)m (g_j), i = 1, …, n. So for q < 1, we have $\sum_{j = 1}^{p} (m (g_{j}))^{q} \geq \sum_{i = 1}^{n} (m (f_{i}))^{q},$ therefore since $\frac{1}{q - 1} < 0$ we get $\begin{matrix} S_{q} (A) & = & \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} (m (f_{i}))^{q}) \\ \leq & \frac{1}{q - 1} (m (1_{F}) - \sum_{j = 1}^{p} (m (g_{j}))^{q}) = S_{q} (B) . \end{matrix}$ But if q > 1, then $\sum_{j = 1}^{p} (m (g_{j}))^{q} \leq \sum_{i = 1}^{n} (m (f_{i}))^{q} .$ Thus similar to the above relations, the assertion holds. □

4 Tsallis conditional entropy of partitions in F

In this section, we define the notion of Tsallis conditional entropy of finite partitions in F. We prove the chain rules for Tsallis conditional entropy of partitions. We also study the notions of Tsallis entropy and Tsallis conditional entropy of independent partitions in F.

Let $A = {f_{1}, . . ., f_{n}}$ and $B = {g_{1}, . . ., g_{p}}$ be partitions in F. Tsallis conditional entropy of $A$ given $B$ is defined by: $S_{q} (A | B) = - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q} {ln}_{q} m (f_{i} | g_{j}),$ (6) where $m (f_{i} | g_{j}) = \frac{m (f_{i} \otimes g_{j})}{m (g_{j})},$ and q > 0, q ≠ 1. Then via Lemma 3.5, (i), we obtain $S_{q} (A | B) = \frac{1}{q - 1} (\sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q} \frac{(m (f_{i} \otimes g_{j}))^{1 - q}}{(m (g_{j}))^{1 - q}} - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}) = \frac{1}{q - 1} (\sum_{i = 1}^{n} \sum_{j = 1}^{p} m (f_{i} \otimes g_{j}) (m (g_{j}))^{q - 1} - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}) = \frac{1}{q - 1} (\sum_{j = 1}^{p} (m (g_{j}))^{q} - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}) .$ Therefore, $S_{q} (A | B)$ can be defined as follows:

Definition 4.1. Let $A = {f_{1}, . . ., f_{n}}$ and $B = {g_{1}, . . ., g_{p}}$ be two partitions in F. Tsallis conditional entropy of $A$ given $B$ is defined as: $S_{q} (A | B) = \frac{1}{q - 1} (\sum_{j = 1}^{p} (m (g_{j}))^{q} - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}),$ where q > 0, q ≠ 1.

Remark 4.2. For each q > 0, q ≠ 1, we have $S_{q} (A | B) \geq 0,$ because by Definition 4.1, we obtain $S_{q} (A | B) = \frac{1}{q - 1} (\sum_{j = 1}^{p} (m (g_{j}))^{q} - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}) = \frac{1}{q - 1} (\sum_{j = 1}^{p} (m (g_{j}))^{q - 1} \sum_{i = 1}^{n} m (f_{i} \otimes g_{j}) - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}) = \frac{1}{q - 1} \sum_{i = 1}^{n} \sum_{j = 1}^{p} m (f_{i} \otimes g_{j}) ((m (g_{j}))^{q - 1} - (m (f_{i} \otimes g_{j}))^{q - 1}) .$ Now if q > 1, since for each i, j, m (g_j) ≥ m (f_i ⊗ g_j), we have (m (g_j)) ^q-1 ≥ (m (f_i ⊗ g_j)) ^q-1, i = 1, …, n, j = 1, …, p. Therefore we conclud for each q > 1, $S_{q} (A | B) \geq 0 .$ But if 0 < q < 1, then since q - 1 <0, and m (g_j) ≥ m (f_i ⊗ g_j), it holds (m (g_j)) ^q-1 ≤ (m (f_i ⊗ g_j)) ^q-1, i = 1, …, n, j = 1, …, p. Therefore we conclud for each 0 < q < 1, $S_{q} (A | B) \geq 0 .$

A chain rule is proved for Tsallis entropy of partitions in F, by the following theorem. The assertion of this theorem will be useful in further theorems.

Theorem 4.3. Let $A$ and $B$ be finite partitions in F. Then $S_{q} (A \lor B) = S_{q} (B) + S_{q} (A | B) .$ Consequently, $S_{q} (A \lor B) \geq max {S_{q} (A), S_{q} (B)} .$

Proof. Assume that $A = {f_{1}, . . ., f_{n}}$ and $B = {g_{1}, . . ., g_{p}} .$ We have $S_{q} (A \lor B) - S_{q} (B) = \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q} - m (1_{F}) + \sum_{j = 1}^{p} (m (g_{j}))^{q}) = S_{q} (A | B) .$ □

In the following theorem, the another chain rule is proved for Tsallis conditional entropy of partitions in F.

Theorem 4.4. Let $A, B$ and $C$ be finite partitions in F. then $S_{q} (A \lor B | C) = S_{q} (A | C) + S_{q} (B | A \lor C) .$

Proof. Let $A = {f_{1}, . . ., f_{n}},$ $B = {g_{1}, . . ., g_{p}}$ and $C = {h_{1}, . . ., h_{r}} .$ We have $S_{q} (A | C) + S_{q} (B | A \lor C) = \frac{1}{q - 1} (\sum_{k = 1}^{r} (m (h_{k}))^{q} - \sum_{i = 1}^{n} \sum_{k = 1}^{r} (m (f_{i} \otimes h_{k}))^{q}) + \frac{1}{q - 1} (\sum_{i = 1}^{n} \sum_{k = 1}^{r} (m (f_{i} \otimes h_{k}))^{q}) - \sum_{i = 1}^{n} \sum_{j = 1}^{p} \sum_{k = 1}^{r} (m (f_{i} \otimes g_{j} \otimes h_{k}))^{q}) = \frac{1}{q - 1} (\sum_{k = 1}^{r} (m (h_{k}))^{q} - \sum_{i = 1}^{n} \sum_{j = 1}^{p} \sum_{k = 1}^{r} (m (f_{i} \otimes g_{j} \otimes h_{k}))^{q}) = S_{q} (A \lor B | C) .$ □

From Remark 4.2 and Theorem 4.4, we have the following corollary.

Corollary 4.5. Let $A, B$ and $C$ be finite partitions in F. Then $(i) S_{q} (A | C) \leq S_{q} (A \lor B | C);$ $(ii) S_{q} (B | A \lor C) \leq S_{q} (A \lor B | C) .$

In the next theorem an inequality is proved.

Theorem 4.6. Let $A$ and $B$ be finite partitions in F and let m (1_F) =1. If q > 1, then $S_{q} (A | B) \leq S_{q} (A) .$

Proof. From Theorems 3.6 and 4.3, we conclude $S_{q} (A | B) = S_{q} (A \lor B) - S_{q} (B) \leq S_{q} (A) .$ □

Two finite partitions $A$ and $B$ in F are called independent if m (f ⊗ g) = m (f) m (g) for all $f \in A$ and $g \in B .$ In the next theorems we observe that, for two independent finite partitions $A$ and $B$ in F, $S_{q} (A \lor B) \neq S_{q} (A) + S_{q} (B)$ necessarily. In this case, also $S_{q} (A | B) \neq S_{q} (A)$ necessarily.

Theorem 4.7 Let $A$ and $B$ be independent finite partitions in F. Then $S_{q} (A \lor B) = S_{q} (B) + \frac{1}{q - 1} (m (1_{F}) + (1 - q) S_{q} (B)) \times (1 - m (1_{F}) + (q - 1) S_{q} (A)) .$

Proof. Since $A, B$ are independent, by Definition 3.3, we may write $S_{q} (A \lor B) = \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} \sum_{j = 1}^{p} (m (f_{i} \otimes g_{j}))^{q}) = \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} (m (f_{i}))^{q} \sum_{j = 1}^{p} (m (g_{j}))^{q}) = \frac{1}{q - 1} (m (1_{F}) - \sum_{j = 1}^{p} (m (g_{j}))^{q} + \sum_{j = 1}^{p} (m (g_{j}))^{q} - \sum_{i = 1}^{n} (m (f_{i}))^{q} \sum_{j = 1}^{p} (m (g_{j}))^{q}) = \frac{1}{q - 1} (m (1_{F}) - \sum_{j = 1}^{p} (m (g_{j}))^{q}) + \frac{1}{q - 1} \sum_{j = 1}^{p} (m (g_{j}))^{q} (1 - \sum_{i = 1}^{n} (m (f_{i}))^{q}) = S_{q} (B) + \frac{1}{q - 1} (m (1_{F}) + (1 - q) S_{q} (B)) (1 - m (1_{F}) + (q - 1) S_{q} (A)) .$ □

Theorem 4.8. Let $A$ and $B$ be independent finite partitions in F. Then (i) $S_{q} (A | B) = \frac{1}{q - 1} (m (1_{F}) + (1 - q) S_{q} (B)) (1 - m (1_{F}) + (q - 1) S_{q} (A));$ (ii) if m (1_F) =1 then $S_{q} (A \lor B) = S_{q} (A) + S_{q} (B) + (1 - q) S_{q} (A) S_{q} (B);$ (iii) if m (1_F) =1 then $S_{q} (A | B) = S_{q} (A) + (1 - q) S_{q} (A) S_{q} (B) .$

Proof. (i) and (ii) Follow from Theorems 4.3 and 4.7. (iii) Follows from Theorem 4.7 and the part (i) of this theorem. □

5 Tsallis entropy of dynamical systems in F

In this section, using the suggested concept of Tsallis entropy of partitions in F, we define Tsallis entropy of a dynamical system. We prove the concavity property of Tsallis entropy of dynamical systems and prove an analogy of the Kolmogorov-Sinai Theorem on generators of dynamical systems for the case of Tsallis entropy.

Let any dynamical system (F, m, u) be given. If $A$ is a partition in F then $u (A) = {u (f_{1}), . . ., u (f_{n})}$ is a partition in F, too. Namely, $\oplus_{i = 1}^{n} u (f_{i})$ exists, since $\oplus_{i = 1}^{n} f_{i}$ exists we have $m (\oplus_{i = 1}^{n} u (f_{i})) = \sum_{i = 1}^{n} m (u (f_{i})) .$ [6]

Lemma 5.1. Let ${a_{n}}_{n = 1}^{\infty}$ be a sequence of nonnegative real numbers such that a_n+m ≤ a_n + a_m for every $n, m \in ℕ .$ Then $lim_{n \to \infty} \frac{1}{n} a_{n}$ exists.

Proof. The proof can be found in [19]. □

In the following theorem, the existence of the limit in Definition 5.3, is shown.

Theorem 5.2. Let (F, m, u) be a dynamical system and $A$ be a partition in F, then $(i) S_{q} (u^{k} A) = S_{q} (A), k = 0, 1, 2, . . .;$ (ii) if m (1_F) =1, then for $q > 1, lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{i = 0}^{n - 1} u^{i} A)$ exists.

Proof. (i) Since m (u (f)) = m (f) for every f ∈ F, we have m (u^k (f_i)) = m (f_i), i = 1, 2, …, n, k = 0, 1, 2, … Thus for every k = 0, 1, 2, …

$\begin{matrix} S_{q} (u^{k} A) & = & \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} m (u^{k} (f_{i}))^{q}) \\ = & \frac{1}{q - 1} (m (1_{F}) - \sum_{i = 1}^{n} m (f_{i})^{q}) \\ = & S_{q} (A) . \end{matrix}$

(ii) Let $a_{n} = S_{q} (\lor_{i = 0}^{n - 1} u^{i} A)$ . According to subadditivity of Tsallis entropy (Theorem 3.6) and the part (i) of this theorem, we get

$\begin{matrix} a_{n + m} & = & S_{q} (\lor_{i = 0}^{n + m - 1} u^{i} A) \\ \leq & S_{q} (\lor_{i = 0}^{n - 1} u^{i} A) + S_{q} (\lor_{i = n}^{n + m - 1} u^{i} A) \\ = & a_{n} + S_{q} (\lor_{i = 0}^{m - 1} u^{n + i} A) \\ = & a_{n} + S_{q} (u^{n} (\lor_{i = 0}^{m - 1} u^{i} A)) \\ = & a_{n} + S_{q} (\lor_{i = 0}^{m - 1} u^{i} A) \\ = & a_{n} + a_{m} . \end{matrix}$ By the previous lemma $lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{i = 0}^{n - 1} u^{i} A)$ exists. □

The second stage and the final stage of the definition of Tsallis entropy of a dynamical system (F, m, u) is given in the next definition.

Definition 5.3. Let (F, m, u) be a dynamical system and $A$ be a partition in F and q > 1 and let m (1_F) =1. Tsallis entropy of u respect to $A$ is defined by: $S_{q} (u, A) = lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{i = 1}^{n} u^{i} A) .$ (7)

Tsallis entropy of u is defined as: $S_{q} (u) = sup_{A} S_{q} (u, A),$ (8) where the supremum is taken over all finite partitions in F.

In the following theorem, the concavity property of $S_{q} (u, A)$ will be studied.

Theorem 5.4. Let (F, m₁, u) and (F, m₂, u) be dynamical systems such that m₁ (1_F) = m₂ (1_F) =1. Then for every partition $A$ and for r ∈ [0, 1], $r S_{q} (u, A, m_{1}) + (1 - r) S_{q} (u, A, m_{2}) \leq S_{q} (u, A, r m_{1} + (1 - r) m_{2}) .$

Proof. Follows from Theorem 3.4 (ii). □

Theorem 5.5. If (F, m, u) is a dynamical system and m (1_F) =1 and $A$ is a partition in F, then (i) $S_{q} (u, A) = S_{q} (u, \lor_{i = 1}^{k} u^{i} A);$ (ii) For $k \in ℕ$ , S_q (u^k) = kS_q (u).

Proof. (i) $\begin{matrix} S_{q} (u, \lor_{i = 1}^{k} u^{i} A) & = & lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{j = 1}^{n} u^{j} (\lor_{i = 1}^{k} u^{i} A)) \\ = & lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{i = 1}^{k + n - 1} u^{i} A) \\ = & lim_{n \to \infty} (\frac{k + n - 1}{n}) (\frac{1}{k + n - 1}) \\ \times & S_{q} (\lor_{i = 1}^{k + n - 1} u^{i} A) \\ = & S_{q} (u, A) . \end{matrix}$ (ii) Let $A$ be an arbitrarary finite partition in F. we may write $\begin{matrix} S_{q} (u^{k}, \lor_{i = 1}^{n} u^{i} A) & = & lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{j = 1}^{n} (u^{k})^{j} (\lor_{i = 1}^{n} u^{i} A)) \\ = & lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{j = 1}^{n} \lor_{i = 1}^{k} u^{kj + i} A) \\ = & lim_{n \to \infty} \frac{1}{n} S_{q} (\lor_{i = 1}^{nk - 1} u^{i} A) \\ = & lim_{n \to \infty} \frac{nk}{n} \frac{1}{nk} S_{q} (\lor_{i = 1}^{nk - 1} u^{i} A) \\ = & {kS}_{q} (u, A) \end{matrix}$ So ${kS}_{q} (u) = k sup_{A} S_{q} (u, A) = sup_{A} S_{q} (u^{k}, \lor_{i = 1}^{n} u^{i} A) \leq sup_{A} S_{q} (u^{k}, A) = S_{q} (u^{k}) .$ Since $A ≺ \lor_{i = 1}^{k} u^{i} A$ , by Theorem 3.7, we get $S_{q} (u^{k}, A) \leq S_{q} (u^{k}, \lor_{i = 1}^{n} u^{i} A) = {kS}_{q} (u, A) .$ (9) □

Definition 5.6. Let (F, m, u) be a dynamical system. A finite partition $C$ in F, is said to be a generator of u, if there exists $r \in ℕ$ such that $A ≺ \lor_{i = 1}^{r} u^{i} C$ for each finite partition $A$ in F.

The main aim of this theorem is to prove an analogue of the Kolmogorov-Sinaj theorem on Tsallis entropy and generators.

Theorem 5.7. Let (F, m, u) be a dynamical system and m (1_F) =1. If $C$ is a generator of u, and q > 1, then $S_{q} (u) = S_{q} (u, A) .$

Proof. Let $A$ be an arbitrary finite partition in F. Since $A$ is a generator, $A ≺ \lor_{i = 1}^{r} u^{i} C .$ By Theorem 3.7, since q > 1, we get $S_{q} (u, A) \leq S_{q} (u, \lor_{i = 1}^{r} u^{i} C) = S_{q} (u, C) .$ Hence $S_{q} (u) = sup_{A} S_{q} (u, A) \leq S_{q} (u, C) .$ On the other hand $S_{q} (u, C) \leq S_{q} (u) .$ □

6 Discussion

The classical Tsallis entropy was discussed (e.g., in [8 , 17]) as an alternative measure of information. We have extended the definitions of Tsallis entropy and Tsallis conditional entropy to an appropriate algebraic structure. In fact, we have introduced a general algebraic theory for the case of Tsallis entropy. In the papers [6, 13], the notions of Shannon entropy and logical entropy of partitions on the appropriate algebraic structure were introduced and studied. In this paper, we have generalized the results of [6, 13], concerning Tsallis entropy. We have defined the notions of Tsallis entropy and Tsallis conditional entropy of finite partitions on the appropriate algebraic structure and proved basic properties of these measures. The properties of subadditivity (in the case of q > 1) and concavity for Tsallis entropy of partitions were proved. We have also studied Tsallis entropy of independent partitions. It was shown that, the basic properties of Shannon entropy and logical entropy of partitions in the appropriate algebraic structure, valid for the case of Tsallis entropy, in the case of q > 1. Using the suggested concept of Tsallis entropy of finite partitions, we have defined Tsallis entropy of dynamical systems and proved the concavity property of Tsallis entropy of dynamical systems. We have also proved the version of Kolmogorov-Sinai theorem for Tsallis entropy in the case of q > 1. So the suggested measure information can be used besides of Shannon entropy and logical entropy of dynamicalsystems.

Footnotes

Acknowledgments

The authors thank the editor and the referees for their valuable comments and suggestions.

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Tsallis entropy of dynamical systems – a general scheme

Abstract

Keywords

1 Introduction

3 Tsallis Entropy of partitions in F

Footnotes

Acknowledgments

References