Elsevier

Fuzzy Sets and Systems

Volume 342, 1 July 2018, Pages 138-152
Fuzzy Sets and Systems

Conditional fuzzy entropy of fuzzy dynamical systems

https://doi.org/10.1016/j.fss.2017.12.011Get rights and content

Abstract

Entropy of fuzzy dynamical systems has been investigated as an isomorphism invariant in Dumitrescu (1995) [13]. In this paper, we will introduce a new invariant called the conditional fuzzy entropy of fuzzy dynamical systems with respect to a finite fuzzy partition and an invariant sub-σ-algebra, which is an extension of fuzzy entropy on fuzzy dynamical systems. This new invariant possesses some basic properties, such as power rule, fuzzy version of Abramov–Rokhlin entropy additive formula and generating sequence.

Introduction

Nowadays, ergodic theory is a large and rapidly developing subject of the modern theory of dynamical systems, which mainly deals with the qualitative study of transformations on a probability measure space in a measure-preserving way. Its initial development was motivated by problems in statistical physics. An old and central problem in ergodic theory is to determine whether or not two given dynamical systems (or measure-preserving transformations) are isomorphic. The usual way to tackle such an isomorphism problem is to look for isomorphism invariants. The classical measure-theoretic entropy in ergodic theory was first introduced by Kolmogorov [16] in 1958 and later by Sinai [29] in the general case. The lecture notes in [24] are classical references. To learn more about the theory related to the entropy, we refer the interested reader to see books [15], [20], [21], [25], [30] and the references therein.

So far, the concept of measure entropy has been proved to be the most successful invariant. On the one hand, isomorphic measure-preserving transformations have the same entropy. On the other hand, Ornstein [17], [18] proved that entropy is a complete invariant for a class of transformations known as Bernoulli shifts, i.e., two Bernoulli shifts with the same entropy are isomorphic. This topic had contributions by many authors. The related results can be found in the references [2], [3], [19].

The theory of fuzzy sets, whose development starts with Zadeh's paper in 1965 [31], has attracted a lot of attention today. It may be seen as an extension of the classical set theory and has been successfully applied to computer and information sciences, intelligent systems, control systems and many other fields.

The concept of fuzzy entropy was introduced based on the idea of substituting partitions, in the classical ergodic theory, by fuzzy partitions. In 1969, Zadeh [32] first introduced the fuzzy entropy of fuzzy events. Dumitrescu [10], [11] and Rybárik [27] introduced the entropy on the σ-algebra of fuzzy sets. Moreover, Dumitrescu [13] defined the entropy of fuzzy dynamical systems as an isomorphism invariant, and gave a fuzzy version of Kolmogorov–Sinai theorem, using the notion of generators of a fuzzy dynamical system [14]. In 2011, Cheng [8] defined and studied the conditional entropy of fuzzy dynamical systems with respect to invariant fuzzy partitions. Recently, Rahimi et al. [22], [23] introduced a new kind of fuzzy version of the local entropy of a continuous dynamical system on a compact metric space and proved some of its properties.

In the classical ergodic theory, the conditional measure-theoretic entropy theory was studied systematically and became a more powerful and flexible tool (see [15], [21], [30] for example). It is natural to ask how to discuss the conditional fuzzy entropy of fuzzy dynamical systems? The aim of this paper is to study this problem. In [13], Dumitrescu proved that two isomorphic fuzzy dynamical systems have the same fuzzy entropy. To study the relationship between the fuzzy entropy of a fuzzy dynamical system and its factor system, we give a reasonable definition of conditional fuzzy entropy for fuzzy dynamical systems. We use the concept of conditional fuzzy entropy to obtain an entropy additive formula, which is viewed as the fuzzy version of the classical Abramov–Rokhlin formula (see e.g. [1]). In [14], the authors defined the generators of a fuzzy dynamical system and given a fuzzy version of Kolmogorov–Sinai theorem, which supplied a tool for calculating the fuzzy entropy. In this paper, we consider a slight modification of the notion of generator. We introduce the fuzzy generating sequence and prove a result, which is often useful to calculate the fuzzy entropy whenever it is difficult to find a generator for a given fuzzy dynamical system. The Pinsker σ-algebra can be viewed as the largest sub-σ-algebra with zero entropy, which became a useful tool in the study of classical ergodic theory and topological dynamical systems (see [15], [30] for example). In this paper, we will introduce the Pinsker σ-algebra of fuzzy dynamical systems, and investigate the relationship between conditional fuzzy entropy relative to the Pinker σ-algebra and the fuzzy entropy of a fuzzy dynamical system.

This paper is organized as follows. In Section 2, we will recall some basic concepts of fuzzy sets and fuzzy measures. In Section 3, we will introduce a new invariant called the conditional fuzzy entropy of fuzzy dynamical systems with respect to a finite fuzzy partition and an invariant sub-σ-algebra. In Section 4, we will investigate some properties of conditional fuzzy entropy of a fuzzy dynamical system, such as power rule, fuzzy version of Abramov–Rokhlin entropy additive formula and generating sequence. In Section 5, we will discuss the Pinsker σ-algebra of fuzzy dynamical systems.

Section snippets

Preliminaries

In this section, we will review some basic concepts about fuzzy sets (see [9], [26], [28], [31]) and fuzzy measures (see [4], [5], [6]).

Conditional entropy of fuzzy dynamical systems

In this section, we will introduce the notion of conditional fuzzy entropy of a fuzzy dynamical system.

Properties of conditional fuzzy entropy

In this section, we will investigate some basic properties of conditional fuzzy entropy. Firstly, we give the following lemma which will be useful in the proof of the main results of this section.

Lemma 4.1

Let (X,F,μ,T) be a fuzzy dynamical system. If Q is a finite fuzzy partition of X, then we havehμ(T)=hμ(T|Q)+hμ(T,Q).

Proof

Using (3) in Lemma 3.8, we havehμ(T,PQ)=hμ(T,Q)+hμ(T,P|Q) for any finite fuzzy partitions P and Q of X. It follows thathμ(T)=supPhμ(T,P)=supPhμ(T,PQ)=supPhμ(T,P|Q)+hμ(T,Q)=hμ(T|Q)+hμ(

Pinsker σ-algebra

In this section, we will introduce the Pinsker σ-algebra of a fuzzy dynamical system, which can be viewed as the largest sub-σ-algebra with zero entropy. Pinsker σ-algebra is an important tool in the study of entropy theory.

Definition 5.1

Let (X,F,μ,T) be a fuzzy dynamical system. DenotePμ(T)={AF:hμ(T,{A,A})=0}. If Pμ(T) is a sub-σ-algebra of F, then we call it the Pinsker σ-algebra of T.

Remark 5.2

It is well known that Pμ(T) is a σ-algebra whenever FP(X) [15], which became an indispensable tool in the classical

Concluding remarks

The concepts of entropy of a fuzzy partition and the entropy of a fuzzy process are successfully introduced by Dumitrescu to define the entropy of a fuzzy dynamical system [11], [12], [13]. The aim of this paper is to study the conditional fuzzy entropy of fuzzy dynamical systems. The definition of the conditional fuzzy entropy is given in three stages: the conditional entropy of finite fuzzy partitions, the conditional entropy of a fuzzy process (Definition 3.7) and the conditional fuzzy

Acknowledgements

The authors would like to thank the editor and anonymous referees for their useful comments and helpful suggestions that improved the manuscript.

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    The authors are supported by NNSF of China (11401288, 11761012) and the Improvement program of Young Teachers of Guangxi Province (KY2016YB405). The first author is partially supported by Project of Outstanding Young Teachers in Higher Education Institutions of Guangxi Province (GXQG022014080), Project of Key Laboratory of Quantity Economics (2016YBKT02) and Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing of Guangxi University of Finance and Economics.

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