On the Notion of Observers

Abstract

By considering the idea and the notion of observers of dynamical systems, it is shown that relative semidynamical systems can be a suitable model for inserting an observer as a realistic fact to systems theory. Multi-dimensional observers and their affect on semidynamical systems are considered. Locally distinguishable semidynamical systems are introduced, and it is proved that this property preserves under relative conjugate relations which preserves observers. For an application of the role of observer relative topological entropy for relative semidynamical systems on compact relative topological spaces are considered.

1 Introduction

State spaces and observable objects are two main points in systems theory, which have essential roles in the realistic models in control theory and quantum physics. In the case of observable systems, one can reverse the time by determining the initial state via its output [11, 13,14,15, 17, 18, 20].

There is another notion in systems theory which is called “observer.” This notion determines different properties of an object in the state space numerically. The first appearance of observer was in considering an extension of Chang’s fuzzy topological spaces [1] in 2002 (see also [16]). This notion has been applied in the theory of semidynamical systems in 2004 [8] via inserting one property of an object of the state space. Later, these kinds of observers were called “one dimensional observers” extended by introducing multi-dimensional observers in 2009 [11]. It is interesting to know that hesitant fuzzy sets, which have been introduced in 2010 [19], are exactly multi-dimensional observers with finite dimensions, which have been introduced in 2009 [11].

In many physical systems, the observers appear in a naturally way. For example, Chua’s diode consisting of two linear capacitors ($C_1, C_2$), a linear inductor (L), two linear resistors ($R_1, R_2$), and piecewise-linear negative resistor (see [3, 22]) determines the system $\dot{X}=AX+\Psi (X)$, with

$$\begin{aligned} X=\left( \begin{array}{c} x_{1}\\ x_{2}\\ x_{3}\end{array}\right) \in \textbf{R}^{3}, A=\left( \begin{array}{ccc} -\frac{G}{C_{1}} &{} \frac{G}{C_{1}} &{} 0 \\ \frac{G}{C_{2}} &{} -\frac{G}{C_{2}} &{} \frac{1}{C_{2}}\\ 0 &{} -\frac{1}{L}&{} -\frac{R_{2}}{L} \end{array}\right) ,~ and ~\Psi (X)=\left( \begin{array}{c} -\frac{f(x_{1})}{C_{1}}\\ 0\\ 0\end{array}\right) , \end{aligned}$$

where $G=\frac{1}{R_1}$ f is the piecewise-linear characteristics. The time one map of this system is a dynamical system, and the time one map of its driven system is an observer dependent dynamical system which we called such systems “relative dynamical systems.” In fact, the driven system of the Chua’s diode is $\dot{Y}=AY+\Lambda \Psi (Y),$ with $\Lambda =\mu (X)A+\nu (X)\Lambda _{1},$ where $\Lambda _{1}=\left( \begin{array}{ccc}\lambda &{} 0 &{} 0 \\ 0 &{} \lambda &{} 0\\ 0&{} 0&{} \lambda \end{array}\right) ,$ and $\mu$ and $\nu$ are two functions from $\textbf{R}^{3}$ to the interval [0, 1], and $\lambda$ is a constant.

The paper is organized as follows: At the beginning, the motivation for the present research is given. In the next section, we recall the definition of observer, and we consider its role in the systems theory. In Section 3, we consider relative semidynamical systems. Distinguishable points and locally distinguishable semidynamical systems are considered in Section 5, and it is proved that locally distinguishable semidynamical systems preserve under relative conjugate relations, which preserve the observer. At last, relative topological entropy as an application of the role of the observer in systems theory is considered.

2 Motivation

On the first step in the motivation of the present work is similar as to the work [12]. In fact, however, we go further by asking questions about relations between observer and observability of a dynamic system. More precisely, it is well known that when a widely understood calculating machine represents a number with a finite decimal expansion, there is no any error in it, but the error appears when a number has decimal part with infinite expansion. Then, it can be concluded that the numerical calculations in the calculating machine were made with an accuracy up to a certain observer. So, the question is how to deaminate this observer. Taking into account that in fact we play with the decimal part, it is natural to restrict ourself to the interval (0, 1) and consider the numerical computations of the calculating machine up to the function ${\mu :(0,1)\rightarrow [0,1]}$ defined as

$$\begin{aligned} \mu (x)=1-\min \{\mid x-a\mid ~ :a\in \textbf{A}\} , \end{aligned}$$

(1)

where $\textbf{A}$ is the set of machine numbers. Such defined function can be called the function $\mu$-observer.

One can see the trace of observers in many physical systems. For example, consider the unified chaotic system in ${\textbf{R}^{3}}$ described as (see [2]):

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{x}=(25\theta +10)(y-x) \\ \dot{y}=(28-35\theta )x-xz+(29\theta -1)y\\ \dot{z}=xy-(\frac{8+\theta }{3})z \end{array}.\right. \end{aligned}$$

(2)

$\theta \in [0;1]$ as an observer determines different physical systems. In fact, in the case $\theta =0$, we have Lorenz system [6]; in the case $\theta = \frac{28}{35}$, we have Lü and Chen system [7]; and in the case $\theta =1$, we have Chen system [7].

Now, the question is as follows: How can this observer affect on the usual notions of the mathematics such as topological and dynamical concepts? For example, what does mean continuity up to an observer? What does mean a dynamical system up to an observer? Or what does mean chaotic behavior up to an observer?

3 A Mathematical Model for an Observer

Since fuzzy sets are strictly associated with the idea of an observer, at the beginning, we recall some ideas and facts about these sets. For more details, see, for example, in [5, 23].

Let X be a nonempty set. Recall that by a completely distributive lattice is meant a complete lattice L such that for any double indexed family $\{x_{j,k}:\;j\in J, k\in K_j\}$ of L it holds

$$\begin{aligned} \forall j\in J \exists k\in K_j\;x_{j,k}=\exists f\in F \forall j\in J\;x_{j,f(j)} \end{aligned}$$

for set F of choice functions f choosing for each index $j\in J$ some index $f(j)\in K_k$, [4].

Then, any map $X\rightarrow L$ is a fuzzy set. Note the definition of the observer function $\mu$ given by (1) implies that it is a fuzzy set. In fact, there is a deference between fuzzy sets and observers. It follows from the following observation: replacing completely distributive lattice L with an isomorphic completely distributive lattice, as a result, one obtains a fuzzy set that has no any difference with the previous one. But two isomorphic completely distributive lattices may have different meanings: it is enough to consider the square $[0,1]\times [0,1]$ in the real two-dimensional space and the cube $[0,1]\times [0,1]\times [0,1]$ in the real three-dimensional space. These completely distributive lattices are isomorphic, but physically, they are completely different.

Suppose that I is an nonempty index set with the cardinality n. Note that n can be finite or infinite.

Definition 1

[12] An n-dimensional observer or a multi-dimensional observer of a set X is a map**

$$\begin{aligned} \mu :X\rightarrow \displaystyle \prod _{i\in I}[0,1]. \end{aligned}$$

(3)

From Definition 1, it follows that an n-dimensional observer determines n attributes of a given object. Also, if I is a finite set, then the resulted multi-dimensional observer is a hesitant fuzzy set (see [19]).

Example 1

Let X be a nonempty set. Consider a map $\mu :X\rightarrow \displaystyle \prod _{i\in I}[0,1]$ and a subset $E\subset X$. Then, one can define the relative probability measure (see [5]) defined as a map

$$\begin{aligned} {m}^f_{\mu }(E):X\rightarrow \displaystyle \prod _{i\in I}[0,1] \end{aligned}$$

(4)

(note that it is a new n-dimensional observer) defined by

$$ {m}^f_{\mu }(E)(x)=\limsup _{n\rightarrow \infty } \frac{1}{n}\sum _{i=0}^{n-1}\chi _{E}(f^i(x))\mu (f^i(x)), $$

where $f: X\rightarrow X$ is a map**. In fact, the map**, f, defines a semidynamical system.

Proposition 1

Suppose that $(X, \mathcal {A}, m)$ is a probability space with the probability measure m and the $\sigma -$algebra $\mathcal {A}$. Moreover, assume f is an ergodic map.¹ Then for given $E\in \mathcal {A}$, we have ${m}^f_{\chi _{E}}(E)=m(E)~a.e.$, where $\chi _{E}$ is the characteristic function of E.

Proof

The equality ${m}^f_{\chi _{E}}(E)=m(E)~a.e.$ follows from the Birkhoff Ergodic Theorem (see [21]). $\square$

In quantum mechanics, there is a concept called observable. In fact, an observable object is a Borel probability measure on the set of real numbers (see [5]). So, observable means something different that observer; for details, see [13].

Definition 2

[12] Let T be a group. An observable object up to an observer $\mu :X\rightarrow [0,1]$ and a dynamical system $(X,\{\varphi ^{t}:X\rightarrow X, t\in T\})$ is a member of the set

$$\begin{aligned} \{{m}^{\varphi ^{t}}_{\mu }(\cdot )(x):P(X)\rightarrow [0,1], P(X)~is ~the ~powerset ~of ~X, ~and ~t\in T\}. \end{aligned}$$

(5)

Remark 1

Note that in the case $X=R$, $\mu =\chi _{R}$ is the characteristic function and each $\varphi ^{t}$ is an ergodic map, then the set (5) contains all real probability measures which in fact they are observable objects in quantum mechanics.

4 Semidynamical Systems from an Observer Point of View

Let

$$\begin{aligned} \mu = \displaystyle \prod _{i\in I} \mu _{i},\quad \eta =\displaystyle \prod _{i\in I} \eta _{i} \end{aligned}$$

(6)

where $\mu$ and $\eta$ are given by (3), and for given i, $\mu _{i}$ and $\eta _{i}$ are 1-dimensional observers of the nonempty set X. Operations of $\cup$ and $\cap$ on these map**s are, respectively, given by

$$\begin{aligned} (\eta \cap \mu )_{i} (x):= & {} inf\{\eta _{i}(x), \mu _{i}(x)\} \\ (\eta \cup \mu )_{i}(x):= & {} sup \{\eta _{i}(x), \mu _{i}(x)\}. \end{aligned}$$

Moreover, the inclusion relation on the set of n-dimensional observers of X is defined as the following:

$$\begin{aligned} \mu \subseteq \eta \quad \text {if}\quad \eta _{i}(x) \le \mu _{i}(x) \end{aligned}$$

for all $x\in X$ and $i\in I$.

Similarly as in [11], corresponding to each observer $\mu$ given by (6) of the set X, a $\mu$-topology (or n-dimensional relative topology) can be associated as a collection $\tau _\mu$ of subsets of $\mu$ which satisfies the following axioms:

• (i) $\mu ,{\chi ^{n}}_\emptyset \in \tau _\mu$, where ${\chi ^{n}}_\emptyset (x)=\displaystyle \prod _{i\in I} 0$ for all $x\in X$;

• (ii) if $\lambda ,\eta \in \tau _\mu$ then $\lambda \cap \eta \in \tau _\mu$;

• (iii) if $(\mu ^{a})_{a\in F}$ is a family of $\tau _\mu$, then $\displaystyle \bigcup _{a\in F}\mu ^{a}\in \tau _\mu$, where F is any family.

Elements of the collection $\tau _\mu$ are called the $\mu$-open observers (or the relative open observers). The triple $(X,n,\tau _\mu )$ is called a relative topological space.

Definition 3

If ${\lambda }\subseteq \mu$, then by the complement of ${\lambda }$ relative to $\mu$ is the n-dimensional observer ${\lambda }^\mu =\mu \setminus {\lambda }$, where the operation “$\setminus$” is taken on the values of map**s $\mu ,{\lambda }$. If ${\lambda }^\mu \in \tau _\mu$, then ${\lambda }$ is said a $\mu$-closed observer.

A $\mu$-closed observer can be constructed via an observer $\beta$ by taking the intersection on the all $\mu$-closed sets $\gamma$ with $\beta \subseteq \gamma$. Such constructed observer, denoted by $\overline{\beta }$, is called the closure of the observer $\beta$. The union on the all $\mu$-open sets $\eta$ such that $\eta \subseteq \beta$ is a $\mu$-open set that is the observer [11]. This observer will be denoted by $\beta ^0$.

Definition 4

[12] A map** f from a relative topological space $(X,n,\tau _\mu )$ to relative topological space $(Y,n, \zeta _\nu )$ is said to be relative continuous map (or a $(\mu , \nu )$-continuous map ) if

$$\begin{aligned} (\gamma of)\cap \mu \in \tau _\mu \end{aligned}$$

(7)

for all $\gamma \in \zeta _\nu$.

We will denote $\gamma of$ by $f^{-1}\gamma$. If a relative continuous map f is a bijection and $f^{-1}$ is also a relative continuous map, then f is known as a relative $(\mu ,\nu )$-homeomorphism.

Proposition 2

[11] Suppose that $(X,n,\tau _\mu )$ and $(Y,n, \zeta _\nu )$ are two relative topological spaces and $g:X\rightarrow Y$ is a $(\mu ,\nu )$-continuous map with $\nu o g=\mu$. Then, if $\gamma$ is a $\nu$-closed observer, then $(\gamma \circ g)\cap \mu$ is a $\mu$-closed observer.

Now, we present the definition of a relative semidynamical system created by a relative continuous map.

Definition 5

Suppose $(X,n,\tau _\mu )$ is a relative topological space. A map** $f:X\rightarrow X$ is said to be a relative semidynamical system if f is a $(\mu , \mu )$-continuous map.

Example 2

Suppose C is the set of complex numbers and $\mu :C^{2}\rightarrow [0,1]^{2}$ be the map**

$$\begin{aligned} \mu (x,y)=\left\{ \begin{array}{cccc} (\frac{1}{4},\frac{1}{4}) &{}if&{} (x,y)\in (C\setminus R)^{2}\cup \{(0,0)\} \\ (\frac{1}{4},0) &{}if&{}(x,y)\in ((C\setminus R)\cup \{0\})\times (R\setminus \{0\})\\ (0,\frac{1}{4}) &{}if&{} (x,y)\in (R\setminus \{0\})\times ((C\setminus R)\cup \{0\})\\ (0,0) &{}if&{} (x,y)\in (R\setminus \{0\})\times (R\setminus \{0\}) \end{array}\right. \end{aligned}$$

For given $a\in [0,\frac{1}{2})$, we define $\eta _{a}:C^{2}\rightarrow [0,1]^{2}$ by

$$\begin{aligned} \eta _{a}(x,y)=\left\{ \begin{array}{cccc} (a^{2},a^{2}) &{}if&{} (x,y)\in (C\setminus R)^{2}\cup \{(0,0)\} \\ (a^{2},0) &{}if&{}(x,y)\in ((C\setminus R)\cup \{0\})\times (R\setminus \{0\})\\ (0,a^{2}) &{}if&{} (x,y)\in (R\setminus \{0\})\times ((C\setminus R)\cup \{0\})\\ (0,0) &{}if&{} (x,y)\in (R\setminus \{0\})\times (R\setminus \{0\}) \end{array}\right. \end{aligned}$$

The set $\tau _{\mu }=\{\mu ,\eta _{a}~:~a\in [0,1)\}$ is a $\mu$ topology for $C^{2}$. We define a relative semidynamical system $h:C^{2}\rightarrow C^{2}$ by

$$\begin{aligned} h(x,y)=\left\{ \begin{array}{cccc} (\frac{x}{5},\frac{x}{5}) &{}if&{} (x,y)\in (C\setminus R)^{2}\cup \{(0,0)\} \\ (\frac{x}{5},5x) &{}if&{}(x,y)\in ((C\setminus R)\cup \{0\})\times (R\setminus \{0\})\\ (5x,\frac{x}{5}) &{}if&{} (x,y)\in (R\setminus \{0\})\times ((C\setminus R)\cup \{0\})\\ (5x,5x) &{}if&{} (x,y)\in (R\setminus \{0\})\times (R\setminus \{0\}) \end{array}\right. \end{aligned}$$

Since $\mu o f=\mu$ and $\eta _{a}oh=\eta _{a}$ for each $a\in [0,\frac{1}{2})$, then h is $(\mu , \mu )$-continuous.

5 Distinguishable Points

Let p be a fixed point for a relative semidynamical system f on $(X,n,\tau _\mu )$.

Definition 6

Let $\mu$ be given by (3) A point p is called sink if there is $\gamma \in \tau _{\mu }$ such that

$$\begin{aligned} \{p\}=\displaystyle \bigcap _{n=0}^{\infty } f^{n}(\{x\in X ~:\frac{\mu (p)_{i}}{2}<\gamma (x)_{i}~for~each~i\in I\}), \end{aligned}$$

where $f^{0}$ is the identity map and $f^n=\underbrace{f\circ \ldots \circ f}_n$ denotes n-folt composition of the map f.

Example 3

With the assumptions of Example 2, the point (0, 0) is a sink for h, because

$$\begin{aligned} \displaystyle \bigcap _{n=0}^{\infty } h^{n}(\{(x,y)\in C^{2} ~:\frac{\mu (0,0)_{i}}{2}=\frac{1}{8}<\eta _{\frac{2}{3}}(x,y)_{i}~for~i=1,2\})= \end{aligned}$$

$$\begin{aligned} \displaystyle \bigcap _{n=0}^{\infty } h^{n}(\{(x,y)~:~(x,y)\in (C\setminus R)^{2}\cup \{(0,0)\}\}=\{(0,0)\}. \end{aligned}$$

Proposition 3

If p and q are two different sinks for a relative semidynamical system f on a relative topological space $(X,n,\tau _\mu )$, then there is $\gamma \in \tau _\mu$ such that they are distinguishable.

Proof

It is enough to show that there is $\gamma \in \tau _\mu$ such that $\gamma (p)\ne \gamma (q)$. If $\mu (p)\ne \mu (q)$, then the thesis is obvious. Suppose $\mu (p)=\mu (q)$. In this case, the reasoning is similar to the reasoning presented in [12]. Namely, since p is a sink, then it is a fixed point, and there is $\gamma \in \tau _{\mu }$ such that

$$\begin{aligned} \{p\}=\displaystyle \bigcap _{n=0}^{\infty } f^{n}(\{x\in X~:\frac{\mu (p)_{i}}{2}<\gamma (x)_{i}~for~each~i\in I\}). \end{aligned}$$

Let q be another sink such that $p\ne q$. Suppose that $\gamma (p)=\gamma (q)$. Then, $f(q)=q$ implies that

$$\begin{aligned} \{q\}\in & {} \displaystyle \bigcap _{n=0}^{\infty } f^{n}(\{x\in X~:\frac{\mu (q)_{i}}{2}<\gamma (x)_{i}~for~each~i\in I\}),\\= & {} \displaystyle \bigcap _{n=0}^{\infty } f^{n}(\{x\in X~:\frac{\mu (p)_{i}}{2}<\gamma (x)_{i}~for~each~i\in I\}) \end{aligned}$$

which is contradicts with $p\ne q$. Thus, $\gamma (p)\ne \gamma (q).$ $\square$

Let f be a relative semidynamical system on a relative topological space $(X,n,\tau _\mu )$.

Definition 7

Two different points $p,q\in X$ are said to be distinguishable by $\mu$-observer with respect to the relative semidynamical system $f:X\rightarrow X$ if there is $\gamma \in \tau _{\mu }$ such that $\gamma (f(p))\ne \gamma (f(q))$. In the other case, these points are indistinguishable. Moreover if $p=q$, then we say that p is indistinguishable from itself.

It is not difficult to see that the indistinguishability relation is the equivalence relation.

Definition 8

The relative semidynamical system $f:X\rightarrow X$ is said to be locally $\mu$-distinguishable at $p \in X$ with respect to a $\mu$-observer if there exists $\lambda \in \tau _{\mu }$ such that for each $p\ne q\in X$ with $\lambda (q)= \lambda (p)$ there is $\gamma \subseteq \lambda$ such that $\gamma (f(q))\ne \gamma (f(p)).$

Roughly speaking, a semidynamical system $f:X\rightarrow X$ is locally $\mu$-distinguishable with respect to a $\mu$-observer if every $p\in X$ can be distinguished from its neighbors with respect to a $\mu$-observer by using systems trajectories remaining close to p.

Example 4

Let $X=(-1,1)$ and $\mu$ be the characteristic function of X, i.e., $\mu =\chi _X$. For given natural numbers n, m, k, l, let us define the function $\lambda _{n,m,k,l}$ as follows:

$$\begin{aligned} \lambda _{n,m,k,l}(t)= \left\{ \begin{array}{ll} e^{[(\frac{t-k}{n})^{2}-1)^{-1}+1]}~if~t\le 0&{} \\ e^{[(\frac{t+l}{m})^{2}-1)^{-1}+1]}~if~t>0&{} \end{array}.\right. \end{aligned}$$

(8)

The graph of function (8) for chosen values of parametres n, m, k, l is presented on Fig. 1. Note also that the set $\tau _{\mu }=\{\chi _{\emptyset }, \mu , \lambda _{n,m,k,l}~:~n,m,k,l\in N\}$ is a relative topology. Then, the map** $f: X\rightarrow X$ defined by $f(t)=\frac{t}{2}$ is a relative semidynamical system. It is not difficult to observe that f is locally $\mu$-distinguishable at each sink $p \in X$.

Fig. 1

The figures of $\lambda _{10,11,2,1}$, $\lambda _{10,10,1,1}$, and $\lambda _{12,9,1,1}$ with the colors red, blue, and green, respectively

Definition 9

Two relative semidynamical systems f and g on the relative topological space $(X,n,\tau _\mu )$ are called $\mu$-conjugate if there is a $(\mu , \mu )$-homeomorphism $\varphi :X\longrightarrow X$ so that the following diagram commutes

$$\begin{aligned} \begin{array}{ccccc} X &{} \underset{\longrightarrow }{f} &{} X \\ \varphi \downarrow &{}&{} \downarrow \varphi \\ X &{} \underset{\longrightarrow }{g} &{} X \end{array}. \end{aligned}$$

(9)

From (9), it follows that if $\mu$ is an observer for the semidynamical system f, then it is also an observer for semidynamical system g and vice versa. Moreover, if semidynamical system f is locally $\mu$-observable, then system g also is and vice versa.

Theorem 4

Suppose two relative semidynamical systems $f:X\longrightarrow X$ and $g:X\longrightarrow X$ are $\mu$-conjugate under a $(\mu , \mu )$-homeomorphism $\varphi :X\longrightarrow X$. Point p is a sink for f if and only if $\varphi (p)$ is a sink for g.

Proof

"$\Rightarrow$" The definition of a sink implies that there is $\gamma \in \tau _{\mu }$ such that

$$\begin{aligned} \{p\}=\displaystyle \bigcap _{n=0}^{\infty } f^{n}(\{x\in X:\frac{\mu (p)_{i}}{2}<\gamma (x)_{i}~for~each~i\in I\}). \end{aligned}$$

(10)

Let $q=\varphi (p)$. Then,

$$\begin{aligned} \{\varphi ^{-1}(q)\}=\displaystyle \bigcap _{n=0}^{\infty } f^{n}(\{x=\varphi ^{-1}(y)\in X:\frac{\mu (\varphi ^{-1}(q))_{i}}{2}<\gamma (\varphi ^{-1}(y))_{i}~for~each~i\in I\}). \end{aligned}$$

(11)

Thus,

$$\begin{aligned} \{q\}= & {} \varphi \{\varphi ^{-1}(q)\} \nonumber \\= & {} \displaystyle \bigcap _{n=0}^{\infty } \varphi \circ f^{n}(\{x=\varphi ^{-1}(y)\in X ~:\frac{\mu (\varphi ^{-1}(q))_{i}}{2}<\gamma (\varphi ^{-1}(y))_{i}~for~each~i\in I\})\nonumber \\= & {} g^{n}o\varphi (\{x=\varphi ^{-1}(y)\in X ~:\frac{\mu (\varphi ^{-1}(q))_{i}}{2}<\gamma (\varphi ^{-1}(y))_{i}~for~each~i\in I\})\nonumber \\= & {} g^{n}(\{y\in X ~:\frac{\mu (q)_{i}}{2}<(\gamma o \varphi ^{-1})(y)_{i}~for~each~i\in I\}). \end{aligned}$$

(12)

Since $\gamma \circ \varphi ^{-1}\in \tau _{\mu }$, then the former equality implies that $q=\varphi (p)$ is a sink for g.

“$\Leftarrow$” Since $q=\varphi (p)$ is a sink for the map g, then there is $\gamma \in \tau _\mu$ such that (10) holds. Then, also (11) and (12) hold. $\square$

Theorem 5

Suppose two relative semidynamical systems $f:X\longrightarrow X$ and $g:X\longrightarrow X$ are $\mu$-conjugate under a $\mu$-homeomorphism $\varphi :X\longrightarrow X$ with $\mu o \varphi =\mu$. Semidynamical system f is locally $\mu$-distinguishable at $p\in X$ if and only if the semidynamical system g is locally $\mu$-distinguishable at $\varphi (p)$.

Proof

Suppose f is locally $\mu$-distinguishable at $p\in X$, then there is $\lambda \in \tau _{\mu }$ such that for each $p\ne q\in X$ with $\lambda (q)= \lambda (p)$ there is $\gamma \subseteq \lambda$ such that $\gamma (f(q))\ne \gamma (f(p)).$ If we take $\theta =(\lambda o (\varphi )^{-1})\cap \mu$, then when $\theta (\varphi (q))=\theta (\varphi (p))$, we have $\lambda (q)= \lambda (p)$, because $\mu o \varphi =\mu$. Thus, there is $\gamma \subseteq \lambda$ such that

$$ (\gamma o\varphi ^{-1})(g(\varphi (q)))\ne (\gamma o\varphi ^{-1})(g(\varphi (p))). $$

Since

$$ \eta =\gamma o\varphi ^{-1}\cap \mu \subseteq \theta \;\;\;\text {and}\;\;\; \eta (g(\varphi (q)))\ne \eta (g(\varphi (p))), $$

then g is locally $\mu$-distinguishable at $\varphi (p)$. The method of the proof ofthe converse is similar. $\square$

6 An Application of $\mu$-Observers

A relative topological space $(X,n,\tau _\mu )$ is said to be a compact relative topological space if each relative open cover $\{\gamma ^{a} ~:~ a\in A\}\subseteq \tau _\mu$ with $\displaystyle \bigcup _{a\in A}\gamma ^{a}=\mu$ has a finite relative open subcover $\{\gamma ^{b} ~:~ b\in B\}$ such that $\displaystyle \bigcup _{b\in B}\gamma ^{b}=\mu$.

Proposition 6

[11] If $(X,n,\tau _\mu )$ is a compact relative topological space and $f: (X,n,\tau _\mu )\rightarrow (Y,n, \zeta _\nu )$ is an onto relative continuous map so that $\nu o f =\mu$, then $(Y,n, \zeta _\nu )$ is a compact relative topological space.

Let $\mu$ be an observer. Suppose $(X,n,\tau _\mu )$ is a compact relative topological space, and $\Theta =\big \{{\lambda }^{i}$: ${\lambda }^{i}\in {\tau }_{\mu }$, $i=1,\cdots ,m\big \}$ is a relative open cover for $\mu$.

Definition 10

[11] The relative topological entropy of the relative cover $\Theta$, denoted by $H(\Theta )$, is defined as

$$ H(\Theta )=log N(\Theta ), $$

where $N(\Theta )$ denotes the smallest cardinal number of sets that can be used as a relative subcover of $\Theta$.

The refinement of a family of relative open covers for $\mu$,

$$ \big \{\Theta ^{r}=\{({{\lambda }}_{r}^{1}),\cdots ,({{\lambda }}_{r}^{Nr})\}:r=1,\cdots ,k\big \} $$

is a relative open cover, $\bigvee _{r=1}^{k}\Theta ^{r}$, defined by

$$ \Big \{ ({{\lambda }}^{i_{1}}_{1})\cap ({{\lambda }}^{i_{2}}_{2})\cap \cdots \cap ({{\lambda }}^{i_{m}}_{k}): ({{\lambda }}^{i_{j}}_{j}) \in \Theta ^{j},j= 1,\cdots ,k \Big \}. $$

Theorem 7

[11] If $f:X\longrightarrow X$ is a relative semidynamical system and $\Theta =\{{\lambda }^{1},\cdots ,{\lambda }^{m}\}$ is a relative open cover for the observer $\mu$, then

$$ f^{-1}\Theta =\big \{(\mu \cap {f^{-1}}{{\lambda }^{1}}),\cdots ,(\mu \cap {f^{-1}}{{\lambda }^{m}})\big \} $$

is a relative open cover for $(\mu \cap f^{-1}\mu )$. Moreover, $H(\Theta )\ge H(f^{-1}\Theta )$ and

$$ \begin{array}{c} \displaystyle \limsup _{m\rightarrow \infty } \frac{1}{m}H\big (\displaystyle \bigvee ^{m-1}_{i=0}f^{-i}\Theta \big ) \end{array} $$

exists.

The relative entropy of a relative semidynamical system f relative to a relative open cover $\Theta$ of the observer $\mu$ is defined by

$$ h(f,\Theta )=\displaystyle \lim _{m\rightarrow \infty } \frac{1}{m}H (\bigvee ^{m-1}_{i=0}f^{-i} \Theta ). $$

The relative topological entropy of f is defined by

$$ h(f)=\sup \{h(f,\Theta ):\Theta \;\text {is a finite relative open cover of}\;\mu \}. $$

It can be shown that the relative topological entropy is an invariant object under the $\mu$-conjugate relations (see [11]).

Example 5

Let define observer $\mu :[0,1]\rightarrow [0,1]$ as

$$ \mu (x)=\left\{ \begin{array}{ccc} 1 &{}if&{} x\in (R\setminus Q)\cap [0,1] \\ \frac{1}{4} &{}if&{}x\in Q\cap [0,1] \end{array}\right. . $$

If $a,b\in R$ with $a<b$, then we define $\gamma _{[a,b]}:[0,1]\rightarrow [0,1]$ by

$$ \gamma (x)=\left\{ \begin{array}{ccc} \mu (x) &{}if&{} x\in [a,b] \\ 0 &{}otherwise&{} \end{array}\right. $$

The set $\tau _{\mu }=\{\chi _{\emptyset },\mu ,\gamma _{[a,b]}~:~a,b\in [0,1]~and~a<b\}$ is a relative topology. With this, relative topology [0, 1] is a compact relative topological space. The map** $f:[0,1]\rightarrow [0,1]$ defined by

$$ f(x)=5x ~(mod~1) $$

is a relative semidynamical system. For finding its relative topological entropy, we take

$\Theta =\{\gamma _{[0, \frac{1}{5}+\frac{1}{5^{2}}]},\gamma _{[\frac{1}{5}-\frac{1}{5^{2}}, \frac{2}{5}+\frac{1}{5^{2}}]},\gamma _{[\frac{2}{5}-\frac{1}{5^{2}}, \frac{3}{5}+\frac{1}{5^{2}}]},\gamma _{[\frac{3}{5}-\frac{1}{5^{2}}, \frac{4}{5}+\frac{1}{5^{2}}]}, \gamma _{[\frac{4}{5}-\frac{1}{5^{2}}, 1]} \}.$ We see that for given a relative open cover $\Lambda$ of [0, 1], there exist $n_{0}$ such that each given member of $\displaystyle \bigvee _{i=0}^{n_{0}-1}f^{-i}\Theta$ is a subset of a member of $\Lambda$. Hence,

$$ h(f)=\lim _{n\rightarrow \infty }\frac{H(\displaystyle \bigvee _{i=0}^{n-1}f^{-i}\Theta )}{n}= \lim _{n\rightarrow \infty }\frac{log (5^{n-1}N(\Theta ))}{n}=log5. $$

7 Conclusion

A model for an observer has been considered. For the chaotic behavior of a relative semidynamical system, the notion of relative topological entropy has been introduced. In fact, a relative semidynamical system is called chaotic if its relative topological entropy be a positive number. The locally distinguishable relative semidynamical systems have been consider, and it has been shown that the local distinguishable property preserves under relative conjugate relations which protect the observer. The role of $\mu$-observer in dynamical system theory has been shown in the aspect of the relative topological entropy.