This is based from the paper titled, "Chaos-Based Simultaneous Compression and Encryption for Hadoop" in Section 2.3.1 download link

The Authors say that given a symbolic sequence, it can be encoded into an initial condition by reverse interval mapping using the inverse map $f^{-1}_T(.)$. Then, starting from the initial condition, if the map $f_T(.)$ is iterated and the symbolic dynamics obtained by state space partition using the intervals computed from the encoding stage, we can obtain the same symbolic sequence.

Question 1 : Is the inverse map chaotic? Would the local and global Lyapunov exponent of $f^{-1}_T(.)$ also identical? The global and local LE of the Tent map are identical, so I thought may be this is also applicable to the inverse map, but I am not quite sure.

Question 2 : In general, if there is a piecewise-linear chaotic map having non-overlapping intervals, and there is an inverse map for it, then would the inverse map be chaotic?

Background information :

Let $f_T(.)$ be the chaotic Tent Map which produces a scalar valued time series where the first iterate is obtained from an initial condition $x[0]$ as $x[1] = f(x[0],\mu)$ where $\mu$ is the control parameter. SO, iteratively, we obtain a an array of values $x[1],x[2],\ldots, x[N]$.

Using conncepts of symbolic dynamics, if $\mathbf{s} = \{s[0],s[1],\ldots,s[n-1]\}$ is the symbolic dynamics obtained from the following rule: if $x[n] >= 0.5$, then $s[n] = 1$, else $s[n]=0$. I remember reading that knowing $x[n]$ and $s[n-1]$ is sufficient to determine $x[n-1]$ and so I can write $$x[n-1] = f^{-1}(s[n-1],x[n])$$ where $f^{-1}(.)$ is the inverse of $f_T(.)$ given a symbol $s[n-1]$. There is a one to one mapping between the symbolic sequence and the initial condition from which the symbolic sequence has been generated. This means that $$x[0] = f^{-1}(x[1]) o f^{-1}(x[1]) o \ldots o f^{-1}(x[n])$$, where $o$ is the functional composition. I am considering the Skew Tent Map and its inverse given below

\begin{equation} \begin{aligned} f^{-1}(I) = x_n = \begin{cases} p \times I, \text{symbol} s_n =0 \\ 1-p \times I, \text{symbol} s_n =1 \label{InverseSkewTentMap} \end{cases} \end{aligned} \end{equation} where $I_{1}$ implies interval when the symbol at $s_i$ is $a_1$ = 0 and $I_{1}$ implies the interval when the symbol at $s_i$ is $a_2= 1$. The Skew Tent map is expressed as \begin{equation} \begin{aligned} f(x) = \begin{cases} x/p, 0\le x <p \\ (1-x)/(1-p), p \le x \le 1 \label{SkewTentMap} \end{cases} \end{aligned} \end{equation} The Skew Tent map is related to the symbols by the choice of the partition point $p$.

Please correct me if the notations are incorrect or is not the standard way of expressing.

This is based from the paper titled, "Chaos-Based Simultaneous Compression and Encryption for Hadoop" in Section 2.3.1 download link

The Authors say that given a symbolic sequence, it can be encoded into an initial condition by reverse interval mapping using the inverse map $f^{-1}_T(.)$. Then, starting from the initial condition, if the map $f_T(.)$ is iterated and the symbolic dynamics obtained by state space partition using the intervals computed from the encoding stage, we can obtain the same symbolic sequence.

This is based from the paper titled, "Chaos-Based Simultaneous Compression and Encryption for Hadoop" in Section 2.3.1 download link

Let $f_T(.)$ be the chaotic Tent Map which produces a scalar valued time series where the first iterate is obtained from an initial condition $x[0]$ as $x[1] = f(x[0],\mu)$ where $\mu$ is the control parameter. SO, iteratively, we obtain a an array of values $x[1],x[2],\ldots, x[N]$.

Question 1 : Is the inverse map chaotic? Would the local and global Lyapunov exponent of $f^{-1}_T(.)$ also identical? The global and local LE of the Tent map are identical, so I thought may be this is also applicable to the inverse map, but I am not quite sure.

Question 2 : In general, if there is a piecewise-linear chaotic map having non-overlapping intervals, and there is an inverse map for it, then would the inverse map be chaotic?

Background information :

Using conncepts of symbolic dynamics, if $\mathbf{s} = \{s[0],s[1],\ldots,s[n-1]\}$ is the symbolic dynamics obtained from the following rule: if $x[n] >= 0.5$, then $s[n] = 1$, else $s[n]=0$. I remember reading that knowing $x[n]$ and $s[n-1]$ is sufficient to determine $x[n-1]$ and so I can write $$x[n-1] = f^{-1}(s[n-1],x[n])$$ where $f^{-1}(.)$ is the inverse of $f_T(.)$ given a symbol $s[n-1]$. There is a one to one mapping between the symbolic sequence and the initial condition from which the symbolic sequence has been generated. This means that $$x[0] = f^{-1}(x[1]) o f^{-1}(x[1]) o \ldots o f^{-1}(x[n])$$, where $o$ is the functional composition. I am considering the Skew Tent Map and its inverse given below

\begin{equation} \begin{aligned} f^{-1}(I) = x_n = \begin{cases} p \times I, \text{symbol} s_n =0 \\ 1-p \times I, \text{symbol} s_n =1 \label{InverseSkewTentMap} \end{cases} \end{aligned} \end{equation} where $I_{1}$ implies interval when the symbol at $s_i$ is $a_1$ = 0 and $I_{1}$ implies the interval when the symbol at $s_i$ is $a_2= 1$. The Skew Tent map is expressed as \begin{equation} \begin{aligned} f(x) = \begin{cases} x/p, 0\le x <p \\ (1-x)/(1-p), p \le x \le 1 \label{SkewTentMap} \end{cases} \end{aligned} \end{equation} The Skew Tent map is related to the symbols by the choice of the partition point $p$.

Please correct me if the notations are incorrect or is not the standard way of expressing.

This is based from the paper titled, "Chaos-Based Simultaneous Compression and Encryption for Hadoop" in Section 2.3.1 download link

The Authors say that given a symbolic sequence, it can be encoded into an initial condition by reverse interval mapping using the inverse map $f^{-1}_T(.)$. Then, starting from the initial condition, if the map $f_T(.)$ is iterated and the symbolic dynamics obtained by state space partition using the intervals computed from the encoding stage, we can obtain the same symbolic sequence.

Question 1 : Is the inverse map chaotic? Would the local and global Lyapunov exponent of $f^{-1}_T(.)$ also identical? The global and local LE of the Tent map are identical, so I thought may be this is also applicable to the inverse map, but I am not quite sure.

Question 2 : In general, if there is a piecewise-linear chaotic map having non-overlapping intervals, and there is an inverse map for it, then would the inverse map be chaotic?

Background information :

Let $f_T(.)$ be the chaotic Tent Map which produces a scalar valued time series where the first iterate is obtained from an initial condition $x[0]$ as $x[1] = f(x[0],\mu)$ where $\mu$ is the control parameter. SO, iteratively, we obtain a an array of values $x[1],x[2],\ldots, x[N]$.

Using conncepts of symbolic dynamics, if $\mathbf{s} = \{s[0],s[1],\ldots,s[n-1]\}$ is the symbolic dynamics obtained from the following rule: if $x[n] >= 0.5$, then $s[n] = 1$, else $s[n]=0$. I remember reading that knowing $x[n]$ and $s[n-1]$ is sufficient to determine $x[n-1]$ and so I can write $$x[n-1] = f^{-1}(s[n-1],x[n])$$ where $f^{-1}(.)$ is the inverse of $f_T(.)$ given a symbol $s[n-1]$. There is a one to one mapping between the symbolic sequence and the initial condition from which the symbolic sequence has been generated. This means that $$x[0] = f^{-1}(x[1]) o f^{-1}(x[1]) o \ldots o f^{-1}(x[n])$$, where $o$ is the functional composition. I am considering the Skew Tent Map and its inverse given below

\begin{equation} \begin{aligned} f^{-1}(I) = x_n = \begin{cases} p \times I, \text{symbol} s_n =0 \\ 1-p \times I, \text{symbol} s_n =1 \label{InverseSkewTentMap} \end{cases} \end{aligned} \end{equation} where $I_{1}$ implies interval when the symbol at $s_i$ is $a_1$ = 0 and $I_{1}$ implies the interval when the symbol at $s_i$ is $a_2= 1$. The Skew Tent map is expressed as \begin{equation} \begin{aligned} f(x) = \begin{cases} x/p, 0\le x <p \\ (1-x)/(1-p), p \le x \le 1 \label{SkewTentMap} \end{cases} \end{aligned} \end{equation} The Skew Tent map is related to the symbols by the choice of the partition point $p$.

Please correct me if the notations are incorrect or is not the standard way of expressing.