Let $f(.)$ be a chaotic 1 D 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 an array of values $x[1],x[2],\ldots, x[N]$.

Using concepts 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 $s[n]=0$. 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 $f(.)$ 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.

Again, in Usama and Zakaria's "Chaos-Based Simultaneous Compression and Encryption for Hadoop"] in Section 2.3.1, the authors apply inverse interval mapping using the inverse function to encode a given a symbolic sequence into an initial condition. A symbolic message can be encoded into an initial condition by reverse interval mapping using the inverse function $f^{-1}(.)$. Then, starting from the initial condition, if the map $f(.)$ 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.

Let $f(.)$ be the Skew Tent Map and its inverse given below

\begin{equation} \begin{aligned} f^{-1}(I) = x[n] = \begin{cases} p \times I_{s[n]}, \text{symbol} s[n] =0 \\ 1-p \times I_{s[n]}, \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 = 0 and $I_{1}$ implies the interval when the symbol at $s[i]$ is = $1$.

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 $p$.

From concepts of symbolic dynamics, [notations and reading material form the book, Hirsh, Smale and Devaney, "differential equations, dynamical systems and an introduction to chaos" chapter 15 download link https://www.math.upatras.gr/~bountis/files/def-eq.pdf

- if the chaotic map maps numbers from real domain onto itself, $f : \mathcal{R} \rightarrow \mathcal{R}$,
- the shift map defined as, $\sigma: \Sigma_2 \rightarrow \Sigma_2$ where $\Sigma_2$ is the alphabet space for two alphabets 0 and 1 and
- the itenarary map, $S: \mathcal{R} \rightarrow \Sigma_2$ having $S^{-1}: \Sigma_2 \rightarrow \mathcal{R}$

Questions :

**Problem :** Is the map $f^{-1}$ the same as $S^{-1}$ ?

What is the meaning of conjugacy?