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.