Based on the paper titled, Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps
and another related paper http://www-connexe.univ-brest.fr/lest/tst/publications/pdf/comm04_compression_chaos.pdf
Authors use a piecewise linear chaotic map for their application which is encoding a bitstream into the chaotic trajectory of the map. In the second paper, they say that the map is a Bernoulli map. I am not familiar with piecewise linear chaotic Map. Is it an extension of the Skew Tent Map or the Tent map? The Tent Map is $$x_{n+1} = a(1-|2 x_n -1|)$$ where $a \in [0,1]$ and $x_n \in [0,1]$
Are the Bernoulli map, Tent Map and the piecewise linear map the same in terms of the derivatives of the map and properties? Can I say that the piecewise Tent Map is obtained by extending the Skew map or the Tent map?
These are three different maps, if you call
$$ x_{n+1} = f(x_n) $$
then the function $f$ is different for all the naps you mentioned, and so is the dynamics they generate. More precisely, the skewed tent map can be reduced to the tent map by a particular choice of parameters
Tent map $$ f(x) = a(1 - |2x - 1|) $$
Bernoulli shift map
$$ f(x) = \begin{cases} 2x, & \mbox{for}\quad 0\le x \le 1/2 \\ 2x - 1, & \mbox{for}\quad 1/2< x \le 1\end{cases} $$
Skew tent map
$$ f(x) = \begin{cases} \nu + (1 - \nu)x/\mu, & \mbox{for}\quad 0\le x \le \mu \\ (1 - x) / (1- \mu), & \mbox{for}\quad \mu< x \le 1\end{cases} $$
Note that the tent map can be recovered by setting $\nu =0$ and $\mu = 1/2$
The figure below shows a plot of $f(x)$ for these three cases