Distinction between Tent Map and pievewise linear chaotic map Based on the paper titled, Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps and another related paper < 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

$$ f(x) = \begin{cases} \
u + (1 - \
u)x/\mu, & \mbox{for}\quad 0\le x \le \mu \\\ (1 - x) / (1- \mu), & \mbox{for}\quad \mu< x \le 1\end{cases} $$

Note that the tent map can be recovered by setting $\
u =0$ and $\mu = 1/2$




The figure below shows a plot of $f(x)$ for these three cases

![enter image description here](