Suppose we have the tent map, defined by $$T_c(x) = \begin{cases} cx & 0\leq x \leq 1/2\\ c-cx& 1/2 \leq x \leq 1. \end{cases}$$
What are the prime period 3 orbits for this map? I have seen them for the tent map when $c=2$, but I cannot find them in terms of $c$ for the life of me.
The line segment of $T_c^3$ have equations and fixed points \begin{array}{c|rl|c} 0&c(c(c(x)))&=c^3x&\\ 1&c(1-c(c(x)))&=c-c^3x&\dfrac{c}{1+c^3}\\ 2&c(c(1-c(x)))&=c^2-c^3x&\dfrac{c^2}{1+c^3}\\ 3&c(1-c(1-c(x)))&=c-c^2+c^3x&\dfrac{c}{1+c+c^2}\\ 4&c(c(c(1-x)))&=c^3-x^3&\dfrac{c^3}{1+c^3}\\ 5&c(1-c(c(1-x)))&=c-c^3+c^3x&\dfrac{c+c^2}{1+c+c^2}\\ 6&c(c(1-c(1-x)))&=c^2-c^3+c^3&\dfrac{c^2}{1+c+c^2}\\ 7&c(1-c(1-c(1-c)))&=c-c^2+c^3-c^3x&\dfrac{c}{1+c} \end{array}
where one can recognize the two 3 cycles $$ \dfrac{c}{1+c^3}\to\dfrac{c^2}{1+c^3}\to\dfrac{c^3}{1+c^3} $$ and $$ \dfrac{c}{1+c+c^2}\to\dfrac{c^2}{1+c+c^2}\to\dfrac{c+c^2}{1+c+c^2} $$ These exist for $$ \dfrac{c}{1+c+c^2}\le \frac12\le\dfrac{c^2}{1+c+c^2}\\\iff\\ 2c\le1+c+c^2\le 2c^2\\\iff\\ -c^2+c-1\le 0 \le c^2-c-1 $$ which is true for $c\ge\frac{1+\sqrt5}2$, see also Prove that the critical value for the appearance of period 3-orbits in the tent map is $\frac{1+\sqrt5}{2}$.
