Prove that the function $$L(x)=\left\{\begin{array}[cl] \displaystyle 3x & \text{if } x\leq \displaystyle\frac{1}{3} \\ \displaystyle\frac{3}{2}-\frac{3}{2}x & \displaystyle\text{if } x>\frac{1}{3} \end{array}\right.$$ is chaotic on $[0,1]$.
Ideally, I want to set up a conjugacy with a function I already know is chaotic. Since trying to directly prove periodic points are dense seems like a fools errand, finding a conjugacy is quite important. I cannot really comprehend how previous conjugacy functions were chosen, and the few examples in my book don’t seem to lay any intuition for finding these conjugacies.
I played around in Desmos trying to come up with functions that could possibly work. One such function I came up with was $\sin(\pi x^{\log_3(2)})$ which came from trying to match up vertices of $\sin(\alpha x^\beta)$ with $L(x)$. After just one iteration of the function the resulting vertices no longer match up, furthermore trying to deal with $x^{\log_3(2)}$ inside of my sin function seems like a complete waste of time.
The biggest problem I keep running into is the non-symmetry of the function about $x=1/3$. I feel like the conjugacy function I am in search of must have this same non-symmetry, based off the fact that previous conjugacies for functions like $Q_c(x)$ and the tent map have symmetric conjugacies.
Any hints appreciated.