Consider the set $[0,1]$ is equipped with the Borel $\sigma$-algebra and Lebesgue measure $\lambda$. Let $0<r<\frac12$, and define $$W(x)=\begin{cases} \frac12(1-\frac xr)&\text{if }0\leq x\leq r,\\\frac12\left(\frac{x-r}{\frac12-r}\right)&\text{if }r< x\leq \frac12,\\W(1-x)&\text{if } \frac12<x\leq1.\end{cases}$$ I'm asked to find a probability measure $\mu$ for this transformation that is dominated by the Lebesgue measure, and to show that this transformation is ergodic (w.r.t. $\mu$, which, of course, would also be implied if we showed $W$ is ergodic w.r.t. $\lambda$).
It was not difficult to find that $\mu=2\cdot\mathbf1_{[0,1/2]}\lambda$ is $W$-invariant, but I found it difficult to show that $W$ is ergodic. I tried looking at the Perron-Frobenius operator, for which I found $$P_W(f)=\left[2rf(r(1-2x))+(1-2r)f(r(1-2x)+x)\right]\mathbf1_{[0,1/2]}(x),$$ and I tried to use this, to show that $W$ is even exact; however, I didn't succeed.
Any help is much appreciated!
EDIT: it seems that $(I,\mathcal B,\mu,W)$ is measure preservingly isomorphic to the tent map $T(x)=2x\wedge(2-2x)$ on $(I,\mathcal B,\lambda)$ through $\psi:[0,\frac12]\to[0,1]$ defined by $$\psi(x)=\begin{cases}1-\frac x{2r}&\text{if }0\leq x\leq r,\\\frac12-\frac{x-r}{1-2r}&\text{if }r< x\leq \frac12,\end{cases}$$ while it is proved in my book that the tent map is isomorphic to the left shift $S$ on $\{0,1\}^\mathbb N$; it is also well-known that $S$ is ergodic, hence $W$ is ergodic (right?). I'm still interested in other solutions, if possible.