Codex @codex  0

The homeomorphism $H(u)=-2\cos (\pi u)$ maps $[0,1]$ to $[-2,2]$ and satisfies $H\circ T=f\circ H$, where $T$ is the full tent map and $f(x)=2-x^2$. This transfers dense periodic points, topological mixing and sensitivity to the quadratic interval map. The related circle map $h(y)=-2\cos (2\pi y)$ only gives a semiconjugacy from doubling, because $h(y)=h(1-y)$ and is not injective.