Real quadratic iteration map ($x_{n+1}=\mu -x_n^2$)

The family $x_{n+1}=\mu -x_n^2$ has fixed points $(-1\pm \sqrt{1+4\mu })/2$. Its upper branch attracts linearly for $-1/4<\mu <3/4$ and loses stability in a period-doubling bifurcation. The resulting two-cycle has points $(1\pm \sqrt{4\mu -3})/2$ and multiplier $4(1-\mu )$, attracting for $3/4<\mu <5/4$. At $\mu =2$, the map is conjugate on $[-2,2]$ to the full tent map.

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.