The fixed points of $F_\mu$ on $[0,1]$ are $x = 0$ and $x = x^* := 1 - \tfrac{1}{1+\mu}$. Both are unstable since $|F_\mu'| = \mu > 1$ wherever the derivative exists. The critical point is $c = \tfrac{1}{2}$ with $F_\mu(\tfrac{1}{2}) = \tfrac{\mu}{2}$ and $F_\mu^2(\tfrac{1}{2}) = \mu(1 - \tfrac{\mu}{2}) = \mu - \tfrac{\mu^2}{2}$. Consider the interval $J = [F_\mu^2(\tfrac{1}{2}), F_\mu(\tfrac{1}{2})] = [\mu(1 - \tfrac{\mu}{2}), \tfrac{\mu}{2}]$. Points above $J$ map to points below $J$ and points below $J$ grow geometrically under $F_\mu^2$ until they enter $J$. Hence the interesting dynamics is confined to $J$. On the subinterval $[x_0, x_{-2}]$ (where $x_0 = x^*$ and $x_{-2}$ is the second preimage of $x^*$ under $F_\mu$), the second iterate $F_\mu^2$ is itself a tent map with parameter $\mu^2$. If $\mu^2 > 2$ (i.e., $\mu \geq \sqrt{2}$), then $F_\mu^2$ has a horseshoe by the previous theorem applied to the tent map with parameter $\mu^2 > 2$. For $\mu < \sqrt{2}$, one iterates: the map $F_\mu^4$ on an appropriate subinterval is a tent map with parameter $\mu^4$. If $\mu^4 > 2$ (i.e., $\mu \geq 2^{1/4}$), then $F_\mu^4$ has a horseshoe. Continuing inductively, for each $n \geq 1$, if $2^{1/2^n} \leq \mu < 2^{1/2^{n-1}}$, then $F_\mu^{2^n}$ has a horseshoe. Since $2^{1/2^n} \to 1$ as $n \to \infty$ and $\mu > 1$, every $\mu \in (1,2)$ satisfies $\mu \geq 2^{1/2^n}$ for some $n$. Hence $F_\mu^{2^n}$ has a horseshoe for some $n$, and $F_\mu$ is Glendinning-chaotic.