For G-spaces one has the property that fixed points of a join of two spaces is the join of the fixed points. I want to prove an analogue for homotopy fixed points. Is it true that 
$$
X^{hG} \ast Y^{hG} \simeq (X\ast Y)^{hG}
$$
where $(-)^{hG}$ means homotopy fixed points.



Homotopy fixed points of a join
It is indeed not true.
Let $G = \mathbb{Z}$ and let $X = Y$ be the set of two elements with the action of $\sigma$, the generator of $\mathbb{Z}$ switching the two points.
For a space $Z$ with a $<\sigma> = \mathbb{Z}$ action, 
a point $z \in Z^{h\mathbb{Z}}$ is the same as a point $z\_0 \in Z$ and a path between $z\_0$ and $\sigma(z\_0)$. Thus we have $X^{hG} = Y^{hG}= \emptyset$ and thus $X^{hG} \star Y^{hG} = \emptyset$. However, $X \star Y \sim S^1$. Since $S^1$ is path connected we have $(S^1)^{hG} \neq \emptyset$.


