Action of $S^1$ on homotopy groups of an $S^1$-space

Question

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed space with a continuous action from $S^1$. Is the induced action of $S^1$ on its homotopy $\pi_{\ast} (X)$ trivial ?

This is motivated by trying to understand the $S^1$ action on $THH$. It seems like the $S^1$ action is trivial on $THH_{\ast}$ but I don't understand why.

Thanks!

Answer 1

Yes. $S^1$ is path connected, so every element of $S^1$ acts by a map homotopic to the identity.