# Number of fixed points of a homeomorphism of $\mathbb{I}\times\mathbb{I}$

Let $f,g$ be homeomorphisms of the topological space $\mathbb{I}\times\mathbb{I}$ such that both of $f,g$ have more than one fixed point. Must $fg$ have more than one fixed point ?

I tried looking for counterexamples for some time but I didn't find.

Thank you

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I'm assuming that by $\mathbb{I}$ you mean a unit interval.
For simplicity, let us assume that $\mathbb I=[-1,1]$. Consider the following maps $$g(x,y)=(-x,y)\ \text{and} \ f(x,y)=(x,-y)$$ Both of them have more than one fixed point (actually uncountably many ) but $(0,0)$ is the only fixed point of $f\circ g$.