Linearization of topologically Anosov homeomorphisms of non compact surfaces of genus zero and finite type

We study the dynamics of {\it topologically Anosov} homeomorphisms of non-compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f\colon S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \mathbb{R}^2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in \cite{cgx}.