This paper deals with classifying the dynamics of {\it Topologically Anosov}
plane homeomorphisms. We prove that a Topologically Anosov homeomorphism
f:R2→R2 is conjugate to a homothety if it is the time
one map of a flow. We also obtain results for the cases when the nonwandering
set of f reduces to a fixed point, or if there exists an open, connected,
simply connected proper subset U such that U⊂Int(f(U)), and such that ∪n≥0fn(U)=R2. In the general
case, we prove a structure theorem for the α-limits of orbits with empty
ω-limit (or the ω-limits of orbits with empty α-limit),
and we show that any basin of attraction (or repulsion) must be unbounded.Comment: 10 page