The second iterate of a map with dense orbit

Suppose that $X$ is a Hausdorff topological space having no isolated points and that $f:X\rightarrow X$ is continuous. We show that if the orbit of a point $x\in X$ under $f$ is dense in $X$ while the orbit of $x$ under $f\circ f$ is not, then the space $X$ decomposes into three sets relative to which the dynamics of $f$ are easy to describe. This decomposition has the following consequence: suppose that $x$ has dense orbit under $f$ and that the closure of the set of points of $X$ having odd period under $f$ has nonempty interior; then $x$ has dense orbit under $f\circ f$.