Homeomorphisms with many recurrent points

Abstract:

Let $X$ be a topological space and $H(X)$ the space of all homeomorphisms of $X$ onto itself with the compact open topology. If $f \in H(X)$ and $p \in X$, then $p$ is a recurrent point of $f$ provided $p$ is in the closure of $\{ {f^n}(p)|n \geqslant 1\}$. It is shown that if $X$ is Hausdorff and $V$ is a nonempty open subset of $X$ homeomorphic to Euclidean $n$-dimensional space with $n \geqslant 1$, then $\{ f \in H(X)|$ the recurrent points of $f$ are dense in $V$ is nowhere dense in $H(X)$.