Abstract. Let $f\colon X\rightarrow X$ be a continuous map on a Hausdorff topological space $X$ without isolated points. We show that if the orbit of a point $x\in X$ under $f$ is dense in $X$ while the orbit of $x$ under $f^N, N>1,$ is not, then the space $X$ decomposes into a family of sets relative to which the behaviour of $f$ is simple to describe. This decomposition solves a problem that P. S. Bourdon posed in 1996 ([3]). A slight variant of our result also provides a new argument for the celebrated theorem of S. Ansari [1]: If $T$ is a hypercyclic operator on a topological vector space $X$ then $T$ and $T^N$ have the same sets of hypercyclic vectors ($N\geq1$).

AMS Subject Classification (1991): 47A16; 37B05

Keyword(s): Dense orbit, Hypercyclic operator, Powers of hypercyclic operators, Ansari's theorem

Received September 21, 2007, and in revised form October 17, 2007. (Registered under 6015/2009.)