Any ideas, hints on the following would be great.
Suppose that $T\colon X \to X$ is continuous, and there exist at least two distinct periodic orbits. Show that if one of the periodic orbits is attracting then $T$ is not topologically transitive.
Note that the definition of attracting is: If $\Gamma = O(x)$ is the periodic orbit then there is and open set $U$ containing $\Gamma$ such that $\omega(x') = \Gamma$ for every $x' \in U$.
It may be assumed that $X$ is compact.