Suppose that is topologically transitive on a metric space X. Prove that either X is an infinite set or X consists of the orbit of a single periodic point. (Conjecture: Senior Project/Master’s...

Suppose that

is topologically transitive on a metric space X. Prove that either X is an infinite set or X consists of the orbit of a single periodic point.

(Conjecture: Senior Project/Master’s Thesis) Let
be a continuous map on a metric space X (an interval I) which is chaotic. Show that if X is connected, then fm is chaotic for all m
.