I recently read a paper about chaos in real dynamical systems. Supposedly, it suffices to show that a continuous function $f:[0,1]\rightarrow [0,1]$ is topologically transitive for its set of periodic points to be dense. For that end, the author proves a straightforward lemma: if $0<m<n$, and $x,f^m(x),f^n(x)\in J$, where $J\subset [0,1]$ is an interval that contains no periodic points of $f$, then $f^m(x)$ must lie between $x$ and $f^n (x)$. However, I believe it's not used properly in the proof of the theorem.
The final lines of the theorem read:
We have that $0<m<n$ and $z,f^n (z)\in U$ while $f^m (z)\not\in U$, and this violates our earlier lemma.
Why is the lemma violated? First of all, the set $U$ could not be an interval. This can be fixed easily, but how can one guarantee that this interval contains no periodic points? (It's just an open set that exists, but in the proof there is nothing to relate it to the interval $J$ with no periodic points.) And even if these issues were fixed, I see no contradiction. The fact that $f^m (z)\not\in U$ only means that it does not lie between $z$ and $f^n (z)$, which is not a contradiction to the lemma, but its contraposition.
Am I missing something here?