 Solution

Given a binary cylinder with finite prefix $b_1\dots b_N$, repeat a finite block beginning with that prefix indefinitely. The resulting binary sequence is periodic under the shift and represents a periodic point of an interval map of $F$ in the cylinder. If the given prefix consists entirely of ones, append a zero to the block so that its value is less than one; this still preserves the prefix. Since cylinders can be taken inside any nonempty open interval, periodic points are dense.