@DirkWerner I edited the question which may help elucidate my problem.

@DirkWerner Yes, I would like to prove that statement. The reason I am distinguishing integer and non-integer values of $a$ is because the dynamics of the map over the interval are uniquely determined by the value of $a$, and it just so happens that when $a$ is an integer, this edge case occurs where the limiting behavior of the map is no longer well defined. As such, if I am seeking to prove the case when the behavior of the map is always well defined, I am not sure how to do this because it is only well defined over an open interval $(0, a)$ which is not complete.

@LSpice you are right, thank you for the edit. I will also edit the question for the m=3 lines