The questions you are asking are fundamental to the theory of one-dimensional dynamical systems. I would suggest starting with an introductory textbook, such as An Introduction to Chaotic Dynamical Systems by Devaney. Books with more in-depth results include Iterated Maps of the Interval as Dynamical Systems by Collet and Eckmann, and One-Dimensional Dynamics by Melo and van Strien.
Jim Belk
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