Benjamin Hutz, in Chapter 10 of his An Experimental Introduction to Number Theory, allows for the optional inclusion of discrete dynamical systems with a number-theoretic flavor in an undergraduate introduction to number theory.
What are the possibilities to start the other way around, with an introductory course on discrete dynamical systems, such as Robert L. Devaney's A First Course in Chaotic Dynamical Systems: Theory and Experiment, and include some relevant material from elementary number theory? What would be good topics or results to include? They would ideally be accessible without extremely lengthy excursions into number theory, but would show how number theory can arise, sometimes unexpectedly, when studying dynamics.
The expected background would be at least first year calculus, preferably with some mathematical maturity gained from differential equations, second year calculus and/or linear algebra, but not necessarily abstract algebra, complex analysis or other higher courses. A prior course in number theory would not be a prerequisite.