Definitely "The arithmetic of Dynamical Systems" by Silverman is THE textbook you should read first. In my opinion, in order to get a grasp of what arithmetic dynamics is about, you firstly need solid bases of algebraic geometry and number theory. Complex dynamics is also going to be pretty helpful if you want to enter the more dynamical aspects of the theory.
Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer.
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arithmetic - What are the formal names of operands and results for ...