Classically, discrete dynamics is the
study of iteration of polynomial and rational maps on the
complex plane or on the Riemann sphere. The arithmetic
properties for the associated dynamical system were studied
by J. Silverman in his paper "Integer points, Diophantine
approximation, and iteration of rational maps" (1993).
Motivated by the analogy between Nevanlinna theory and
Diophantine approximation discovered by C. F. Osgood and P.
Vojta, M. Ru and E. Yi studied the complex hyperbolic
properties for the associated dynamical systems in their
paper "Nevanlinna Theory and Iteration of Rational Maps"
(2005). Finiteness in Diophantine approximation corresponds
to constancy of a meromorphic function in Nevanlinna
Theory. This book discusses the iteration of rational maps
in relation to Diophantine approximation, as well as
Nevanlinna theory, based on the above mentioned