Polynomial Dynamics: Algebraic and Arithmetic Properties

We outline some recent results and pose several open questions
about the algebraic properties of univariate and multivariate polynomial
iterations. These properties include the degree growth, irreducibility,
linear and multiplicative independence and similar.

In turn, these algebraic properties form a background for studying
the arithmetic properties of the elements in the orbits. These properties include
periodicity, distribution (which links them to pseudorandom
number generators), multiplicative indepence, etc.