  Let X be a smooth projective surface defined over an uncountable
algebraically closed field k and let k(X) be its field of rational functions.
Let s be an automorphism of X. This paper proves there is a non-negative
integer n and elements a and b in k(X) such that the subalgebra of the skew
Laurent extension k(X)[t, t^{-1};s] generated by at^n and bt^n is a free
algebra if and only if the spectral radius for the action of s^* on the
Neron-Severi group of X is >1. Thus, when s is an automorphism of a smooth
complex projective surface X, C(X)[t,t^{-1};s] has a free subalgebra on >1
variables if and only if the topological entropy of s is positive. Furthermore,
if s is an automorphism of k(X), then k(X)[t,t^{-1};s] contains a free
subalgebra if and only if the dynamical degree of s is >1; in this situation, s
might not be induced by an automorphism of any smooth projective surface. These
results are used to show that certain twisted homogeneous coordinate rings of
smooth projective surfaces contain free subalgebras.
