  Let \sigma be an automorphism of a commutative k-algebra R. The skew
polynomial ring R[t;\sigma] is generated by R and an indeterminate t subject to
the relations ta=\sigma(a)t for all a in R. For certain R and appropriate
\sigma there are elements a and b in R such that the subalgebra of R[t;\sigma]
generated by at and bt is a free algebra. If \sigma is an automorphism of the
polynomial ring k[x,y], then the subalgebra of k[x,y][t;\sigma] generated by xt
and yt is free if and only if \sigma is not conjugate to an elementary
automorphism. If \sigma is an automorphism of k[x,x^{-1},y,y^{- 1}] of the form
\sigma(x)=x^ay^b and \sigma(y)=x^cy^d, then the subalgebra of k[x,x^{-1},y,y^{-
1}][t;\sigma] generated by xt and yt is free if the spectral radius of the 2x2
matrix {{a b} \\ {c d}} is >2; indeed, k[x,x^{-1},y,y^{- 1}][t;\sigma] contains
a free subalgebra if and only if the spectral radius of 2x2 matrix {{a b} \\ {c
d}} is >1.
