Free subalgebras of the skew polynomial rings k[x,y][t;σ] and k[x,x⁻¹,y,y⁻¹][t;σ]

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.