  We examine the completely isometric automorphisms of a free product of
noncommutative disc algebras. It will be established that such an automorphism
is given simply by a completely isometric automorphism of each component of the
free product and a permutation of the components. This mirrors a similar fact
in topology concerning biholomorphic automorphisms of product spaces with nice
boundaries due to Rudin, Ligocka and Tsyganov.
  This paper is also a study of multivariable dynamical systems by their
semicrossed product algebras. A new form of dynamical system conjugacy is
introduced and is shown to completely characterize the semicrossed product
algebra. This is proven by using the rigidity of free product automorphisms
established in the first part of the paper.
  Lastly, a representation theory is developed to determine when the
semicrossed product algebra and the tensor algebra of a dynamical system are
completely isometrically isomorphic.
