Renormalization group applications in area-preserving nontwist maps and relativistic quantum field theory

In this thesis, we apply the ideas of the renormalization group to two different areas of physics. Extending the work of del-Castillo-Negrete, Greene and Morrison on the standard nontwist map, we study the break-up of an invariant torus with winding number different from the inverse golden mean, and interpret the result within the renormalization group framework. We construct a renormalization operator on the space of commuting map pairs, and study its fixed point. In addition, we present preliminary results about a new map that we call the piecewise-linear standard nontwist map. In the second part of this thesis, we extend to fields defined on Minkowski spacetime a functional integral formalism, developed in Euclidean Field Theory to study the long-distance behavior of scalar field theories. A new aspect of this approach is the use of an independent scaling variable. To compute some of the resulting integrals, we work out explicit formulae for the Fourier transform of Lorentz invariant functions in pseudo-spherical coordinates.