Piecewise Rational Affine Maps, Multidimensional Generalized Shifts, and Computation over the Real Numbers

In this paper we study the computational power of certain classes of dynamical systems defined by the iteration of piecewise rational affine maps on the compact set [0,1]n as a model of computation over the real numbers. To achieve this aim we use symbolic dynamics to study the dynamics of this class of piecewise rational affine maps and in the process introduce multidimensional generalized shifts, which are an extension of the generalized shift of Moore to higher dimensional spaces. We show that multidimensional generalized shifts are not equivalent (conjugate) to (one-dimensional) generalized shifts.We then establish that computable analysis, which allows to extend computability based on Turing machines to real numbers, is computationally equivalent to multidimensional generalized shifts, both at a computability and at a computational complexity (polynomial time) level. This implies that a certain class of dynamical systems defined by the iteration of rational piecewise affine maps on the unit ball can thus compute any function computable over the real numbers in the computable analysis sense.These results highlight that several distinct models of computation over the real numbers, such as computable analysis, Claude Shannon's General Purpose Analog Computer, certain classes of piecewise rational affine maps, and the generalized shift, are all computationally equivalent.