Studies on spaces of initial conditions for nonautonomous mappings of the plane

We study nonautonomous mappings of the plane by means of spaces of initial conditions. First we introduce the notion of a space of initial conditions for nonautonomous systems and we study the basic properties of general equations that have spaces of initial conditions. Then, we consider the minimization of spaces of initial conditions for nonautonomous systems and we show that if a nonautonomous mapping of the plane with a space of initial conditions, and unbounded degree growth, has zero algebraic entropy, then it must be one of the discrete Painlev\'{e} equations in the Sakai classification.