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

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