Fractional Nottale's Scale Relativity and emergence of complexified gravity

Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale's Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.