Fractional Susceptibility in Quadratic Dynamics

This lecture discusses the fractional susceptibility function within the context of the quadratic family of dynamical systems. The instructor begins by introducing the concept of fractional response and its relevance to understanding dynamical systems. They present a toy model of piecewise expanding maps, highlighting the connection between fractional susceptibility and linear response. The discussion includes results from previous research, emphasizing the importance of the SRB measure and its differentiability. The instructor then delves into the quadratic family, illustrating how the susceptibility function behaves under various parameters. They explain the paradoxes that arise when examining the dependence of the SRB measure on these parameters, particularly in cases of non-structural stability. The lecture concludes with conjectures regarding the behavior of the fractional susceptibility function, particularly its holomorphic properties and the implications of certain parameter choices. Overall, the lecture provides a comprehensive overview of the interplay between fractional susceptibility and dynamical systems, supported by mathematical rigor and examples.