On the Theory of Chaotic Rank One Attractors

(1) We felt that there was a geometric structure (namely the structure of critical regions) that, if added, could make more transparent the overall purpose of the different components of the very complicated tour de force analysis of Benedicks and Carleson. It could also induce non-trivial technical simplifications. We also felt that the various technical details skipped in [3], considering the profoundness of the theory in terms of impact, should be filled with care. (2) We thought it highly desirable to have a theory on non-uniformly hyperbolic maps that could be applied to the analysis of concrete differential equations. Based on what was available to us at the time, it was decided that we could achieve such a goal in two steps. First, we could introduce a flexible setting on non-uniformly hyperbolic maps and construct a comprehensive chaos theory by generalizing previous analysis based on [3]. Second, we could try to find concrete equations of either practical or historical implications, to which we would hopefully apply our new theory.