Absolutely continuous invariant probability measures (ACIPs) are a fundamental object in the study of multimodal interval maps since they capture statistical properties of orbits of points under chaotic regime. Two questions arise naturally: do there exist sufficient conditions that guarantee the existence of ACIPs? For a given family of maps, how many parameters satisfy such conditions?
In the last few decades, a plethora of sufficient analytic conditions have been developed: Collet–Eckmann condition, Nowicki–van Strien summability condition, Martens–Nowicki condition, and the BRSS large derivative condition, just to name a few. On the other hand, it has been shown that topological properties can sometimes be sufficient as well, for example Misiurewicz, or Topological Collet–Eckmann. In the special case of families of unimodal maps, most of these conditions are satisfied by almost every non-hyperbolic parameter, but unfortunately, the same cannot be said for maps with more than one critical point.
In this talk, we will present the problem at hand, and discuss progress made in establishing a new condition that bridges analytical and topological conditions for real multimodal maps. We will also explore possible applications of this new existence condition. This is currently a work in progress.
