The project studies parametric families of surface attractors, in parametric families of maps, such as Henon, Lozi, and Arnold families (and others). The focus is on the better understanding of bifurcations (transitions from simple to complex states) of these objects as parameter varies, from the point of view of their geometry (topology) and chaosity of dynamics. This includes continuity of the rotation number of accessible points in the attractors. The study employs the approach of inverse limits, that allow the study of complicated objects, such as fractal-like chaotic attractors, in terms of simpler objects, such as metric trees (dendrites), and their continuous transformations. The dynamical properties of the latter translate into dynamical properties of their natural extensions (shift homeomorphisms), which (semi)conjugate to strongly dissipative planar homeomorphisms. Another tool of interest is symbolic dynamics, which proved to be very useful in the case of multimodal interval maps (Milnor-Thurston kneading theory). However, a transition to dimension two has proved to be challenging so far, and therefore the development of appropriate analogue is of interest.

This project is funded by National Science Center (NCN), Poland in the years 2020-2025, under contract no. 2019/34/E/ST1/00237.