A 1D smooth map on an interval is unimodal if it maps the interval into itself by folding it once (at the unique critical point). Analogously, a 2D smooth diffeomorphism on a square is Hénon-like if it maps the square into itself by squeezing it along the vertical direction to a thin strip, then bending it into a “C”-shape.
Joint with S. Crovisier, M. Lyubich and E. Pujals, we extended the celebrated renormalization theory of 1D unimodal maps to the 2D setting, so that it can be applied to the study of Hénon-like maps. In this talk, I will give an outline of our main results. This includes renormalization convergence, the uniqueness of the “2D critical point”, and the robustness of the required regularity conditions on the maps (so that they are finite-time checkable).
The talk will mostly NOT overlap with the talk I gave at the Renormalization and Friends workshop last summer.