Renormalization in Dimension Two - Lecture 1

Explore the concept of renormalization in two-dimensional dynamical systems through this lecture by Jonguk Yang from the University of Zurich. Delve into the extension of renormalization techniques from one-dimensional cases to higher-dimensional settings, combining this approach with the theory of non-uniformly partially hyperbolic systems. Examine the generalization of renormalization theory for unimodal interval maps to a specific class of diffeomorphisms in two dimensions. Discover the key step of identifying the higher-dimensional analog of a "critical point" and learn about two-dimensional a priori bounds, which provide uniform control on the geometry of dynamics at arbitrarily small scales. Cover topics including the proof of a priori bounds for unimodal interval maps in one dimension, quantitative Pesin theory, the definition of critical orbit and unimodality in two dimensions, linear ordering on the renormalization limit set for 2D unimodal diffeomorphisms, and the regularity of the first return map. Gain insights into this cutting-edge research based on joint work with S. Crovisier, M. Lyubich, and E. Pujals.