Learn about renormalization theory in higher dimensions through this mathematics lecture that explores the extension of one-dimensional techniques to two-dimensional dynamical systems. Discover how combining renormalization approaches with non-uniformly partially hyperbolic systems theory enables the analysis of diffeomorphisms in dimension two. Explore key concepts including the identification of higher-dimensional critical points, two-dimensional a priori bounds, and uniform control of dynamics at small scales. Examine topics like quantitative Pesin theory, critical orbits in 2D, linear ordering on renormalization limit sets, and the regularity of first return maps. Gain insights from research conducted jointly with S. Crovisier, M. Lyubich and E. Pujals that bridges fundamental concepts from one-dimensional unimodal interval maps to more complex two-dimensional systems.