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Explore the first lecture in a mini-course series that delves into extending renormalization theory from one-dimensional to two-dimensional dynamical systems. Learn how self-similarity at smaller scales can reveal crucial insights about combinatorial, topological, and geometric aspects of dynamics. Discover how the theory of non-uniformly partially hyperbolic systems combines with renormalization approaches to generalize unimodal interval maps to diffeomorphisms in dimension two. Master key concepts including the higher-dimensional analog of critical points, two-dimensional a priori bounds, and uniform control of dynamics at small scales. Study the proof of a priori bounds for unimodal interval maps, quantitative Pesin theory, critical orbit definition in 2D, linear ordering on renormalization limit sets, and the regularity of first return maps.
