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Explore the concept of Birkhoff sums as distributions in this 43-minute lecture by Daniel Smania from ICMC-USP, presented at the Simons Semester on Dynamics. Delve into regularity results for various dynamical systems exhibiting hyperbolicity, including hyperbolic linear maps on the torus and piecewise expanding maps on the interval. Discover applications to deformation theory and learn about a method for proving that topological classes of one-dimensional dynamical systems are often finite codimension smooth manifolds. Understand the crucial step of identifying infinitesimal deformations with primitives of Birkhoff sums, which enables the use of ergodic properties in piecewise expanding maps to study the regularity of infinitesimal deformations. This lecture presents joint work with Clodoaldo Ragazzo, offering insights into the intersection of dynamical systems, ergodic theory, and deformation theory.
