We discuss a remarkable phenomenon which occurs in one-dimensional dynamics, that is, for maps acting on an interval or a circle. Such maps are often not structurally stable, however their topological class is an infinite dimensional smooth manifold with finite codimension. Consequently, the deformation theory of such systems is quite rich. Recent developments suggest that the study of the existence, uniqueness and regularity of solutions for certain cohomological equations is crucial for deeper understanding of these phenomena, and ergodic theory plays an important role in this study.
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Daniel Smania (2025) studied this question.
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