Williams' work from the 1960s and 1970s provides a thorough understanding of hyperbolic one-dimensional attractors through their representation as inverse limits. In fact, point in a uniformly ...hyperbolic attractor has a neighbourhood that is homeomorphic to a Cantor set of open arcs. In order to understand the topology of non-uniformly hyperbolic attractors better, we study the existence and prevalence of points with more complicated local structures in simple models of planar attractors, focusing on unimodal inverse limits setting. Such points whose neighbourhoods are not homeomorphic to the product of a Cantor set and an open arc are called folding points. We distinguish between various types of folding points and study how the dynamics of the underlying unimodal map affects their structures. Specifically, we characterise unimodal inverse limit spaces for which every folding point is an endpoint.
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