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Learn about one-parameter families of hyperbolic iterated function systems (IFS) on the line in this 47-minute lecture that explores advanced mathematical concepts and recent developments in the field. Discover a novel technique that extends the traditional transversality method, essential for studying these complex systems. Explore how strictly contracting smooth self-mappings work within hyperbolic IFS, understand the conditions for determining attractor dimensions, and examine the geometric and dimensional properties of measures on symbolic space. Delve into the challenges of studying parameter-dependent measures and learn how the extended transversality technique enables the analysis of dimension and absolute continuity in these systems. Gain insights into applications involving Gibbs measures, place-dependent probabilities, Bernoulli convolutions, and Blackwell measure for binary channels, based on recent research conducted in collaboration with Bárány, Solomyak, and Śpiewak.
