One-Parameter Families of Hyperbolic Iterated Function Systems on the Line - Lecture 2A

Explore a 45-minute lecture from the Simons Semester on Dynamics series focusing on one-parameter families of hyperbolic iterated function systems (IFS) on the line. Delve into advanced mathematical concepts surrounding self-similar measures' dimensions and absolute continuity, with particular emphasis on hyperbolic IFS. Learn about a novel technique that extends the traditional transversality method, developed in collaboration with Bárány, Solomyak, and Śpiewak. Examine how strictly contracting smooth self-mappings of compact intervals behave when studying dimension of attractors, especially in cases without separation conditions. Understand the complexities of parameter-dependent measures and their natural projections, including applications to Gibbs measures with Hölder continuous potentials and systems with place-dependent probabilities. Discover how this extended transversality technique enables the study of absolute continuity for projected measures when the ratio of entropy over Lyapunov exponent exceeds 1, with practical applications to place-dependent Bernoulli convolutions and Blackwell measure for binary channel.