MLC Along the Real Line

The problem of local connectivity of the Mandelbrot set goes back to the 80s, and is now closely linked to the Renormalization Theory of quadratic polynomials. A key task is to establish a priori bounds (compactness) for the quadratic-like renormalization operator. Working in the near degenerate regime, we prove such bounds for maps with real combinatorics. As a consequence, real combinatorial classes are singletons on the real line. We also obtain a uniform control of the shapes of real-symmetric copies of the Mandelbrot set, as well as their universal scaling properties. Joint work in progress with Jeremy Kahn and Misha Lyubich.

Please report video problems to itsupport@slmath.org.

See more of our Streaming videos on our main VMath Videos page.