01 · The idea
In everyday language
Take a complex number and repeatedly apply a function such as a polynomial. Nearby starting points may remain close, escape or develop radically different futures. The resulting dynamical picture depends on the function’s parameters. McMullen used complex analysis, geometry and renormalisation to investigate these behaviours and their rigidity. Renormalisation isolates a smaller dynamical system and compares it with the original at a new scale; geometric control makes this comparison more than a visual analogy. His broader programme connects these questions with hyperbolic three-manifolds, revealing shared structures across subjects that at first seem unrelated.
Why it matters
The work strengthened links between complex dynamics and low-dimensional geometry, giving rigorous ways to study self-similarity, parameter spaces and the rigidity of intricate dynamical patterns.
