Many parallels between complex dynamics and hyperbolic geometry have
emerged in the past decade. Building on work of Sullivan and Thurston,
this book gives a unified treatment of the construction of fixed-points
for renormalization and the construction of hyperbolic 3- manifolds
fibering over the circle.
Both subjects are studied via geometric limits and rigidity. This
approach shows open hyperbolic manifolds are inflexible, and yields
quantitative counterparts to Mostow rigidity. In complex dynamics, it
motivates the construction of towers of quadratic-like maps, and leads
to a quantitative proof of convergence of renormalization.