In the parabolic situation the analytic properties of the spectrum will be derived by means of infinite ergodic theory, in particular revealing phase transitions depending on the Poincare exponent and the maximal rank of the cusps. This set is the union of the closure of the forward orbit of the critical point with the set of parabolic cycles. Tools include real and complex dynamics, holomorphic box maps and quasiconformal functions. Abstract Invariants such as entropy, Hausdorff dimension, physical measures, etc are rarely available in closed form for nonlinear dynamical systems. We will consider various strategies for efficiently approximating these objects.
Rational maps satisfying decay of geometry enjoy good properties, including a connecting lemma.
In both cases, the mating will be realised by a third rational map or quasi-Fuchsian group. We describe a new method for estimating rates of mixing for certain non-uniformly hyperbolic dynamical systems. This enables one to prove that many systems which were previously believed to have sub-exponential decay of correlations actually do so. Our main tool is a general theorem about certain sequences of bounded linear operators acting on Banach spaces. Abstract We study a natural extension of the renormalization operator for multimodal maps.