Finite type coarse expanding conformal dynamics

Hausdorff Dimension and Conformal Dynamics II: Geometrically Finite Rational Maps

This paper investigates several dynamically defined dimensions for rational maps f on the Riemann sphere, providing a systematic treatment modeled on the theory for Kleinian groups. We begin by defining the radial Julia set Jrad(f), and showing that every rational map satisfies H. dimJrad(f) = α(f) where α(f) is the minimal dimension of an f -invariant conformal density on the sphere. A rationa...

Coarse Grained Molecular Dynamics

Conformal fluid dynamics

We present a conformal theory of a dissipationless relativistic fluid in 2 space-time dimensions. The theory carries with it a representation of the algebra of 2-D area-preserving diffeomorphisms in the target space of the complex scalar potentials. A complete canonical description is given, and the central charge of the current algebra is calculated. The passage to the quantum theory is discus...

Coarse Dynamics for Coarse Modeling: An Example From Population Biology

Networks have become a popular way to concisely represent complex nonlinear systems where the interactions and parameters are imprecisely known. One challenge is how best to describe the associated dynamics, which can exhibit complicated behavior sensitive to small changes in parameters. A recently developed computational approach that we refer to as a database for dynamics provides a robust an...

Renormalization Ideas in Conformal Dynamics

1. Introduction How to look at a dynamical system f at a small scale? You should take a small piece of the phase space, consider the rst return map to this piece, and then rescale it to \the original size". The new dynamical system is called the renormalization Rf of the original one. It may happen that Rf looks \similar" to f, and then you can try to repeat this procedure, and construct the se...