§0. Introduction Polynomial maps g : C → C are the simplest holomorphic maps with interesting dynamical behavior. The study of such maps has had an important influence on the field of dynamical systems. On the other hand the traditional focus of the field of dynamical systems has been in a different direction: invertible maps or diffeomorphisms. Thus we are led to study polynomial diffeomorphis...

Studying dynamical systems is key to understanding a wide range of phenomena ranging from planetary movement to climate patterns to market dynamics. Various numerical tools have been developed to address specific questions about dynamical systems, such as predicting the weather or planning the trajectory of a satellite. However, the theory of computation behind these problems appears to be very...

Northern New England State University (NNESU) is a four-year comprehensive state-supported institution located in a small city near the Canadian border. The university is comprised of three colleges: Arts and Sciences, Business and Finance, and Education and Professional Studies. It is the largest employer in the surrounding community. Enrollment at NNESU is comprised of approximately 6,000 und...

1.1. Overview. We will describe recent developments in several intimately related problems of complex and real one-dimensional dynamics: rigidity of polynomials and local connectivity of the Mandelbrot set, measure of Julia sets, and attractors of quasi-quadratic maps. A combinatorial basis for this study is provided by the Yoccoz puzzle. The main problem is to understand the geometry of the pu...

We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism. Introduction. In this paper we recall from [U2] the ...

Given a quadratic polynomial q : Ĉ→ Ĉ and a representation G : Ĉ→ Ĉ of C2 ∗C5 in PSL(2,C) satisfying certain conditions, we will construct a 4 : 4 holomorphic correspondence on the sphere (given by a polynomial relation p(z,w)) that mates the two actions: The sphere will be partitioned into two completely invariant sets Ω and Λ. The set Λ consists of the disjoint union of two sets, Λ+ and Λ−, e...

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent h equal to the Hausdorff dimension of the Julia set. Both h-dimensional Hausdorff and packing measures are finite and positive on the Julia set and ...