Dynamics of hyperbolic correspondences

This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

The Projection Pressure for Asymptotically Weak Separation Condition and Bowen's Equation

Let{Si}i=1lbe a weakly conformal iterated function system onRdwith attractorK. Letπbe the canonical projection. In this paper we define a new concept called “projection pressure”Pπ(φ)forφ∈C(Σ)and show the variational principle about the projection pressure under AWSC. Furthermore, we check that the zero of “projection pressure” still satisfies Bowen's equation. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractorK.

THE AVERAGE DENSITY OF SELF-CONFORMAL MEASURES

The paper calculates the average density of the normalized Hausdorff measure on the fractal set generated by a conformal iterated function system. It equals almost everywhere a positive constant given by a truncated generalized s-energy integral, where s is the corresponding Hausdorff dimension. As a main tool a conditional Gibbs measure is determined. The appendix proves an appropriate extension of Birkhoff's ergodic theorem which is also of independent interest.

Parabolic iterated function systems

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.