Feliks Przytycki

On the transfer operator for rational functions on the Riemann sphere

Let T be a rational function of degree 2 on the Riemann sphere. Denote L the transfer operator of a HH older-continuous function on its Julia set J = J(T) satisfying P(T;) > sup z2J (z). We study the behavior of fL n : n 1g for HH older-continuous functions and show that the sequence is (uniformly) norm-bounded in the space of HH older-continuous functions for suuciently small exponent. As a consequence we obtain that the density of the equilibrium measure for with respect to the exppP(T;) ? ]-conformal measure is HH older-continuous. We also prove that the rate of convergence of L n to this density in sup-norm is O ? exp(? p n). >From this we deduce the central limit theorem for .

On the Perron-Frobenius-ruelle operator for rational maps on the riemann sphere and for Hölder continuous functions

AbstractLet be a rational function on the Riemann sphere, φ be a Hölder continuous function on the Julia set denote the Perron-Frobenius-Ruelle operator on the space of continuous functions:

Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials

\mathcal{P}_\varphi (\psi {\text{)(}}x{\text{) = }}\sum\limits_{y \in f^{ - 1} (x)} {\psi {\text{(}}y{\text{)exp}} \varphi {\text{(}}y{\text{)}}}

Equality of pressures for rational functions

.Suppose that topological pressureP=P (f, φ) satisfiesP>sup φ. Then for every the family (expP)−nPφn (ψ) is equicontinuous and there exists a probability measure ν on and a function such that ψ0 > 0 and for every(expP)∫ψdn and

When do two rational functions have the same Julia set

(\exp P)^{ - n} \mathcal{P}_\varphi ^n (\psi {\text{)}} \to \psi _0 \cdot \smallint \psi d\eta

Rigidity of Conformal Iterated Function Systems

. The measure ψ0·η is unique equilibrium (Gibbs) state for φ.This theorem was proved recently by M. Denker and M. Urbański. We give here a significantly different proof of it, less ergodic but going deeper into holormophic dynamics.We discuss also modulus of continuity of ψ0, in particular we prove, it is bounded by