We consider the limit set of generalised iterated function systems. Under the assumption of a natural potential, the so called cylinder function, we prove the existence of the invariant probability measure satisfying the equilibrium state. We motivate this approach by showing that for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set itself.
In 1994 J. Lewis obtained a purely harmonic proof of the classical Little Picard Theorem by showing that if the joint value distribution of two entire harmonic functions satisfies certain restrictions then they are necessarily constant. We generalize Lewis'theorem and the harmonic Liouville theorem in terms of the range of a harmonic map in the plane.
Let R be a compact surface and let Γ be a Jordan curve which separates R into two connected components Σ 1 and Σ 2 .A harmonic function h 1 on Σ 1 of bounded Dirichlet norm has boundary values H in a certain conformally invariant non-tangential sense on Γ.We show that if Γ is a quasicircle, then there is a unique harmonic function h 2 of bounded Dirichlet norm on Σ 2 whose boundary values agree with those of h 1 .Furthermore, the resulting map from the Dirichlet space of Σ 1 into Σ 2 is bounded with respect to the Dirichlet semi-norm.
The paper is devoted to the study of sharp versions of mixed A(p)-A(q) weighted estimates for the dyadic maximal function M-d on R-n. For given parameters 1 < p < infinity and 1 <= q <= infinity, if a weight w satisfies Muckenhoupt's condition A(p), then we have the sharp A(p)-A(q) bound parallel to M-d parallel to(Lp(w)-> Lp(w)) <= p(1+1/p)/p - 1 (q/q - 1)((q-1)/p) [w](Ap)(1/p)[w(1/(1-p))](Aq)(1/p) (for q is an element of {1, infinity}, the constant is understood as an appropriate limit). Actually, a wider class of related sharp two-weight estimates for M-d is established. The results hold true in a more general context of maximal operators on probability spaces associated with a tree-like structure.
The article investigates mappings whose inverses distort the modulus of paths similarly to the Poletsky inequality.It is proved that the classes of such mappings form equicontinuous families if the majorant corresponding to the distortion of the module is integrable in the domain of their definition.Under additional conditions on the geometry of the domain of definition and the image domain these families are equicontinuous, not only at inner, but also at boundary points.In addition, the question of removability of the isolated singularities for such mappings is resolved.
