This paper is motivated by Brolin's theorem. The phenomenon we wish to demonstrate is as follows: if F is a holomorphic correspondence on P 1 , then (under certain conditions) F admits a measure µF such that, for any point z drawn from a "large" open subset of P 1 , µF is the weak * -limit of the normalised sums of point masses carried by the pre-images of z under the iterates of F . Let † F denote the transpose of F . Under the condition dtop(F ) > dtop( † F ), where dtop denotes the topological degree, the above phenomenon was established by Dinh and Sibony. We show that the support of this µF is disjoint from the normality set of F . There are many interesting correspondences on P 1 for which dtop(F ) ≤ dtop( † F ). Examples are the correspondences introduced by Bullett and collaborators. When dtop(F ) ≤ dtop( † F ), equidistribution cannot be expected to the full extent of Brolin's theorem. However, we prove that when F admits a repeller, equidistribution in the above sense holds true.On the dynamics of multivalued maps between complex manifolds: results of perhaps the broadest scope are established in [11]. We borrow from [11] the following definition.Definition 1.1. Let X 1 and X 2 be two compact complex manifolds of dimension k. We say that Γ is a holomorphic k-chain in X 1 × X 2 if Γ is a formal linear combination of the

In this note we show a large deviation result for the periodic points of a hyperbolic rational map on the Riemann sphere. This extends the well known equidistribution result of Lyubich in this setting. We also consider convergence results for more general weighted averages of orbital measures with respect to Hölder continuous functions.

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In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on P 1 . Results on the iterative dynamics of such a correspondence can now be applied to the study of the rational semigroup. We focus on an invariant measure for the aforementioned correspondence -known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to analyse the distribution of repelling fixed points of a finitely generated rational semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia set of such a semigroup.

In this paper, we consider the family of hyperbolic quadratic polynomials parametrised by a complex constant; namely P c (z) = z 2 + c with |c| < 1 and the family of hyperbolic cubic polynomials parametrised by two complex constants; namely P (a 1 ,a 0 ) (z) = z 3 + a 1 z + a 0 with |a i | < 1, restricted on their respective Julia sets. We compute the Lyapunov characteristic exponents for these polynomial maps over corresponding Julia sets, with respect to various Bernoulli measures and obtain results pertaining to the dependence of the behaviour of these exponents on the parameters describing the polynomial map. We achieve this using the theory of thermodynamic formalism, the pressure function in particular.