No abstract

This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measuretheoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance properties of the supports of the measures mentioned.

In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh-Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh-Sibony measure, which might be of independent interest. We then use the F -functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.

This paper studies ergodic properties of certain measures arising in the dynamics of holomorphic correspondences. These measures, in general, are not invariant in the classical sense of ergodic theory. We define a notion of ergodicity, and prove a version of Birkhoff's ergodic theorem in this setting. In fact, we strengthen this classical result in the setting of rational maps on the Riemann sphere with the Lyubich measure. We also show the existence of ergodic measures when a holomorphic correspondence is defined on a compact complex manifold. Lastly, we give an explicit class of dynamically interesting measures that are ergodic as in our definition.

For a domain Ω in the complex plane, we consider the domain Sn(Ω) consisting of those n × n complex matrices whose spectrum is contained in Ω. Given a holomorphic self-map Ψ of Sn(Ω) such that Ψ(A) = A and the derivative of Ψ at A is identity for some A ∈ Sn(Ω), we investigate when the map Ψ would be spectrum-preserving. We prove that if the matrix A is either diagonalizable or non-derogatory then for most domains Ω, Ψ is spectrum-preserving on Sn(Ω). Further, when A is arbitrary, we prove that Ψ is spectrumpreserving on a certain analytic subset of Sn(Ω).