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The dynamics of holomorphic correspondences of P1: invariant measures and the normality set

Bharali, Gautam and Sridharan, Shrihari (2016) The dynamics of holomorphic correspondences of P1: invariant measures and the normality set. In: COMPLEX VARIABLES AND ELLIPTIC EQUATIONS, 61 (12). pp. 1587-1613.

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Official URL: http://dx.doi.org/10.1080/17476933.2016.1185419

Abstract

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 mu F such that, for any point z drawn from a ` large' open subset of P-1, mu F is the weak*- limit of the normalized 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) > d(top)((+) F), where dtop denotes the topological degree, the above phenomenon was established by Dinh and Sibony. We show that the support of this mu F is disjoint from the normality set of F. There are many interesting correspondences on P-1 for which d(top)(F) <= d(top)((+) F). Examples are the correspondences introduced by Bullett and collaborators. When d(top)(F) = d(top)((+) 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.

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the TAYLOR & FRANCIS LTD, 2-4 PARK SQUARE, MILTON PARK, ABINGDON OR14 4RN, OXON, ENGLAND
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 20 Jan 2017 04:31
Last Modified: 20 Jan 2017 04:31
URI: http://eprints.iisc.ac.in/id/eprint/55942

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