Dynamics of Meromorphic Maps with Small Topological Degree I: From Cohomology to Currents

We consider the dynamics of a meromorphic map on a compact Kähler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which iterates of the map expand the cohomology class of a Kähler form. Our goal in this article and its seque...

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Bibliographic Details
Published in:Indiana University mathematics journal 2010-01, Vol.59 (2), p.521-561
Main Authors: Diller, Jeffrey, Dujardin, Romain, Guedj, Vincent
Format: Article
Language:English
Subjects:
Automorphisms
Entropy
Ergodic theory
Exact sciences and technology
General mathematics
General, history and biography
Global analysis, analysis on manifolds
Hyperplanes
Mathematical surfaces
Mathematical theorems
Mathematics
Polynomials
Sciences and techniques of general use
Topological theorems
Topology
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We consider the dynamics of a meromorphic map on a compact Kähler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which iterates of the map expand the cohomology class of a Kähler form. Our goal in this article and its sequels is to carry out a program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, using the linear action of the map on cohomology to construct and analyze invariant currents with special geometric structure. We also give some examples and consider in more detail the special cases where the surface is irrational or the self-intersections of the invariant currents vanish.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2010.59.4023