Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0803.0955

Dynamics of meromorphic maps with small topological degree I: from cohomology to currents

Jeffrey Diller·Romain Dujardin·Vincent Guedj
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some examples and identify some additional properties of maps on irrational surfaces and of maps whose invariant cohomology classes have vanishing self-intersection.

Explore related subjects

math.CVmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jeffrey Diller, Romain Dujardin, Vincent Guedj. 2009-07-09. Dynamics of meromorphic maps with small topological degree I: from cohomology to currents. https://arxiv.org/abs/0803.0955

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions

The celebrated Ohsawa--Takegoshi extension theorem for $L^2$ holomorphic functions on bounded pseudoconvex domains in $\mathbb C^n$ is a fundamental result in the fields of complex analysis and algebraic geometry. In 1995, Ohsawa conjectured that the theorem holds more generally on bounded complete Kähler domains in $\mathbb C^n$. Recently, Chen, Wu and Wang confirmed this conjecture in a special case. In this paper, we extend their result to the case of holomorphic sections of twisted canonical bundles over relatively compact complete Kähler domains in Stein manifolds. As an application, we establish a Hartogs-type extension theorem for plurisubharmonic functions across compact complete pluripolar sets. This result complements a classical theorem of Shiffman and may be regarded as a plurisubharmonic analogue of the Skoda--El Mir extension theorem, thereby filling a gap that appears to have remained open in the literature since at least 1985.

math.CV↗
arXiv2026

The real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives

We apply the methods of simultaneous uniformization and composition operators on Besov spaces to the Teichmüller space $T^Z$ of circle diffeomorphisms with Zygmund continuous derivatives. As consequences, we obtain the following: (1) a new proof of the correspondence between quasiconformal self-homeomorphisms of the unit disk with complex dilatations of linear decay order and their quasisymmetric extensions to the unit circle with regularity in the Zygmund continuously differentiable class; (2) a real-analytic equivalence of $T^Z$ with the real Banach space of Zygmund continuous functions on the unit circle.

math.CV↗
arXiv2026

The Oka principle for holomorphic fibre bundles of Holder-Zygmund classes on strongly pseudoconvex domains

Let \(\overline Ω\) be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let \(h:Z\to \overline Ω\) be a fibre bundle of Hölder-Zygmund class \(Λ^r\), \(r>0\), which is holomorphic over \(Ω\). Assuming that the fibre is an Oka manifold, we prove that every continuous section \(f_0:\overline Ω\to Z\) is homotopic to a section \(f_1:\overline Ω\to Z\) of class \(Λ^r(\overline Ω)\) which is holomorphic on \(Ω\). We also establish the parametric h-principle in this context. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of Hölder-Zygmund classes on such domains.

math.CV↗