Corpus ID: 119156658

# The Kobayashi distance in holomorphic dynamics and operator theory

```@article{Abate2015TheKD,
title={The Kobayashi distance in holomorphic dynamics and operator theory},
author={M. Abate},
journal={arXiv: Complex Variables},
year={2015}
}```
• M. Abate
• Published 2015
• Mathematics
• arXiv: Complex Variables
These are the notes of a short course I gave in the school "Aspects m\'etriques et dynamiques en analyse complete", Lille, May 2015. The aim of this notes is to describe how to use a geometric structure (namely, the Kobayashi distance) to explore and encode analytic properties of holomorphic functions and maps defined on complex manifolds. We shall first describe the main properties of the Kobayashi distance, and then we shall present applications to holomorphic dynamics in taut manifolds… Expand
3 Citations
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#### References

SHOWING 1-10 OF 83 REFERENCES
Characterizing polynomial domains by their automorphism group
In this paper we study the automorphism group of bounded convex domains with smooth boundary. In particular, we show that such a domain is biholomorphic to a weighted homogeneous polynomial domain ifExpand
Hyperbolic complex spaces
In the three decades since the introduction of the Kobayashi distance, the subject of hyperbolic complex spaces and holomorphic mappings has grown to be a big industry. This book gives aExpand
Characterizing domains by the limit set of their automorphism group
In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weightedExpand
Theorems of Denjoy–Wolff type
• Mathematics
• 2013
Using the Kobayashi distance, we first establish a version of the Denjoy–Wolff theorem for a bounded and strictly convex domain in \$\${{\mathbb{C}}^k}\$\$ . Next, we prove analogous results forExpand
A characterization of hyperbolic manifolds
In this note we prove that a complex manifold X is Kobayashi hyperbolic if and only if the space Hol(A, X) of holomorphic maps of the unit disk A into X is relatively compact (with respect to theExpand
Invariant distances and metrics in complex analysis
• Mathematics
• 2000
C onstructing a distance that is invariant under a given class of mappings is one of the fundamental tools for the geometric approach in mathematics. The idea goes back to Klein and even to Riemann.Expand
Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C[n] with smooth boundary
The Carathe'odory and Kobayashi distance functions on a bounded domain G in Cn have related infinitesimal forms. These are the Caratheodory and Kobayashi metrics. They are denoted by F(z, t) OengthExpand
THE DENJOY-WOLFF THEOREM FOR CONDENSING HOLOMORPHIC MAPPINGS
• Mathematics
• 1999
If B is the open unit ball of a strictly convex Banach space (X, ‖·‖) and f: B→B is holomorphic, condensing with respect to α‖·‖, and fixed-point-free, then there exists ξ∈∂B such that the sequenceExpand
Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings
• Mathematics
• 1987
A classical theorem of Carath6odory [8] states that every biholomorphic map f: D~ ~D2 between domains in the complex plane C bounded by simple closed Jordan curves extends to a homeomorphism of/31Expand
Product formulas, nonlinear semigroups, and accretive operators
A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency isExpand