## Thesis etd-10042020-221411 |

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Thesis type

Tesi di laurea magistrale

Author

BERTELLOTTI, ALESSANDRO

URN

etd-10042020-221411

Thesis title

The Wolff-Denjoy theorem: old and new approaches.

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Abate, Marco

**controrelatore**Prof. Frigerio, Roberto

Keywords

- complex analysis
- complex dynamics
- complex geometry
- Gromov-hyperbolicity
- holomorphic dynamics
- Kobayashi distance
- metric geometry
- metric spaces
- taut manifolds
- Wolff-Denjoy theorem

Graduation session start date

23/10/2020

Availability

Full

Summary

This thesis is about the Wolff-Denjoy theorem and some of its generalizations. The classical Wolff-Denjoy theorem describes the behavior of the sequence of the iterations of a holomorphic function f:D->D without fixed points, where D is the unit disk of C. Over the years the theorem has been generalized in various way. The first and more natural generalization is that to the case of bounded domains of Cn, for example strictly convex domains and strongly pseudoconvex domains. In this cases if the sequence (f^k) of the iterations of a holomorphic function f:D->D

is compactly divergent, then the sequence converges, uniformly on compact subsets, to a constant in the boundary of the domain D. In the first part of this work we review and introduce some classical tools from complex analysis and geometry, such as the Schwarz lemma, the Kobayashi distance and taut manifolds that allows us to prove the generalizations said above. In recent years the approach toward such kind of problems is changed. Indeed the idea is to prove appropriate generalizations of the Wolf-Denjoy theorem in the case of metric spaces and Gromov-hyperbolic spaces in order to deduce from these results some new generalizations of the theorem in the setting of several complex variables. In this light the second part of the thesis is devoted to introduce the concepts and techniques from metric geometry and continuous dynamical systems necessary to prove these versions of the Wolff-Denjoy theorem, such as geodesics, compactifications and Gromov-hyperbolicity. In the last part of our work we return to the study of domains of Cn under this new light, and we give some sufficient conditions that allow us to deduce some other generalizations of the Wolff-Denjoy theorem from the metric case. Here we use some estimates for the behavior of the Kobayashi distance and metric near the boundary of the domain of interest. In the last chapter we examine the problem of describe the domains of Cn that are Gromov-hyperbolic, question that is still open. In particular we describe a recent result due to Andrew Zimmer that furnishes a characterization of the Gromov-hyperbolic convex bounded domains that satisfies a finiteness condition at the boundary and a more classical result about the Gromov-hyperbolicity of strongly pseudoconvex domains.

is compactly divergent, then the sequence converges, uniformly on compact subsets, to a constant in the boundary of the domain D. In the first part of this work we review and introduce some classical tools from complex analysis and geometry, such as the Schwarz lemma, the Kobayashi distance and taut manifolds that allows us to prove the generalizations said above. In recent years the approach toward such kind of problems is changed. Indeed the idea is to prove appropriate generalizations of the Wolf-Denjoy theorem in the case of metric spaces and Gromov-hyperbolic spaces in order to deduce from these results some new generalizations of the theorem in the setting of several complex variables. In this light the second part of the thesis is devoted to introduce the concepts and techniques from metric geometry and continuous dynamical systems necessary to prove these versions of the Wolff-Denjoy theorem, such as geodesics, compactifications and Gromov-hyperbolicity. In the last part of our work we return to the study of domains of Cn under this new light, and we give some sufficient conditions that allow us to deduce some other generalizations of the Wolff-Denjoy theorem from the metric case. Here we use some estimates for the behavior of the Kobayashi distance and metric near the boundary of the domain of interest. In the last chapter we examine the problem of describe the domains of Cn that are Gromov-hyperbolic, question that is still open. In particular we describe a recent result due to Andrew Zimmer that furnishes a characterization of the Gromov-hyperbolic convex bounded domains that satisfies a finiteness condition at the boundary and a more classical result about the Gromov-hyperbolicity of strongly pseudoconvex domains.

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