This thesis deals with specific features of the theory of holomorphic dynamics in dimension 2 and then sets out to study analogous questions in higher dimensions, e.g. dealing with normal forms for rigid germs, and examples of Kato 3-folds. The local dynamics of holomorphic maps around critical points is still not completely understood, in dimension 2 or higher, due to the richness of the geometry of the critical set for all iterates. In dimension 2, the study of the dynamics induced on a suitable functional space (the valuative tree) allows a classification of such maps up to birational conjugacy, reducing the problem to the special class of rigid germs, where the geometry of the critical set is simple. In some cases, from such dynamical data one can construct special compact complex surfaces, called Kato surfaces, related to some conjectures in complex geometry.
Orlando Innamorato of Matteo Maria Boiardo. Translated into prose from the Italian of Francesco Berni and interspersed with extracts in the same stanza as the original by William Stewart Rose. Orlando Innamorato (Orlando in Love) is an epic poem written by the Italian Renaissance author Matteo Maria Boiardo. The poem is a romance concerning the heroic knight Orlando (Roland). The beautiful Angelica, daughter of the king of Cataio (Cathay), comes to Charlemagne's court for a tournament in which both Christians and pagans can participate. She offers herself as a prize to whoever will defeat her brother, Argalia, who in the consequent competition fighting imprisons many Christians. But then Ferraguto (aka Ferrau) kills Argalia and Angelica flees, chased by many paladins, especially Orlando and Rinaldo. Stopping in the Ardenne forest, she drinks at the Stream of Love (making her fall in love with Rinaldo), while Rinaldo drinks at the fount of hate (making him conceive a passionate hatred of Angelica): first reversal. She asks the magician Malagigi to kidnap Rinaldo, and the magician brings him to an enchanted island, while she returns to Cataio where she is besieged by king Agricane, another of her admirers, in the fortress of Albracca. Orlando comes to kill Agricane and to free her, and he succeeds. Afterwards, Rinaldo tries to convince him to return to France to fight alongside Charlemagne: consequently, Orlando and Rinaldo duel furiously. In the meantime the Saracen king Agramante has invaded France with a massive army (along with Rodomonte, Ferrau, Gradasso, and many others), to avenge his father Troiano, previously killed by Orlando. Rinaldo rushes back to France, chased by Angelica in love with him, in turn chased by Orlando. Back in the Ardenne forest, this time Rinaldo and Angelica drink at the opposite founts: second reversal. Orlando and Rinaldo duel again for Angelica, and Charlemagne decides to entrust her to the old and wise duke Namo, offering her to the one who will fight most valorously against the infidels. In the meantime, the Saracen paladin Ruggiero and Rinaldo's sister, Bradamante, fall in love. The poem stops there abruptly, with Boiardo's narrator explaining that he can write no more because Italy has been invaded by French troops headed by king Charles VIII. (Ariosto's Orlando Furioso will resume from that point.)
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