Periodic points of holomorphic maps via lefschetz numbers

Nüria Fagella, Jaume Llibre

Research output: Contribution to journalArticleResearchpeer-review

32 Citations (Scopus)

Abstract

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension n and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of CP(n) of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem. ©2000 American Mathematical Society.
Original languageEnglish
Pages (from-to)4711-4730
JournalTransactions of the American Mathematical Society
Volume352
Issue number10
Publication statusPublished - 1 Dec 2000

Keywords

  • Holomorphic maps
  • Lefschetz fixed noint theorv
  • Periodic points
  • Set of periods

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