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Roots of iterates of maps

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
66
Issue
2
Identifiers
DOI: 10.1016/0166-8641(95)00024-b
Keywords
  • Roots Of Map Iterates
  • Nielsen Numbers
  • Irreducible Roots
  • Primitive Roots Of Unity
  • Semidynamical System
  • Recurrence Number
  • Root-Essential Map
  • H-Space
  • Coordinate Of A Root Class
Disciplines
  • Mathematics

Abstract

Abstract Let f : X → X be a selfmap of a path-connected space and let a ϵ X. We call x ϵ X an irreducible root of, f n , at a if f n ( x) = a but f m ( x) ≠ a for all m < n, where f n denotes the nth iterate of f. A lower bound N I n ( f; a) for the number of irreducible roots of f n is defined that is homotopy invariant under suitable hypotheses and, in many cases, can be calculated algebraically from the homomorphism of the fundamental group of X induced by f. In particular, this is true when X is a compact orientable manifold without boundary and f has nonzero degree. The geometric content of these calculations is described in concrete examples in terms of the discrete semidynamical system determined by f. We apply the theory of roots of iterates to primitive roots of unity in H-spaces.

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