Periodic point free continuous self-maps on graphs and surfaces

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Abstract

We prove the following three results. We denote by Per (f) the set of all periods of a self-map f.Let G{double-struck} be a connected compact graph such that dimℚH 1(G{double-struck},ℚ)=r, and let f:G{double-struck}→G{double-struck} be a continuous map. If Per(f)=θ, then the eigenvalues of f *1 are 1 and 0, this last with multiplicity r-1, where f *1 is the induced action of f on the first homological space.Let M{double-struck} g,b be an orientable connected compact surface of genus g≥0 with b≥0 boundary components, and let f:M{double-struck} g,b→M{double-struck} g,b be a continuous map. The degree of f is d if b=0. If Per(f)=θ, then the eigenvalues of f *1 are 1, d and 0, this last with multiplicity 2g-2 if b=0; and 1 and 0, this last with multiplicity 2g+b-2 if b>0.Let ℕ g,b be a non-orientable connected compact surface of genus g≥1 with b≥0 boundary components, and let f:ℕ g,b→ℕ g,b be a continuous map. If Per(f)=θ, then the eigenvalues of f *1 are 1 and 0, this last with multiplicity g+b-2.The tools used for proving these results can be applied for studying the periodic point free continuous self-maps of many other compact absolute neighborhood retract spaces. © 2012 Elsevier B.V.
Original languageEnglish
Pages (from-to)2228-2231
JournalTopology and its Applications
Volume159
Issue number8
DOIs
Publication statusPublished - 15 May 2012

Keywords

  • Connected compact graph
  • Connected compact surface
  • Lefschetz number
  • Lefschetz zeta function
  • Non-orientable surface
  • Orientable surface
  • Periodic point free

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