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A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres

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Abstract

Let X be a compact differentiable manifold. A \(C^1\) map \(f:X\rightarrow X\) is called transversal if for all positive integers m, the graph of \(f^m\) intersects transversally the diagonal of \(X\times X\) at (xx) for any x fixed point of \(f^m\). In the present article, we describe the periodic structure of transversal maps on the n-dimensional torus. In particular, we give conditions on the eigenvalues of the induced linear map on the first homology, in order that all sufficiently large odd numbers are periods of the map. We present similar results for transversal maps on products of spheres of the same dimension. Later we generalize these results for transversal self-maps on rational exterior spaces of rank n.

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Acknowledgements

The author thanks the anonymous referees for useful suggestions that made possible to improve the quality and readability of the article.

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Correspondence to Víctor F. Sirvent.

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Sirvent, V.F. A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres. Qual. Theory Dyn. Syst. 19, 45 (2020). https://doi.org/10.1007/s12346-020-00356-7

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Keywords

  • Transversal maps
  • Lefschetz numbers
  • Periodic point
  • Product of spheres
  • Torus
  • Rational exterior spaces

Mathematics Subject Classification

  • 37C25
  • 37C30
  • 37E15