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Homotopy minimal period self-maps on flat manifolds with cyclic holonomies. (English) Zbl 1337.37020

Let \(X\) be a flat \(m\)-dimensional manifold and \(f:X\to X\). The authors study the homotopical minimal period of \(f\), i.e., the set \[ \mathrm{Hper}(f):=\bigcap_{g\simeq f}\{n\in\mathbb N|\mathrm{Fix}(g^n)\setminus\bigcup_{k<n}\mathrm{Fix}(g^k)\not=\emptyset\}. \] Let the finite group \(G\) be the holonomy group of \(X\), i.e., there is a free action of a group \(\Gamma\) on \(X\) such that there is an exact sequence \(1\to\mathbb Z^m\to\Gamma\to G\to 1\) and \(\Gamma\) is a subgroup of \(O(m)\rtimes\mathbb R^m\) which is a subgroup of \(M(m,\mathbb Z)\rtimes\mathbb R^m\) where \(M(m,\mathbb Z)\) denotes the \((m,m)\) matrices with integer entries. Then \(f\) has a lifting to the universal covering \(\mathbb R^m\) of \(X\) of the form \((U,\mu)\in M(m,\mathbb Z)\rtimes\mathbb R^m\). With these notations the authors prove the following theorem: Assume that \(G\cong\mathbb Z/p\mathbb Z\) with a prime \(p\) such that \(p\equiv3\mod 4\). Let \((C,\alpha)\) be a generator of \(G\) and assume that:
(1)
\(UC=C^rU\) for some \(r\) with \(1\leq r\leq p-1\),
(2)
\(\det(I-U^{p-1})\not=0\) if \(r^2\not\equiv 1\mod p\) and \(\det(I-U^{2p})\not=0\) if \(r^2\equiv1\mod p\), and
(3)
\(x^{p-2}\nmid| xI-U|\).
The conclusion is that \(\mathrm{Hper}(f^2)\) is infinite. Moreover, one has that \[ \liminf_{n\to\infty}\frac{\#\{k\leq n|\;k\in\mathrm{Hper}(f^2)\}}{n/\log n}\geq \frac{1}{\phi(p(p-1)/2} \] where \(\phi\) is the Euler \(\phi\)-function.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
55M20 Fixed points and coincidences in algebraic topology
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