## 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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