## Minimal sets of periods for torus maps via Nielsen numbers.(English)Zbl 0843.55004

For a map $$f : X \to X$$, the composite maps $$f^n$$ are defined by setting $$f^1(x) = f(x)$$ and then $$f^n(x) = f(f^{n-1} (x))$$. A natural number $$n \in \mathbb{N}$$ is the periodic set $$\text{Per} (f)$$ if $$f$$ has a periodic point of period $$n$$, that is, a solution $$x$$ to $$f^n (x) = x$$ such that $$f^k(x) \neq x$$ for all $$k < n$$. The minimal periodic set $$\text{MPer}(f)$$ of the title is the intersection of $$\text{Per}(g)$$ for all $$g$$ homotopic to $$f$$. If $$X$$ is a finite polyhedron, the Nielsen numbers $$N(f^n)$$ can be used to study $$\text{Per} (f)$$. The authors show that if the sequence $$\{N(f^{(n)})\}$$ grows fast enough, in a sense they make precise, then $$\text{Per} (f) = \mathbb{N}$$ and, more generally, if the $$N(f^n)$$ grow rapidly for $$n$$ sufficiently large, then $$\text{Per} (f)$$ is cofinite in $$\mathbb{N}$$. In the case that $$X = S^1 \times \dots \times S^1 = T^m$$, the $$m$$-torus, then $$N(f^n)$$ can be calculated readily from a knowledge of $$f_{1^*}$$, the endomorphism of the integer homology group $$H_1(T^m)$$ induced by $$f$$. Specifically, letting $$A$$ be an integer matrix corresponding to $$f_{1*}$$, then $$N(f^n) = |\text{det} (I - A^n)|$$. For $$f_A : T^m \to T^m$$ the map homotopic to $$f$$ covered by the linear map $$A : \mathbb{R}^m \to \mathbb{R}^m$$, the authors verify a conjecture of B. Halpern [Pac. J. Math. 83, 117-133 (1979; Zbl 0438.58023)] that $$\text{MPer}(f)$$ consists of those $$n \in \text{Per}(f_A)$$ such that $$N(f^n) \neq 0$$. Using this result and techniques from number theory, the authors prove that if $$f : T^m \to T^m$$ is a map such that $$A$$ has some nonzero eigenvalue, but no eigenvalue that is a root of unity, then $$\text{Per}(f)$$ is cofinite in $$\mathbb{N}$$. If, for $$f : T^m \to T^m$$, the sequence $$\{N(f^n)\}$$ is strictly increasing, then $$\text{Per} (f) = \mathbb{N}$$. For maps of the circle, the determination of $$\text{MPer }(f)$$ in terms of the degree $$d$$ of the map has been known for some years. For instance, if $$d = -2$$ then $$\text{MPer}(f) = \mathbb{N} \smallsetminus \{2\}$$ whereas $$\text{MPer}(f) = \mathbb{N}$$ if $$d < - 2$$. This paper contains the complete determination of $$\text{MPer}(f)$$ for $$f : T^2 \to T^2$$ in terms of the determinant $$d$$ and trace $$t$$ of the $$2$$-by-$$2$$ matrix $$A$$ defined above. For instance, if $$t \neq 0$$ and $$t + d = -1$$, then $$\text{MPer}(f)$$ consists of the odd natural numbers whereas if $$t + d = 0$$ or $$-2$$, then $$\text{MPer} (f) = \mathbb{N} \smallsetminus \{2\}$$.

### MSC:

 55M20 Fixed points and coincidences in algebraic topology

### Keywords:

torus map; periodic point; Nielsen number

Zbl 0438.58023
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