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 \(\text{MPer} (f) = \mathbb{N} \smallsetminus \{2\}\).


55M20 Fixed points and coincidences in algebraic topology


Zbl 0438.58023
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