On the Lefschetz zeta function for quasi-unipotent maps on the \(n\)-dimensional torus. II: The general case. (Q306156)

The authors derive an explicit formula for the Lefschetz zeta function for any quasi-unipotent map on the \(n\)-dimensional torus.NEWLINENEWLINEA map \(f\) on the \(n\)-dimensional torus, \(T^n\), is said to be quasi-unipotent if the eigenvalues of \(f_{*k}\) are roots of the unity, for \(0 \leq k \leq n\), where \(f_{*k}\) denotes the homomorphism on the \(k\)-th rational homology group of \(T^n\) induced by \(f\), i.e., \(f_{*k} : H_k(T^n, \mathbb{Q}) \rightarrow H_k(T^n, \mathbb{Q})\).NEWLINENEWLINEA lemma due to Gauss claims that \( Cf_{*1}(t)\), or the characteristic polynomial of \( Cf_{*1}(t),\) is a monic polynomial with integer coefficients and it is equal to the product of cyclotomic polynomials. The main theorem of the paper reads as follows:NEWLINENEWLINETheorem. Let \(f : T^n \rightarrow T^n\) be a quasi-unipotent map of the \(n\)-dimensional torus. Suppose that NEWLINE\[NEWLINECf_{*1} (t)=F_{m_1} (t) \cdots F_{m_r} (t),NEWLINE\]NEWLINE and \(n = \varphi(m_1)+ \cdots +\varphi(m_r), \) where \(F_k(t)\) is \(k\)-th cyclotomic polynomial and \(\varphi\) is the Euler's function. Let \(m\) be the least common multiple of \(m_1,\ldots , m_r. \) Then:NEWLINENEWLINE (a) The Lefschetz numbers \(L(f^ k)\) are constant for all \(k \in S_{d,m},\) for each divisor \(d\) of \(m\), where NEWLINE\[NEWLINE S_{d,m} := \{k \in N | (k,m) = d\} .NEWLINE\]NEWLINE Moreover, this constant value, denoted by \(F_d\), is given by NEWLINENEWLINE\[NEWLINE F_d = \prod^r_{i=1}\left(F_{m_i/(d,m_i)}(1)\right)\frac{\varphi (m_i)}{\varphi(m_i/(d,m_i))}.NEWLINE\]NEWLINE NEWLINE(b) The Lefschetz zeta function of \( f\) is given by NEWLINENEWLINE\[NEWLINE\zeta_f (t) = \prod_{d\mid m} \, (1-t^d)^{-s_d},NEWLINE\]NEWLINE NEWLINEwhere NEWLINENEWLINE\[NEWLINE s_d = \frac{1}{d} \left(\sum_{k|d} F_k \mu(d/k)\right).NEWLINE\]NEWLINE NEWLINEThe product is taken over all divisors \( d\) of \( m\) and the sum in the exponent \(s_d\) is taken over all divisors \(k\) of \(d\). In particular, the exponents \(s_d\) are integers.NEWLINENEWLINEThe Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse-Smale diffeomorphisms on the \(n\)-dimensional torus. This set is then completely described, for different families containing infinitely many Morse-Smale diffeomorphisms. It is also shown that for any given odd integer, there are Morse-Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of that given number. The results of the present article generalize previous results by \textit{P. Berrizbeitia} and \textit{V. F. Sirvent} [J. Difference Equ. Appl. 20, No. 7, 961--972 (2014; Zbl 1305.37021)].