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Integer sequences and periodic points. (English) Zbl 1026.11022

For a set \(X\) and a map \(T: X\to X\), denote by \(\text{Per}_n (T)\) the set of points \(x\in X\) of period \(n\) under \(T\), i.e., \(T^n(x)=x\) and there is no positive integer \(m<n\) with \(T^m(x)=x\). A sequence \(u=(u_n)_{n=1}^{\infty}\) of integers is said to be realizable if there are a set \(X\) and a map \(T: X\to X\) such that \(|u_n|=|\text{Per}_n (T)|\) for every \(n\geq 1\). In the present paper, the authors prove some results about realizable sequences.
Their first result reads as follows. Let \(V\) be the vector space of realizable sequences that satisfy a given linear recurrence \(u_n=c_1u_{n-1}+\cdots +c_ku_{n-k}\) with \(c_1,\ldots ,c_k\in{\mathbb Z}\) and \(c_k\not= 0\). Let \(f(X)=X^k-c_1X^{k-1}-\cdots -c_k\) be the associated characteristic polynomial. Suppose that \(f\) has no multiple roots, that \(f\) has precisely \(l\) irreducible factors in \({\mathbb Z}[X]\) and that \(f\) has a dominant root, i.e, a root which is larger than the absolute values of the other roots. Then \(\dim V\leq l\), and equality holds if the dominant root is either greater than or equal to the sum of the absolute values of the other roots, or greater than the sum of the absolute values of its conjugates. This extends a result for binary recurrence sequences by Y. Puri and T. Ward [J. Integer Seq. 4, 01.2.1 (2001; Zbl 1004.11013)].
In their second result the authors give some concrete examples of realizable sequences. For instance, let \(f(X)\in {\mathbb Z}[X]\) be a monic polynomial of degree \(d\) and let \(\alpha_1,\ldots,\alpha_d\) be its roots. Define the Lehmer-Pierce sequence \(\Delta_n(f)=\prod_{i=1}^d |\alpha_i^n-1|\). Then \((\Delta_n(f))_{n=1}^{\infty}\) is realizable. Further, if \(b_n\) is the denominator of the Bernoulli number \(B_{2n}\) (defined by \(t/(e^t-1)=\sum_{n=0}^{\infty} B_nt^n/n!\)), then \((b_n)_{n=1}^{\infty}\) is realizable.

MSC:

11B37 Recurrences
37P35 Arithmetic properties of periodic points
11B68 Bernoulli and Euler numbers and polynomials
37B40 Topological entropy

Citations:

Zbl 1004.11013

Software:

OEIS
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