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Time-changes preserving zeta functions. (English) Zbl 1431.37022

An integer sequence \((a_{n})\) is called realizable if there is a map \(T: X \to X\) with the property that \[ a_{n} = \mathrm{Fix}_{(X,T)}(n) = |\{x \in X | T^{n}(x) = x \}| \] for all \(n \leq 1.\) Simple arguments showing that realizable sequences can be added and multiplied may be seen using disjoint unions and products of dynamical systems.
For a map \(T: X \to X\) with \(\mathrm{Fix}_{(X, T)}(n) < \infty \) for all \(n \leq 1,\) define \[ \mathcal{P}(X, T) = \{h : \mathbb{N} \to \mathbb{N}| (\mathrm{Fix}_{(X, T)}(h(n))) \text{ is a realizable sequence}\} \] to be the set of realizability-preserving time-changes for \((X,T).\) Also define \[ \mathcal{P} = \displaystyle \bigcap_{\{(X,T)\}}\mathcal{P}(X,T) \] to be the monoid of universally realizability-preserving time-changes, where the intersections is taken over all systems \((X,T)\) for which \(\mathrm{Fix}_{(X,T)}(n) < \infty \) for all \(n \leq 1.\) In this paper the authors prove the following results:
Theorem. A polynomial lies in \(\mathcal{P}\) if and only if it is a monomial.
Theorem. The monoid \(\mathcal{P}\) is uncountable.
Other property and examples related with the above results are presented.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37P35 Arithmetic properties of periodic points
11B05 Density, gaps, topology

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References:

[1] Carnevale, Angela; Voll, Christopher, Orbit Dirichlet series and multiset permutations, Monatsh. Math., 186, 2, 215-233 (2018) · Zbl 1406.37024
[2] Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas, Recurrence sequences, Mathematical Surveys and Monographs 104, xiv+318 pp. (2003), American Mathematical Society, Providence, RI · Zbl 1033.11006
[3] Everest, G.; van der Poorten, A. J.; Puri, Y.; Ward, T., Integer sequences and periodic points, J. Integer Seq., 5, 2, Article 02.2.3, 10 pp. (2002) · Zbl 1026.11022
[4] Luca, F.; Ward, T., An elliptic sequence is not a sampled linear recurrence sequence, New York J. Math., 22, 1319-1338 (2016) · Zbl 1367.11020
[5] pm P. Moss, The arithmetic of realizable sequences, Ph.D. thesis, University of East Anglia, 2003.
[6] Pakapongpun, Apisit; Ward, Thomas, Functorial orbit counting, J. Integer Seq., 12, 2, Article 09.2.4, 20 pp. (2009) · Zbl 1254.37020
[7] MR3194906 Apisit Pakapongpun and Thomas Ward, Orbits for products of maps, Thai J. Math. 12 (2014), no. 1, 33-44. · Zbl 1390.30004
[8] Puri, Yash; Ward, Thomas, Arithmetic and growth of periodic orbits, J. Integer Seq., 4, 2, Article 01.2.1, 18 pp. (2001) · Zbl 1004.11013
[9] MR1866354 Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quart. 39 (2001), no. 5, 398-402. · Zbl 1018.37009
[10] OEIS N. J. A. Sloane (editor), The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org. · Zbl 1044.11108
[11] Windsor, A. J., Smoothness is not an obstruction to realizability, Ergodic Theory Dynam. Systems, 28, 3, 1037-1041 (2008) · Zbl 1143.37007
[12] Ycart, Bernard, A case of mathematical eponymy: the Vandermonde determinant, Rev. Histoire Math., 19, 1, 43-77 (2013) · Zbl 1287.01012
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