Abstract:We associate to any dynamical system with finitely many periodic orbits of each period a collection of possible time-changes of the sequence of periodic point counts for that specific system that preserve the property of counting periodic points for some system. Intersecting over all dynamical systems gives a monoid of time-changes that have this property for all such systems. We show that the only polynomials lying in this monoid are the monomials, and that this monoid is uncountable. Examples give some insight into how the structure of the collection of maps varies for different dynamical systems.
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- Sawian Jaidee
- Affiliation: Department of Mathematics, Faculty of Science, Khon Kaen University, Thailand
- MR Author ID: 772773
- Email: firstname.lastname@example.org
- Patrick Moss
- Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
- Email: email@example.com
- Tom Ward
- Affiliation: Department of Mathematics, Ziff Building 13.01, University of Leeds, LS2 9JT, United Kingdom
- MR Author ID: 180610
- Email: firstname.lastname@example.org
- Received by editor(s): September 24, 2018
- Received by editor(s) in revised form: October 19, 2018, and January 29, 2019
- Published electronically: June 10, 2019
- Additional Notes: The first author thanks Khon Kaen University for funding a research visit to Leeds University, and thanks the Research Visitors’ Centre at Leeds for their hospitality.
The third author thanks the Lorentz Centre, Leiden, where much of the work was done.
- Communicated by: Nimish Shah
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 4425-4438
- MSC (2010): Primary 37P35, 37C30, 11N32
- DOI: https://doi.org/10.1090/proc/14574
- MathSciNet review: 4002553
Dedicated: To Graham Everest $($1957–2010$)$, in memoriam