Counting finite orbits for the flip systems of shifts of finite type

Azmeer Nordin; Mohd Salmi Md Noorani

doi:10.3934/dcds.2021046

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2021, Volume 41, Issue 10: 4515-4529. Doi: 10.3934/dcds.2021046

This issue Previous Article Next Article Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

Counting finite orbits for the flip systems of shifts of finite type

Azmeer Nordin^, and
Mohd Salmi Md Noorani

Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

^* Corresponding author
^* Corresponding author

Received: July 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: October 2021

This work is supported by the research grant FRGS/1/2019/STG06/UKM/01/3 by the Ministry of Higher Education, Malaysia

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For a discrete system $ (X,T) $, the flip system $ (X,T,F) $ can be regarded as the action of infinite dihedral group $ D_\infty $ on the space $ X $. Under this action, $ X $ is partitioned into a set of orbits. We are interested in counting the finite orbits in this partition via the prime orbit counting function. In this paper, we prove the asymptotic behaviour of this counting function for the flip systems of shifts of finite type. The proof relies mostly on combinatorial calculations instead of the usual approach via zeta function. Here, we are able to obtain more precise asymptotic result for this $ D_\infty $-action on shifts of finite type as compared to other group actions on systems available in the literature.

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Mathematics Subject Classification: Primary: 37C35, 37C85; Secondary: 37B10.

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