2021
DOI: 10.3934/dcds.2021046
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Counting finite orbits for the flip systems of shifts of finite type

Abstract: For a discrete system (X, T), the flip system (X, T, F) can be regarded as the action of infinite dihedral group D∞ 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 funct… Show more

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Cited by 3 publications
(2 citation statements)
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“…While we focus on the approach via zeta function, there are other methods to obtain the orbit growth of a system, such as using orbit Dirichlet series [13], orbit monoids [14] and estimates on the number of periodic points [15][16][17]. Furthermore, similar research problem has been studied for group actions on dynamical systems, and some recent results include the orbit growths of nilpotent group shifts [18], algebraic flip systems [19] and flip systems for shifts of finite type [20]. Since our introduction on this subject is rather short, we encourage readers to explore those papers above, and additionally the expository chapters by us [21] and Ward [22].…”
Section: Introductionmentioning
confidence: 99%
“…While we focus on the approach via zeta function, there are other methods to obtain the orbit growth of a system, such as using orbit Dirichlet series [13], orbit monoids [14] and estimates on the number of periodic points [15][16][17]. Furthermore, similar research problem has been studied for group actions on dynamical systems, and some recent results include the orbit growths of nilpotent group shifts [18], algebraic flip systems [19] and flip systems for shifts of finite type [20]. Since our introduction on this subject is rather short, we encourage readers to explore those papers above, and additionally the expository chapters by us [21] and Ward [22].…”
Section: Introductionmentioning
confidence: 99%
“…For a detailed exposure on the topic of orbit growth, we encourage the interested readers to refer to our survey paper in [19]. We shall also mention that there is a similar research problem of counting finite orbits for group actions, and some asymptotic results have been obtained for finitely-generated torsion-free nilpotent group shifts [20], algebraic flip systems [21] and flip systems of shifts of finite type [22]. Now, we focus our attention to shift dynamical systems.…”
Section: Introductionmentioning
confidence: 99%