October  2021, 41(10): 4515-4529. doi: 10.3934/dcds.2021046

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

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

* Corresponding author

Received  August 2020 Revised  January 2021 Published  October 2021 Early access  March 2021

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

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.

Citation: Azmeer Nordin, Mohd Salmi Md Noorani. Counting finite orbits for the flip systems of shifts of finite type. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4515-4529. doi: 10.3934/dcds.2021046
References:
[1]

T. Adachi, Markov families for Anosov flows with an involutive action, Nagoya Math. J., 104 (1986), 55-62.  doi: 10.1017/S0027763000022674.

[2]

S. Akhatkulov, M. S. M. Noorani and H. Akhadkulov, An analogue of the prime number, Mertens' and Meissel's theorems for closed orbits of the Dyck shift, AIP Conf. Proc., 1830 (2017), 070022, 1-9. doi: 10.1063/1.4980971.

[3]

F. AlsharariM. S. M. Noorani and H. Akhadkulov, Estimates on the number of orbits of the Dyck shift, J. Inequal. Appl., 2015 (2015), 372-384.  doi: 10.1186/s13660-015-0899-6.

[4]

F. AlsharariM. S. M. Noorani and H. Akhadkulov, Analogues of the prime number theorem and Mertens' theorem for closed orbits of the Motzkin shift, Bull. Malays. Math. Sci. Soc., 40 (2017), 307-319.  doi: 10.1007/s40840-015-0144-y.

[5]

M. Artin and B. Mazur, On periodic points, Ann. Math., 81 (1965), 82-99.  doi: 10.2307/1970384.

[6]

G. EverestR. MilesS. Stevens and T. Ward, Orbit-counting in non-hyperbolic dynamical systems, J. Reine Angew. Math., 608 (2007), 155-182.  doi: 10.1515/CRELLE.2007.056.

[7]

G. EverestR. MilesS. Stevens and T. Ward, Dirichlet series for finite combinatorial rank dynamics, Trans. Amer. Math. Soc., 362 (2010), 199-227.  doi: 10.1090/S0002-9947-09-04962-9.

[8]

G. H. Hardy and E. M. Wright, An Introduction to Theory of Numbers, eds. D. R. HeathBrown and J. H. Silverman, 6$^th$ edition, Oxford University Press, Oxford, 2008.

[9]

Y.-O. KimJ. Lee and K. K. Park, A zeta function for flip systems, Pacific J. Math., 209 (2003), 289-301.  doi: 10.2140/pjm.2003.209.289.

[10]

Y.-O. Kim and S. Ryu, On the number of fixed points of a sofic shift-flip system, Ergod. Theory Dyn. Syst., 35 (2015), 482-498.  doi: 10.1017/etds.2013.57.

[11] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.
[12]

R. Miles, Orbit growth for algebraic flip systems, Ergod. Theory Dyn. Syst., 35 (2015), 2613-2631.  doi: 10.1017/etds.2014.38.

[13]

R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc., 137 (2009), 1499-1507.  doi: 10.1090/s0002-9939-08-09649-4.

[14]

A. Nordin and M. S. M. Noorani, Orbit growth of periodic-finite-type shifts via Artin-Mazur zeta function, Mathematics, 8 (2020), 1-19.  doi: 10.3390/math8050685.

[15]

A. Nordin, M. S. M. Noorani and S. C. Dzul-Kifli, Counting closed orbits in discrete dynamical systems, in Dynamical Systems, Bifurcation Analysis and Applications (eds. M. Mohd, N. Abdul Rahman, N. Abd Hamid and Y. Mohd Yatim), Springer, Singapore, 2019, 147-171. doi: 10.1007/978-981-32-9832-3_9.

[16]

A. NordinM. S. M. Noorani and S. C. Dzul-Kifli, Orbit growth of Dyck and Motzkin shifts via Artin-Mazur zeta function, Dyn. Syst., 35 (2020), 655-667.  doi: 10.1080/14689367.2020.1770201.

[17]

A. Pakapongpun and T. Ward, Functorial orbit counting, J. Integer Seq., 12 (2009), 1-20. 

[18]

W. Parry, An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions, Isr. J. Math., 45 (1983), 41-52.  doi: 10.1007/BF02760669.

[19]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque, 187-188 (1990), 268 pp.

[20]

M. Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergod. Theory Dyn. Syst., 1 (1981), 107-133.  doi: 10.1017/S0143385700001206.

[21]

S. Waddington, The prime orbit theorem for quasihyperbolic toral automorphisms, Monatshefte für Math., 112 (1991), 235-248.  doi: 10.1007/BF01297343.

show all references

References:
[1]

T. Adachi, Markov families for Anosov flows with an involutive action, Nagoya Math. J., 104 (1986), 55-62.  doi: 10.1017/S0027763000022674.

[2]

S. Akhatkulov, M. S. M. Noorani and H. Akhadkulov, An analogue of the prime number, Mertens' and Meissel's theorems for closed orbits of the Dyck shift, AIP Conf. Proc., 1830 (2017), 070022, 1-9. doi: 10.1063/1.4980971.

[3]

F. AlsharariM. S. M. Noorani and H. Akhadkulov, Estimates on the number of orbits of the Dyck shift, J. Inequal. Appl., 2015 (2015), 372-384.  doi: 10.1186/s13660-015-0899-6.

[4]

F. AlsharariM. S. M. Noorani and H. Akhadkulov, Analogues of the prime number theorem and Mertens' theorem for closed orbits of the Motzkin shift, Bull. Malays. Math. Sci. Soc., 40 (2017), 307-319.  doi: 10.1007/s40840-015-0144-y.

[5]

M. Artin and B. Mazur, On periodic points, Ann. Math., 81 (1965), 82-99.  doi: 10.2307/1970384.

[6]

G. EverestR. MilesS. Stevens and T. Ward, Orbit-counting in non-hyperbolic dynamical systems, J. Reine Angew. Math., 608 (2007), 155-182.  doi: 10.1515/CRELLE.2007.056.

[7]

G. EverestR. MilesS. Stevens and T. Ward, Dirichlet series for finite combinatorial rank dynamics, Trans. Amer. Math. Soc., 362 (2010), 199-227.  doi: 10.1090/S0002-9947-09-04962-9.

[8]

G. H. Hardy and E. M. Wright, An Introduction to Theory of Numbers, eds. D. R. HeathBrown and J. H. Silverman, 6$^th$ edition, Oxford University Press, Oxford, 2008.

[9]

Y.-O. KimJ. Lee and K. K. Park, A zeta function for flip systems, Pacific J. Math., 209 (2003), 289-301.  doi: 10.2140/pjm.2003.209.289.

[10]

Y.-O. Kim and S. Ryu, On the number of fixed points of a sofic shift-flip system, Ergod. Theory Dyn. Syst., 35 (2015), 482-498.  doi: 10.1017/etds.2013.57.

[11] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.
[12]

R. Miles, Orbit growth for algebraic flip systems, Ergod. Theory Dyn. Syst., 35 (2015), 2613-2631.  doi: 10.1017/etds.2014.38.

[13]

R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc., 137 (2009), 1499-1507.  doi: 10.1090/s0002-9939-08-09649-4.

[14]

A. Nordin and M. S. M. Noorani, Orbit growth of periodic-finite-type shifts via Artin-Mazur zeta function, Mathematics, 8 (2020), 1-19.  doi: 10.3390/math8050685.

[15]

A. Nordin, M. S. M. Noorani and S. C. Dzul-Kifli, Counting closed orbits in discrete dynamical systems, in Dynamical Systems, Bifurcation Analysis and Applications (eds. M. Mohd, N. Abdul Rahman, N. Abd Hamid and Y. Mohd Yatim), Springer, Singapore, 2019, 147-171. doi: 10.1007/978-981-32-9832-3_9.

[16]

A. NordinM. S. M. Noorani and S. C. Dzul-Kifli, Orbit growth of Dyck and Motzkin shifts via Artin-Mazur zeta function, Dyn. Syst., 35 (2020), 655-667.  doi: 10.1080/14689367.2020.1770201.

[17]

A. Pakapongpun and T. Ward, Functorial orbit counting, J. Integer Seq., 12 (2009), 1-20. 

[18]

W. Parry, An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions, Isr. J. Math., 45 (1983), 41-52.  doi: 10.1007/BF02760669.

[19]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque, 187-188 (1990), 268 pp.

[20]

M. Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergod. Theory Dyn. Syst., 1 (1981), 107-133.  doi: 10.1017/S0143385700001206.

[21]

S. Waddington, The prime orbit theorem for quasihyperbolic toral automorphisms, Monatshefte für Math., 112 (1991), 235-248.  doi: 10.1007/BF01297343.

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