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Counting closed orbits in discrete dynamical systems. (English) Zbl 1447.37034

Mohd, Mohd Hafiz (ed.) et al., Dynamical systems, bifurcation analysis and applications. Collected papers of the SEAMS school 2018 on dynamical systems and bifurcation analysis, DySBA, Penang, Malaysia, August 6–13, 2018. Singapore: Springer. Springer Proc. Math. Stat. 295, 147-171 (2019).
This survey chapter introduces the main definitions to study the analogy between prime numbers and closed periodic orbits: the Artin-Mazur dynamical zeta function and various closed orbit counting asymptotics. Some examples of how analytic properties of associated generating functions may be used to deduce asymptotics are given.
Particular attention is paid to how these ideas may be brought to bear on several specific classes of shift dynamical systems, including shifts of finite type, countable state Markov shifts, Dyck shifts, and Motzkin shifts. The latter two families of shifts are of particular interest because their dynamical zeta functions are algebraic but not rational.
For the entire collection see [Zbl 1443.37004].

MSC:

37C35 Orbit growth in dynamical systems
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37A44 Relations between ergodic theory and number theory
37B10 Symbolic dynamics
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