scholarly journals On the Zeta Function of a Periodic-Finite-Type Shift

2013 ◽  
Vol E96.A (6) ◽  
pp. 1024-1031 ◽  
Author(s):  
Akiko MANADA ◽  
Navin KASHYAP
Keyword(s):  
Zeta Function ◽  
Finite Type ◽  
Type Shift

scholarly journals Orbit Growth of Periodic-Finite-Type Shifts via Artin–Mazur Zeta Function

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 685
Author(s):  
Azmeer Nordin ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Zeta Function ◽  
Finite Type ◽  
Discrete Dynamical System ◽  
Closed Orbits ◽  
Counting Functions ◽  
Type Shift

The prime orbit and Mertens’ orbit counting functions describe the growth of closed orbits in a discrete dynamical system in a certain way. In this paper, we prove the asymptotic behavior of these functions for a periodic-finite-type shift. The proof relies on the meromorphic extension of its Artin–Mazur zeta function.

scholarly journals Markov measures determine the zeta function

1987 ◽  
Vol 7 (2) ◽  
pp. 303-311 ◽  
Author(s):  
Selim Tuncel
Keyword(s):  
Zeta Function ◽  
Finite Type ◽  
Periodic Orbits ◽  
Point Of View ◽  
Subshifts Of Finite Type ◽  
Markov Measures ◽  
Markov Equivalence

AbstractWith the purpose of understanding when two subshifts of finite type are equivalent from the point of view of their spaces of Markov measures we propose the notion of Markov equivalence. We show that a Markov equivalence must respect the cycles (periodic orbits) of the subshifts. In particular, Markov equivalent subshifts of finite type have the same zeta function.

Periodic points and finite group actions on shifts of finite type

1993 ◽  
Vol 13 (3) ◽  
pp. 485-514 ◽  
Author(s):  
Ulf-Rainer Fiebig
Keyword(s):  
Finite Group ◽  
Zeta Function ◽  
Finite Type ◽  
Ordered Set ◽  
Zeta Functions ◽  
Orbit Space ◽  
Periodic Points ◽  
Shift Of Finite Type ◽  
Finite Group Actions ◽  
Periodic Data

AbstractLet G be an abstract finite group. For an action α of G on a shift of finite type (SFT) S we introduce the periodic data of α, a computable finite-ordered set of complex polynomials. We show that two actions of G on possibly different SFTs are conjugate on periodic points iff their periodic data coincide. For each subgroup H of G the points fixed by α|H (the restriction of α to H) form a subsystem of S, which is of finite type. Our result shows that the zeta functions of these subsystems determine the conjugacy class (on periodic points) of α up to finitely many possibilities.The orbit space of a finite skew action on an SFT S, endowed with the homeomorphism induced by S, is shown to have a zeta function equal to the zeta function of an SFT which is a left-closing quotient of S. We show that this zeta function equals the zeta function of S iff the skew action is inert with respect to a certain power of S.Finally we consider functions of the periodic data as for example gyration numbers.

A Comparative Study of Periods in a Periodic-Finite-Type Shift

2009 ◽  
Vol 23 (3) ◽  
pp. 1507-1524 ◽  
Author(s):  
Akiko Manada ◽  
Navin Kashyap
Keyword(s):  
Comparative Study ◽  
Finite Type ◽  
Type Shift

scholarly journals Periodic-Finite-Type Shift Spaces

2011 ◽  
Vol 57 (6) ◽  
pp. 3677-3691 ◽  
Author(s):  
Marie-Pierre Beal ◽  
Maxime Crochemore ◽  
Bruce E. Moision ◽  
Paul H. Siegel
Keyword(s):  
Finite Type ◽  
Shift Spaces ◽  
Type Shift

scholarly journals Finite group extensions of shifts of finite type: -theory, Parry and Livšic

2016 ◽  
Vol 37 (4) ◽  
pp. 1026-1059 ◽  
Author(s):  
MIKE BOYLE ◽  
SCOTT SCHMIEDING
Keyword(s):  
Finite Group ◽  
Zeta Function ◽  
Finite Type ◽  
Finite Abelian Group ◽  
Conjugacy Classes ◽  
Topological Conjugacy ◽  
Shift Of Finite Type ◽  
Group Extensions ◽  
Dynamical Zeta Function ◽  
Periodic Data

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group$G$, Parry showed how to define a$G$-extension$S_{A}$from a square matrix over$\mathbb{Z}_{+}G$, and classified the extensions up to topological conjugacy by the strong shift equivalence class of$A$over$\mathbb{Z}_{+}G$. Parry asked, in this case, if the dynamical zeta function$\det (I-tA)^{-1}$(which captures the ‘periodic data’ of the extension) would classify the extensions by$G$of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic$\text{K}$-theory group$\text{NK}_{1}(\mathbb{Z}G)$is non-trivial (e.g. for$G=\mathbb{Z}/n$with$n$not square-free) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of$\text{NK}_{1}(\mathbb{Z}G)$, for every non-trivial abelian$G$we show that there exists a shift of finite type with an infinite family of mixing non-conjugate$G$extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for$G$(not necessarily abelian), and extend all the above results to the non-abelian case. There is other work on basic invariants. The constructions require the ‘positive$K$-theory’ setting for positive equivalence of matrices over$\mathbb{Z}G[t]$.

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