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.

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$\mathbb{Z}G[t]$.

scholarly journals PERIODIC ORBITS OF A DYNAMICAL SYSTEM RELATED TO A KNOT

2011 ◽  
Vol 20 (03) ◽  
pp. 411-426 ◽  
Author(s):  
LILYA LYUBICH
Keyword(s):  
Dynamical System ◽  
Finite Group ◽  
Finite Type ◽  
Periodic Orbits ◽  
Weak Topology ◽  
Commutator Subgroup ◽  
Alexander Polynomial ◽  
Shift Of Finite Type ◽  
A Finite Group

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom (K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σx of Hom (K, Σ) onto itself: σxρ(a) = ρ(xax-1) ∀ a ∈ K, ρ ∈ Hom (K, Σ). As proven in [5], the dynamical system ( Hom (K, Σ), σx) is a shift of finite type. In the case when Σ is abelian, Hom (K, Σ) is finite. In this paper we calculate the periods of orbits of ( Hom (K, ℤ/p), σx), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.

Very Exceptional Group

2014 ◽  
Vol 1006-1007 ◽  
pp. 1071-1075
Author(s):  
Xiao Yu Liang ◽  
Xin Zhang
Keyword(s):  
Finite Group ◽  
Zeta Function ◽  
Nilpotent Group ◽  
Galois Extension ◽  
Prime Order ◽  
Zeta Functions ◽  
Number Fields ◽  
Galois Groups ◽  
Exceptional Group ◽  
A Finite Group

<p>A finite group is called exceptional if for a Galois extension of number fields with the Galois groups , the zeta function of between and does not appear in the Brauer-Kuroda relation of the Dedekind zeta functions. Furthermore, a finite group is called very exceptional if its nontrivial subgroups are all exceptional. In this paper，a Nilpotent group is very exceptional if and only if it has a unique subgroup of prime order for each divisor of .</p>

scholarly journals Finite group actions on shifts of finite type

1985 ◽  
Vol 5 (1) ◽  
pp. 1-25 ◽  
Author(s):  
R. L. Adler ◽  
B. Kitchens ◽  
B. H. Marcus
Keyword(s):  
Finite Group ◽  
Finite Type ◽  
Commutation Relation ◽  
Group Actions ◽  
Topological Conjugacy ◽  
Positive Entropy ◽  
Group Automorphism ◽  
Subshift Of Finite Type ◽  
Finite Group Actions ◽  
Shift Transformation

AbstractA continuous ℤ⊗TG action on a subshift of finite type consists of a subshift of finite type with its shift transformation, together with a group, G, of homeomorphisms of the subshift and a group automorphism T, so that the commutation relation σ ° g = Tg ° ∑A is any positive entropy subshift of finite type, G is any finite group and T is any automorphism of G then there is a non-trivial ℤ⊗TG action on ∑A. We then classify all such actions up to ‘almost topological‘ conjugacy.

scholarly journals An invariant of finite group actions on shifts of finite type

2005 ◽  
Vol 25 (06) ◽  
pp. 1985
Author(s):  
DANIEL S. SILVER ◽  
SUSAN G. WILLIAMS
Keyword(s):  
Finite Group ◽  
Finite Type ◽  
Group Actions ◽  
Finite Group Actions

ZETA FUNCTIONS FOR HIGHER-DIMENSIONAL SHIFTS OF FINITE TYPE

2009 ◽  
Vol 19 (11) ◽  
pp. 3671-3689 ◽  
Author(s):  
WEN-GUEI HU ◽  
SONG-SUN LIN
Keyword(s):  
Zeta Function ◽  
Finite Type ◽  
Rotational Symmetry ◽  
Infinite Product ◽  
Three Dimensional ◽  
Zeta Functions ◽  
Trace Operator ◽  
Periodic Patterns ◽  
Type D ◽  
Taylor Series Expansions

This work investigates zeta functions for d-dimensional shifts of finite type, d ≥ 3. First, the three-dimensional case is studied. The trace operator Ta1,a2;b12and rotational matrices Rx;a1,a2;b12and Ry;a1,a2;b12are introduced to study [Formula: see text] -periodic patterns. The rotational symmetry of Ta1,a2;b12induces the reduced trace operator τa1,a2;b12and then the associated zeta function ζa1,a2;b12= ( det (I-sa1a2τa1,a2;b12))-1. The zeta function ζ is then expressed as [Formula: see text], a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL3(ℤ). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a family of identities in number theory. The methods used herein are also valid for d-dimensional cases, d ≥ 4, and can be applied to thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions.

Perturbations of multidimensional shifts of finite type

2010 ◽  
Vol 31 (2) ◽  
pp. 483-526 ◽  
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Symbolic Dynamics ◽  
Upper And Lower Bounds ◽  
Shift Of Finite Type ◽  
Strongly Irreducible ◽  
Dimensional Cube

AbstractIn this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

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