Shift Automorphism Research Articles

Shift dynamics of the groups of Fibonacci type

Abstract We study the shift dynamics of the groups G = G n ⁢ ( x 0 ⁢ x m ⁢ x k - 1 ) G=G_{n}(x_{0}x_{m}x_{k}^{-1}) of Fibonacci type introduced by Johnson and Mawdesley. The main result concerns the order of the shift automorphism of 𝐺 and determining whether it is an outer automorphism, and we find the latter occurs if and only if 𝐺 is not perfect. A result of Bogley provides that the aspherical presentations determine groups admitting a free shift action by Z n \mathbb{Z}_{n} on the nonidentity elements of 𝐺, from which it follows that the shift is an outer automorphism of order 𝑛 when 𝐺 is nontrivial. The focus of this paper is therefore on the non-aspherical cases, which include for example the Fibonacci and Sieradski groups. With few exceptions, the fixed-point and freeness problems for the shift automorphism are solved, in some cases using computational and topological methods.

-ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS

AbstractThis paper is a continuation of the paper, Matsumoto [‘Subshifts,$\lambda $-graph bisystems and$C^*$-algebras’,J. Math. Anal. Appl.485 (2020), 123843]. A$\lambda $-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift$\Lambda $, there exists a$\lambda $-graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra${\mathcal {F}}_{\mathcal {L}}$with shift automorphism$\rho _{\mathcal {L}}$from a$\lambda $-graph bisystem$({\mathcal {L}}^-,{\mathcal {L}}^+)$, and define a$C^*$-algebra${\mathcal R}_{\mathcal {L}}$by the crossed product. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If$\lambda $-graph bisystems come from two-sided subshifts, these$C^*$-algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the$C^*$-algebra${\mathcal R}_{\mathcal {L}}$and the K-theory formulas of the$C^*$-algebras${\mathcal {F}}_{\mathcal {L}}$and${\mathcal R}_{\mathcal {L}}$. The K-group for the AF-algebra${\mathcal {F}}_{\mathcal {L}}$is regarded as a two-sided extension of the dimension group of subshifts.

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

We study a class M of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analyzing their shift extensions we show that the groups in the class M are coherent, subgroup separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all metacyclic. Many of the groups in M are virtually free, some are free products of metacyclic groups and free groups, and some have geometric dimension two. We classify the finite groups that occur in M, giving extensive details about the metacyclic structures that occur, and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three. We show that any finite group in the class M that has fixed point free shift automorphism must be cyclic.

On shift dynamics for cyclically presented groups

A group defined by a finite presentation with cyclic symmetry admits a shift automorphism that is periodic and word-length preserving. It is shown that if the presentation is combinatorially aspherical and orientable, in the sense that no relator is a cyclic permutation of the inverse of any of its shifts, then the shift acts freely on the non-identity elements of the group presented. For cyclic presentations defined by positive words of length at most three, the shift defines a free action if and only if the presentation is combinatorially aspherical and the shift itself is fixed point free if and only if the group presented is infinite.

EVERY SHIFT AUTOMORPHISM VARIETY HAS AN INFINITE SUBDIRECTLY IRREDUCIBLE MEMBER

AbstractA shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra.

Structure of the string R-matrix

By requiring invariance directly under the Yangian symmetry, we rederive Beisert's quantum R-matrix, in a form that carries explicit dependence on the representation labels, the braiding factors and the spectral parameters ui. In this way, we demonstrate that there exists rewriting of its entries, such that the dependence on the spectral parameters is purely of a difference form. Namely, the latter enter only in the combination u1 − u2, as indicated by the shift automorphism of the Yangian. When recasted in this fashion, the entries exhibit a cleaner structure, which allows us to spot new interesting relations among them. This permits us to package them into a practical tensorial expression, where the nondiagonal entries are taken care of by explicit combinations of symmetry algebra generators.

EQUATIONAL COMPLEXITY OF THE FINITE ALGEBRA MEMBERSHIP PROBLEM

We associate to each variety of algebras of finite signature a function on the positive integers called the equational complexity of the variety. This function is a measure of how much of the equational theory of a variety must be tested to determine whether a finite algebra belongs to the variety. We provide general methods for giving upper and lower bounds on the growth of equational complexity functions and provide examples using algebras created from graphs and from finite automata. We also show that finite algebras which are inherently nonfinitely based via the shift automorphism method cannot be used to settle an old problem of Eilenberg and Schützenberger.

The Rohlin property for shifts of finite type

We show that an automorphism of a unital AF C * -algebra with a certain approximate Rohlin property has the Rohlin property. This generalizes a result of Kishimoto. Using this we show that the shift automorphism on the bilateral C * -algebra associated with an aperiodic irreducible shift of finite type has the Rohlin property.

ROHLIN PROPERTY FOR SHIFT AUTOMORPHISMS

We prove that the shift automorphism of the two-sided infinite tensor product of M2 ⊕ M3 has the Rohlin property. This extends the known results on the shift on UHF algebras.

Anyons and the Bose-Fermi duality in the finite-temperature thirring model

Solutions to the Thirring model are constructed in the framework of algebraic QFT. It is shown that for all positive temperatures there are fermionic solutions only if the coupling constant is $\lambda=\sqrt{2(2n+1)\pi}, n\in {\bf N}$. These fermions are inequivalent and only for $n=1$ they are canonical fields. In the general case solutions are anyons. Different anyons (which are uncountably many) live in orthogonal spaces and obey dynamical equations (of the type of Heisenberg's Urgleichung) characterized by the corresponding values of the statistic parameter. Thus statistic parameter turns out to be related to the coupling constant $\lambda$ and the whole Hilbert space becomes non-separable with a different Urgleichung satisfied in each of its sectors. This feature certainly cannot be seen by any power expansion in $\lambda$. Moreover, since the latter is tied to the statistic parameter, it is clear that such an expansion is doomed to failure and will never reveal the true structure of the theory. The correlation functions in the temperature state for the canonical dressed fermions are shown by us to coincide with the ones for bare fields, that is in agreement with the uniqueness of the $\tau$-KMS state over the CAR algebra ($\tau$ being the shift automorphism). Also the $\alpha$-anyon two-point function is evaluated and for scalar field it reproduces the result that is known from the literature.

Chaos in the Gyldén problem

We consider the Gyldén problem—a perturbation of the Kepler problem via an explicit function of time. For certain general classes of planar periodic perturbations, after proving a Poincaré–Melnikov-type criterion, we find a manifold of orbits in which the dynamics is given by the shift automorphism on the set of bi-infinite sequences with infinitely many symbols. We achieve the main result by computing the Melnikov integral explicitly.

QUANTUM MECHANICS IN AFC*-SYSTEMS

Motivated from the chemical potential theory, we study quantum statistical thermodynamics in AF C*-systems generalizing usual one-dimensional quantum lattice systems. Our systems are C*-algebras [Formula: see text] which have a localization [Formula: see text] of finite-dimensional subalgebras indexed by finite intervals of Z and an automorphism γ acting as a right shift on the localization. Model examples are supplied from derived towers (string algebras) for type II1 factor-subfactor pairs. Given a (γ-invariant) interaction and a specific tracial state, we formulate the Gibbs conditions and the variational principle for (γ-invariant) states on [Formula: see text], and investigate the relationship among these conditions and the KMS condition for the time evolution generated by the interaction. Special attention is paid to C*-systems of gauge invariance (typical model in the chemical potential theory) and to C*-systems considered as quantum random walks on discrete groups. The CNT-dynamical entropy for the shift automorphism γ is also discussed.

The embedding structure and the shift operator of the U(1) lattice current algebra

The structure of block-spin embeddings of the U(1) lattice current algebra is described. For an odd number of lattice sites, the inner realizations of the shift automorphism areclassified. We present a particular inner shift operator which admits a factorization involving quantum dilogarithms analogous to the results of Faddeev and Volkov.

Dynamical entropy of generalized quantum Markov chains

We compute the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of shift automorphism of generalized quantum Markov chains as defined by Accardi and Frigerio. For any generalized quantum Markov chain defined via a finite set of conditional density amplitudes, we show that the dynamical entropy is equal to the mean entropy.

Q-Superoscillator Transformation Matrix and q-Supercoherent States11The project supported partially by Foundation of Science and Technology Committee of Hunan.

In this paper, we considered a q-superoscillator model and constructed a transformation matrix which transforms the usual algebra of q-boson and q-fermion operators of the q-superoscillator into a nonlinear shift automorphism algebra. The q-supercoherent states for the q-superoscillator model is introduced by utilizing the transformation matrix. They are compared with the coherent states defined as eigenstates of the annihilation operators. Matrix elements of the transformation matrix on the Fock space bases are evaluated as well.

Generalized q-fermion oscillators and q-coherent states

The algebra of q-fermion operators, developed earlier by two of the present authors is re-examined. It is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q=1. It is shown that it is possible to introduce generalized q-oscillators defined for - infinity (q<or=1. In the range - infinity (q(0, these coincide with the q-boson operators and for 0(q<or=1 they coincide with q-fermions. The ordinary bosons and fermions may be identified with the limits q=-1 and +1 respectively. Generalized q-fermion coherent states are constructed by utilizing a nonlinear shift automorphism of the algebra of q-fermion operators. These are compared with the coherent states defined as eigenstates of annihilation operator. Matrix elements of the shift operator in the Fock space basis are evaluated.

Restricted Non-Commutative Bernoulli Shifts

Constructing non-commutative Bernoulli shifts one starts with a measure theoretic Bernoulli shift and an equivalence relation on the measure space. There is a doubly infinite increasing sequence of measure preserving ergodic equivalence relations giving an increasing sequence of pairwise isomorphic von Neumann algebras invariant under the Bernoulli shift automorphism. We consider the index of Jones for those subalgebras and give an explicit example of conjugate non-commutative Bernoulli shifts.

Dynamics of entire functions near the essential singularity

AbstractWe show that entire functions which are critically finite and which meet certain growth conditions admit ‘Cantor bouquets’ in their Julia sets. These are invariant subsets of the Julia set which are homeomorphic to the product of a Cantor set and the line [0, ∞). All of the curves in the bouquet tend to ∞ in the same direction, and the map behaves like the shift automorphism on the Cantor set. Hence the dynamics near ∞ for these types of maps may be analyzed completely. Among the entire maps to which our methods apply are exp (z), sin (z), and cos (z).

Symbolic dynamics and nonlinear semiflows

For a transverse homoclinic orbit γ of a mapping (not necessarily invertible) on a Banach space, it is shown that the mapping restricted to orbits near γ is equivalent to the shift automorphism on doubly infinite sequences on finitely many symbols. Implications of this result for the Poincare map of semiflows are given.

Shift automorphisms in the H�non mapping

We investigate the global behavior of the quadratic diffeomorphism of the plane given byH(x,y)=(1+y−Ax2,Bx). Numerical work by Henon, Curry, and Feit indicate that, for certain values of the parameters, this mapping admits a “strange attractor”. Here we show that, forA small enough, all points in the plane eventually move to infinity under iteration ofH. On the other hand, whenA is large enough, the nonwandering set ofH is topologically conjugate to the shift automorphism on two symbols.