Extensions Of Automorphisms Research Articles

Extensions of automorphisms of self-similar groups

Abstract In this work, we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric. We obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist. We study in detail the case of the lamplighter group L k = Z k ≀ Z \mathcal{L}_{k}=\mathbb{Z}_{k}\wr\mathbb{Z} .

On tameness of almost automorphic dynamical systems for general groups

Let ( X , G ) be a minimal equicontinuous dynamical system, where X is a compact metric space and G some topological group acting on X. Under very mild assumptions, we show that the class of regular almost automorphic extensions of ( X , G ) contains examples of tame but non-null systems as well as non-tame ones. To do that, we first study the representation of almost automorphic systems by means of semicocycles for general groups. Based on this representation, we obtain examples of the above kind in well-studied families of group actions. These include Toeplitz flows over G-odometers where G is countable and residually finite as well as symbolic extensions of irrational rotations.

Skew derivations on generalized Weyl algebras

A wide class of skew derivations on degree-one generalized Weyl algebras R(a,φ) over a ring R is constructed. All these derivations are twisted by a degree-counting extensions of automorphisms of R. It is determined which of the constructed derivations are Q-skew derivations. The compatibility of these skew derivations with the natural Z-grading of R(a,φ) is studied. Additional classes of skew derivations are constructed for generalized Weyl algebras given by an automorphism φ of a finite order. Conditions that the central element a that forms part of the structure of R(a,φ) needs to satisfy for the orthogonality of pairs of aforementioned skew derivations are derived. In addition local nilpotency of constructed derivations is studied. General constructions are illustrated by description of all skew derivations (twisted by a degree-counting extension of the identity automorphism) of generalized Weyl algebras over the polynomial ring in one variable and with a linear polynomial as the central element.

Spectra of Automorphic Extensions of Finite Simple Exceptional Groups of Lie Type

The spectrum ω (G) of a finite group G is the set of orders of elements of G. Let S be a simple exceptional group of type E6 or E7. We describe all finite groups G such that S ≤ G ≤ Aut S and ω (G) = ω (S) and completes the study of the recognition-by-spectrum problem for all simple exceptional groups of Lie type.

On spectra of automorphic extensions of finite simple groups F4(q) and 3D4(q)

The spectrum [Formula: see text] of a finite group [Formula: see text] is the set of the orders of its elements. Let [Formula: see text] be one of the groups [Formula: see text], [Formula: see text]. We describe the spectra of [Formula: see text] and determine [Formula: see text] such that [Formula: see text] and [Formula: see text]. Combining this result with earlier work, we determine all finite groups [Formula: see text] with [Formula: see text].

Extension of automorphisms of rational smooth affine curves

We provide the existence, for every complex rational smooth affine curve $\Gamma$, of a linear action of $\mathrm{Aut}(\Gamma)$ on the affine 3-dimensional space $\mathbb{A}^3$, together with a $\mathrm{Aut}(\Gamma)$-equivariant closed embedding of $\Gamma$ into $\mathbb{A}^3$. It is not possible to decrease the dimension of the target, the reason for this obstruction is also precisely described.

Extending Automorphisms and Derivations onto Ore-Extensions

We study the question wether an automorphism σ of a field K can be extended to an automorphism τ of the field of fractions $${Q=K(x; \alpha, \delta)}$$ of the Ore-extension $${R=K[x; \alpha, \delta]}$$ (Sect. 3) and wether a σ-derivation $${\varepsilon}$$ of K can be extended to a τ-derivation of Q (Sect. 4), and determine all extensions of σ and $${\varepsilon}$$ . Until now these question have only been discussed under special assumptions (for example in [7] and [11]). In particular, little is known on extensions of derivations. The result is in each case a criterion for extendability (Lemmata 3.2 and 4.3). The characterization of all extensions of automorphisms σ from K to Q is well understood (Corollary 3.5 and Theorem 3.12). This is in contrast to the characterization of the extensions of σ-derivations $${\varepsilon}$$ , which can only be described satisfactorily under additional assumptions (Theorems 4.8, 4.9, 4.10). We obtain the set of all extensions of σ or $${\varepsilon}$$ easily from a particular extension and the normalizer $${N(\sigma^{-1} \alpha \, \sigma)}$$ or $${N(\alpha \, \sigma)}$$ (Corollaries 3.3 and 4.4). These normalizers will be described in Sect. 2 by minimal elements of R.

A non-autonomous bifurcation theory for deterministic scalar differential equations

In the extension of the concepts of saddle-node, transcritical and pitchfork bifurcations to the non-autonomous case, one considers the change in the number and attraction properties of the minimal sets for the skew-product flow determined by the initial one-parametric equation. In this work conditions on the coefficients of the equation ensuring the existence of a global bifurcation phenomenon of each one of the types mentioned are established. Special attention is paid to show the importance of the non-trivial almost automorphic extensions and pinched sets in describing the dynamics at the bifurcation point.

OLD AND NEW RESULTS ON STRANGE NONCHAOTIC ATTRACTORS

Classical and new results concerning the topological structure of skew-products semiflows, coming from nonautonomous maps and differential equations, are combined in order to establish rigorous conditions giving rise to the occurrence of strange nonchaotic attractors on 𝕋d × ℝ. A special attention is paid to the relation of these sets with the almost automorphic extensions of the base flow. The scope of the results is clarified by applying them to the Harper map, although they are valid in a much wider context.

Almost Automorphic and Almost Periodic Dynamics for Quasimonotone Non-Autonomous Functional Differential Equations

The occurrence of almost automorphic dynamics for monotone non-autonomous recurrent finite-delay functional differential equations is analyzed. Topological methods are used to ensure its presence in the case of existence of semicontinuous semi-equilibria. When these semi-equilibria are continuous and strong, the presence of almost automorphic extensions is persistent under small perturbations. The above method provides a minimal set isomorphic to the base in the case of a convex semiflow. Some examples show the applicability of these results.

A Note on non-autonomous scalar functional differential equations with small delay

We prove that the minimal sets in the skew-product semiflows generated from a non-autonomous scalar functional differential equation with a small delay are all almost automorphic extensions of the base. This result is not true for arbitrary delay equations. The point is that, for a small delay, so-called special solutions exist and permit us to tackle the problem by means of some related scalar ODE's for which the study is much simpler. To cite this article: A.I. Alonso et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Canonical extension of actions of locally compact quantum groups

In this paper, we generalize the notion of the canonical extension of automorphisms of von Neumann algebras to the case of actions of locally compact quantum groups (in the sense of Kustermans and Vaes). Various expected properties will be shown to hold for this new canonical extension. As an application, we describe the flow of weights of the crossed product of a type III factor by some special action of a discrete Kac algebra.

Canonical extension of endomorphisms of type III factors

We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions. Following the automorphism case, we introduce two notions for endomorphisms of type III factors: modular endomorphisms and Connes-Takesaki modules. Several applications to compact groups of automorphisms and subfactors of type III factors are given from the viewpoint of ergodic theory.

Invariant Measures and Natural Extensions

AbstractWe study ergodic properties of a family of intervalmaps that are given as the fractional parts of certain real Möbius transformations. Included are the maps that are exactly n-to-1, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.

Embeddings into P(N)/fin and extension of automorphisms

Given a Boolean algebra ${\mathbb B}$ and an embedding $e:{\mathbb B}\to {\cal P} ({\mathbb N})/{\rm fin}$ we consider the possibility of extending each or some automorphism of ${\mathbb B}$ to the whole $ {\cal P}({\mathbb N})/{\rm fin}$. Among other t

EXTENSION OF AUTOMORPHISMS OF A SUBFACTOR TO THE SYMMETRIC ENVELOPING ALGEBRA

In this article, we consider an extension of automorphisms of a subfactor to the symmetric enveloping algebra introduced by Popa. We discuss the relation between this extension and invariants of automorphisms introduced by Kawahigashi. We also compute the symmetric enveloping algebras arising from orbifold subfactors.

On the extension of automorphisms of polynomial rings

On the extension of automorphisms of polynomial rings

Extensions of automorphisms and gauge symmetries

We characterize the automorphisms of aC *-algebra which extend to automorphisms of the crossed product by a compact group dual. The case where the inclusion is equipped with a group of automorphisms commuting with the dual action is also treated. These results are applied to the analysis of broken gauge symmetries in Quantum Field Theory to draw conclusions on the structure of the degenerate vacua on the field algebra.

Inner Extensions of Automorphisms of Irrational Rotation Algebras to AF-Algebras

Let $A_\theta$ be an irrational rotation algebra. In the present paper we will show that automorphisms of $A_\theta$ with some properties can be extended to inner automorphisms of an AF-algebra. In other words, there are a monomorphism $\rho$ of $A_\theta$ into an AF-algebra $B$ and a unitary element $w\in B$ such that $\rho(\alpha(x))=w\rho(x)w^*$ for any $x\in A_\theta$.

More on the Schur group of a commutative ring

The Schur group of a commutative ring, R, with identity consists of all classes in the Brauer group of R which contain a homomorphic image of a group ring RG for some finite group G. It is the purpose of this article to continue an investigation of this group which was introduced in earler work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well‐known Schur index in the global or local field case.