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A weak solenoid is a foliated space defined as the inverse limit of finite coverings of a closed compact manifold $M$. The monodromy of a weak solenoid defines an equicontinuous minimal action on a Cantor space $X$ by the fundamental group $G$ of $M$. The discriminant group of this action is an obstruction to this action being homogeneous. The discriminant vanishes if the group $G$ is abelian, but there are examples of actions of nilpotent groups for which the discriminant is non-trivial. The action is said to be stable if the discriminant group remains unchanged for the induced action on sufficiently small clopen neighborhoods in $X$. If the discriminant group never stabilizes as the diameter of the clopen set $U$ tends to zero, then we say that the action is unstable, and the weak solenoid which defines it is said to be wild. In this work, we show two main results in the course of our study of the properties of the discriminant group for Cantor actions. First, the tail equivalence class of the sequence of discriminant groups obtained for the restricted action on a neighborhood basis system of a point in $X$ defines an invariant of the return equivalence class of the action, called the asymptotic discriminant, which is consequently an invariant of the homeomorphism class of the weak solenoid. Second, we construct uncountable collections of wild solenoids with pairwise distinct asymptotic discriminant invariants for a fixed base manifold $M$, and hence fixed finitely-presented group $G$, which are thus pairwise non-homeomorphic. The study in this work is the continuation of the seminal works on homeomorphisms of weak solenoids by Rogers and Tollefson in 1971, and is dedicated to the memory of Jim Rogers.
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Assuming positive entropy we prove a measure rigidity theorem for higher rank actions on tori and solenoids by commuting automorphisms. We also apply this result to obtain a complete classification of disjointness and measurable factors for these actions.
The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations.
Helical solenoids have been proposed as an option for a Helical Cooling Channel for muons in a proposed Muon Collider. Helical solenoids can provide the required three main field components: solenoidal, helical dipole, and a helical gradient. In general terms, the last two are a function of many geometric parameters: coil aperture, coil radial and longitudinal dimensions, helix period and orbit radius. In this paper, we present design studies of a Helical Solenoid, addressing the geometric tunability limits and auxiliary correction system.
Given a collection of pairwise co-prime integers $% m_{1},ldots ,m_{r}$, greater than $1$, we consider the product $Sigma =Sigma _{m_{1}}times cdots times Sigma _{m_{r}}$, where each $Sigma _{m_{i}}$ is the $m_{i}$-adic solenoid. Answering a question of D. P. Bellamy and J. M. L ysko, in this paper we prove that if $M$ is a subcontinuum of $Sigma $ such that the projections of $M$ on each $Sigma _{m_{i}}$ are onto, then for each open subset $U$ in $Sigma $ with $Msubset U$, there exists an open connected subset $V$ of $Sigma $ such that $Msubset Vsubset U$; i.e. any such $M$ is ample in the sense of Prajs and Whittington [10]. This contrasts with the property of Cartesian squares of fixed solenoids $Sigma _{m_{i}}times Sigma _{m_{i}}$, whose diagonals are never ample [1].
A self-consistent general relativistic configuration describing a finite cross-section magnetic flux tube is constructed. The cosmic solenoid is modeled by an elastic superconductive surface which separates the Melvin core from the surrounding flat conic structure. We show that a given amount $Phi$ of magnetic flux cannot be confined within a cosmic solenoid of circumferential radius smaller than $frac{sqrt{3G}}{2pi c^2}Phi$ without creating a conic singularity. Gauss-Codazzi matching conditions are derived by means of a self-consistent action. The source term, representing the surface currents, is sandwiched between internal and external gravitational surface terms. Surface superconductivity is realized by means of a Higgs scalar minimally coupled to projective electromagnetism. Trading the magnetic London phase for a dual electric surface vector potential, the generalized quantization condition reads: $e/{hc} Phi + 1/e Q=n$ with $Q$ denoting some dual electric charge, thereby allowing for a non-trivial Aharonov-Bohm effect. Our conclusions persist for dilaton gravity provided the dilaton coupling is sub-critical.
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