scholarly journals ORIENTATION OF PIECEWISE POWERS OF A MINIMAL HOMEOMORPHISM

2021 ◽  
pp. 1-31
Author(s):  
COLIN D. REID
Keyword(s):  
Minimal System ◽  
Full Group ◽  
Cantor Space

Abstract We show that, given a compact minimal system $(X,g)$ and an element h of the topological full group $\tau [g]$ of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where G is any virtually polycyclic subgroup of $\tau [g]$ . We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau [g]$ of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that $h \in \tau [g]$ .

scholarly journals Ext and OrderExt Classes of Certain Automorphisms of C*-Algebras Arising from Cantor Minimal Systems

2001 ◽  
Vol 53 (2) ◽  
pp. 325-354 ◽  
Author(s):  
Hiroki Matui
Keyword(s):  
Transformation Group ◽  
Minimal System ◽  
Full Group ◽  
C Algebras ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
Cantor Minimal Systems

AbstractGiordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

scholarly journals Full groups of minimal homeomorphisms and Baire category methods

2014 ◽  
Vol 36 (2) ◽  
pp. 550-573
Author(s):  
TOMÁS IBARLUCÍA ◽  
JULIEN MELLERAY
Keyword(s):  
Baire Category ◽  
Full Group ◽  
Group Topology ◽  
Cantor Space ◽  
Polish Group ◽  
Minimal Homeomorphisms

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space$X$, showing that these groups do not admit a compatible Polish group topology and, in the case of$\mathbb{Z}$-actions, are coanalytic non-Borel inside$\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside$\text{Homeo}(X)$.

scholarly journals A Tits alternative for topological full groups

2019 ◽  
Vol 41 (2) ◽  
pp. 622-640
Author(s):  
NÓRA GABRIELLA SZŐKE
Keyword(s):  
Free Group ◽  
Free Action ◽  
The Other ◽  
Full Group ◽  
Finitely Generated ◽  
Cantor Space ◽  
Cyclic Groups ◽  
Finitely Generated Groups ◽  
Tits Alternative ◽  
The One

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.

Toward living nanomachines

2018 ◽  
Author(s):  
Christof Mast ◽  
Friederike Möller ◽  
Moritz Kreysing ◽  
Severin Schink ◽  
Benedikt Obermayer ◽  
...  
Keyword(s):  
Molecular Evolution ◽  
Thermal Gradient ◽  
Molecular Level ◽  
Minimal System ◽  
Thermal Gradients ◽  
Temperature Oscillations ◽  
Fluid Convection ◽  
Inanimate Matter ◽  
High Concentrations ◽  
Periodic Temperature

How does inanimate matter become transformed into animate matter? Living systems evolve by replication and selection at the molecular level and this chapter considers how to establish a synthetic, minimal system that can support molecular evolution and thus life. Molecular evolution cannot be explained by starting with high concentrations of activated chemicals that react toward their chemical equilibrium; persistent non-equilibria are required to maintain continuous reactivity and we especially consider thermal gradients as an early driving force for Darwinian molecular evolution. The temperature difference across water-filled compartments implements a laminar fluid convection with periodic temperature oscillations that allow for the melting and replication of DNA. Simultaneously, dissolved molecules are moved along the thermal gradient by an effect called thermophoresis. The combined result is an efficient molecule trap that exponentially favors long over short DNA and thus maintains complexity. Future experiments will reveal how thermal gradients could actively drive the Darwinian process of replication and selection.

scholarly journals Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

2021 ◽  
Author(s):  
Jiahao Qiu ◽  
Jianjie Zhao
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Minimal System ◽  
Combinatorial Properties ◽  
Return Times ◽  
Residual Set ◽  
Closed Sets ◽  
Open Set ◽  
Geometric Progressions ◽  
Set Of Points

AbstractIn this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

Archaeal DNA uracil repair via direct strand incision: A minimal system reconstituted from purified components

DNA Repair ◽  
2010 ◽  
Vol 9 (4) ◽  
pp. 438-447 ◽  
Author(s):  
Lars Schomacher ◽  
K. Anke Schürer ◽  
Elena Ciirdaeva ◽  
Paul McDermott ◽  
James P.J. Chong ◽  
...  
Keyword(s):  
Minimal System ◽  
Uracil Repair

QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

2014 ◽  
Vol 7 (3) ◽  
pp. 439-454 ◽  
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Real Line ◽  
Topological Spaces ◽  
Topological Semantics ◽  
Cantor Space ◽  
Quantified Modal Logic ◽  
Propositional Modal Logic ◽  
The Family ◽  
Modal Logic S4 ◽  
Rational Line

AbstractIn the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

Feasibility of a One Degree-of-Freedom Linear Robot for Bed-Bound Stroke Rehabilitation

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Nicholas Berezny ◽  
Dar Dowlatshahi ◽  
Mojtaba Ahmadi
Keyword(s):  
Haptic Feedback ◽  
Field Work ◽  
Minimal System ◽  
Ease Of Use ◽  
User Engagement ◽  
Leg Extension ◽  
Resistive Forces ◽  
Limb Rehabilitation ◽  
Hospital Equipment ◽  
Healthy Participants

Abstract A 1DOF lower limb rehabilitation robot is presented for delivering leg extension therapy to bed-bound stroke patients. Such a compact and minimal system may be beneficial in terms of compatibility with pre-existing hospital equipment, ease-of-use, safety, and cost. A set of design criteria was created based on the literature and on previous field work at a local hospital. The device uses admittance control to apply assistive or resistive forces, and can also use haptic feedback to increase user engagement. A pilot study on six healthy participants was used to determine the feasibility of such a minimal system in administering assistance or resistance through the leg extension exercise. Results indicate that a single DOF is capable of decreasing trajectory error with assistance and increasing user effort with resistance. Observations confirm that the minimal system is effective; however, extending the robot with additional DOFs so that it can target multiple bed-bound exercises may help to increase therapy duration.

Sign in / Sign up

Export Citation Format

Share Document