scholarly journals Entropy, products, and bounded orbit equivalence

2021 ◽  
pp. 1-39
Author(s):  
DAVID KERR ◽  
HANFENG LI
Keyword(s):  
Infinite Groups ◽  
Locally Finite ◽  
Orbit Equivalent ◽  
Amenable Subgroup ◽  
Metrizable Spaces ◽  
Sofic Groups ◽  
Entropy Inequalities ◽  
Uniquely Ergodic ◽  
Sofic Measure ◽  
Measure Preserving Actions

Abstract We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.

scholarly journals On infinite groups in which all abelian subgroups are locally cyclic

2021 ◽  
Author(s):  
Costantino Delizia ◽  
Chiara Nicotera
Keyword(s):  
Finite Groups ◽  
Abelian Subgroup ◽  
Periodic Case ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Periodic Groups ◽  
Abelian Subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows

2000 ◽  
Vol 20 (6) ◽  
pp. 1687-1710 ◽  
Author(s):  
RICHARD GJERDE ◽  
ØRJAN JOHANSEN
Keyword(s):  
Dimension Group ◽  
Orbit Equivalent ◽  
Bratteli Diagrams ◽  
Novel Approach ◽  
Dimension Groups ◽  
Substitution System ◽  
Zero Entropy ◽  
Factor Map ◽  
Cantor Minimal Systems ◽  
Uniquely Ergodic

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

scholarly journals Characterizations of sn-metrizable spaces

2003 ◽  
Vol 74 (88) ◽  
pp. 121-128 ◽  
Author(s):  
Ying Ge
Keyword(s):  
Metric Space ◽  
Metrizable Space ◽  
Finite Point ◽  
Locally Finite ◽  
Metrizable Spaces

We give some characterizations of sn-metrizable spaces. We prove that a space is an sn-metrizable space if and only if it has a locally-finite point-star sn-network. As an application of the result, a space is an sn-metrizable space if and only if it is a sequentially quotient, ? (compact), ?-image of a metric space.

scholarly journals so-metrizable spaces and images of metric spaces

2021 ◽  
Vol 19 (1) ◽  
pp. 1145-1152
Author(s):  
Songlin Yang ◽  
Xun Ge
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Metrizable Space ◽  
Open Mapping ◽  
Generalized Metric Space ◽  
Locally Finite ◽  
Generalized Metric Spaces ◽  
Covering Mapping ◽  
Generalized Metric ◽  
Metrizable Spaces

Abstract so-metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so-metrizable space if it has a σ \sigma -locally finite so-network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so-metrizable spaces by images of metric spaces. This paper gives such characterizations for so-metrizable spaces. More precisely, this paper introduces so-open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so-metrizable space if and only if it is an so-open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so-open, σ \sigma -image of a metric space. In addition, it is shown that so-open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so-metrizable spaces and establish some equivalent relations among so-open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.

scholarly journals Preservations of so-metrizable spaces

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 801-807 ◽  
Author(s):  
Shou Lin ◽  
Ying Ge
Keyword(s):  
Topological Group ◽  
Affirmative Answer ◽  
Metrizable Space ◽  
Regular Space ◽  
Locally Finite ◽  
Open Network ◽  
Covering Mappings ◽  
Metrizable Spaces ◽  
Closed Mappings ◽  
Closed Image

A space is called an so-metrizable space if it is a regular space with a ?-locally finite sequentially open network. This paper proves that so-metrizable spaces are preserved under perfect mappings and under closed sequence-covering mappings, which give an affirmative answer to a question on preservations of so-metrizable spaces under some closed mappings. Also, we prove that the closed image of an so-metrizable space is an so-metrizable space if it is a topological group.

On the structure of groups whose non-abelian subgroups are subnormal

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Leonid Kurdachenko ◽  
Sevgi Atlıhan ◽  
Nikolaj Semko
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Abelian Subgroups

AbstractThe main aim of this article is to examine infinite groups whose non-abelian subgroups are subnormal. In this sense we obtain here description of such locally finite groups and, as a consequence we show several results related to such groups.

Locally Finite Products of Totally Permutable Nilpotent Groups

2009 ◽  
Vol 16 (03) ◽  
pp. 535-540 ◽  
Author(s):  
Maria De Falco ◽  
Francesco de Giovanni ◽  
Carmela Musella
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Infinite Groups ◽  
Locally Finite ◽  
Finite Products ◽  
A Finite Group

A group G=AB is said to be totally factorized by its subgroups A and B if XY=YX for all subgroups X of A and Y of B. It is known that any finite group totally factorized by supersoluble subgroups is supersoluble, and that a finite group totally factorized by nilpotent subgroups is abelian-by-nilpotent. This latter result is extended here to certain classes of infinite groups.

scholarly journals Infinite groups admitting a faithful irreducible representation

2018 ◽  
Vol 17 (01) ◽  
pp. 1850005
Author(s):  
Fernando Szechtman ◽  
Anatolii Tushev
Keyword(s):  
Irreducible Representation ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated. We consider locally finite groups and groups whose socle is essential.

Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form

1999 ◽  
Vol 19 (4) ◽  
pp. 1093-1109 ◽  
Author(s):  
WILLIAM A. VEECH
Keyword(s):  
Riemann Surface ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Quadratic Differential ◽  
Probability Measures ◽  
Locally Finite ◽  
Horocycle Flow ◽  
Closed Riemann Surface ◽  
Borel Probability ◽  
Uniquely Ergodic

We introduce a set, $Q({\bf T})$, of Borel probability measures on the circle such that each $\mu\in Q({\bf T})$ obeys the conclusion of the Kerckhoff–Masur–Smillie theorem [3]: if $q$ is a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface, then for each $\mu\in Q({\bf T})$ and $\mu$-a.e. $\zeta\in{\bf T}$, $\zeta q$ has uniquely ergodic vertical foliation. As an example, the normalized Cantor–Lebesgue measure belongs to $Q({\bf T})$. The analysis also yields an analogue, for the Teichmüller horocycle flow, of a theorem of Dani: every locally finite ergodic invariant measure for the Teichmüller horocycle flow is finite.

scholarly journals Topological orbit equivalence classes and numeration scales of logistic maps

2011 ◽  
Vol 32 (5) ◽  
pp. 1501-1526
Author(s):  
MARÍA ISABEL CORTEZ ◽  
JUAN RIVERA-LETELIER
Keyword(s):  
Logistic Map ◽  
Natural Extension ◽  
Equivalence Classes ◽  
Orbit Equivalence ◽  
Orbit Equivalent ◽  
Uniquely Ergodic

AbstractWe show that every uniquely ergodic minimal Cantor system is topologically orbit equivalent to the natural extension of a numeration scale associated to a logistic map.

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