Boundary dynamics for bireversible and for contracting automaton groups

We study the dynamics of the action of an automaton group on the set of infinite words, and more precisely the discontinuous points of the map which associates to a point its set of stabilizers — the singular points. We show that, for any Mealy automaton, the set of singular points has measure zero. Then we focus our attention on several classes of automata. We characterize those contracting automata generating groups without singular points, and apply this characterization to the Basilica group. We prove that potential examples of reversible automata generating infinite groups without singular points are necessarily bireversible. We also provide some conditions for such examples to exist. Finally, we study some dynamical properties of the Schreier graphs in the boundary.

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Faithful group actions and Schreier graphs

Carpathian Mathematical Publications ◽

10.15330/cmp.9.2.202-207 ◽

2018 ◽

Vol 9 (2) ◽

pp. 202-207

Author(s):

M. Fedorova

Keyword(s):

Group Action ◽

Finite Groups ◽

Finite Automaton ◽

Finite Automata ◽

Group Actions ◽

Sufficient Condition ◽

Free Products ◽

Schreier Graphs ◽

Products Of Groups ◽

Dynamical Properties

Each action of a finitely generated group on a set uniquely defines a labelled directed graph called the Schreier graph of the action. Schreier graphs are used mainly as a tool to establish geometrical and dynamical properties of corresponding group actions. In particilar, they are widely used in order to check amenability of different classed of groups. In the present paper Schreier graphs are utilized to construct new examples of faithful actions of free products of groups. Using Schreier graphs of group actions a sufficient condition for a group action to be faithful is presented. This result is applied to finite automaton actions on spaces of words i.e. actions defined by finite automata over finite alphabets. It is shown how to construct new faithful automaton presentations of groups upon given such a presentation. As an example a new countable series of faithful finite automaton presentations of free products of finite groups is constructed. The obtained results can be regarded as another way to construct new faithful actions of groups as soon as at least one such an action is provided.

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From the Oort cloud to Halley-type comets

International Astronomical Union Colloquium ◽

10.1017/s0252921100031638 ◽

1999 ◽

Vol 173 ◽

pp. 327-338 ◽

Cited By ~ 5

Author(s):

J.A. Fernández ◽

T. Gallardo

Keyword(s):

Total Population ◽

Complex Function ◽

Dynamical Evolution ◽

Oort Cloud ◽

Capture Process ◽

Influx Rate ◽

Short Period ◽

Dynamical Properties ◽

Orbital Periods ◽

Probability Of Capture

AbstractThe Oort cloud probably is the source of Halley-type (HT) comets and perhaps of some Jupiter-family (JF) comets. The process of capture of Oort cloud comets into HT comets by planetary perturbations and its efficiency are very important problems in comet ary dynamics. A small fraction of comets coming from the Oort cloud − of about 10−2− are found to become HT comets (orbital periods < 200 yr). The steady-state population of HT comets is a complex function of the influx rate of new comets, the probability of capture and their physical lifetimes. From the discovery rate of active HT comets, their total population can be estimated to be of a few hundreds for perihelion distancesq <2 AU. Randomly-oriented LP comets captured into short-period orbits (orbital periods < 20 yr) show dynamical properties that do not match the observed properties of JF comets, in particular the distribution of their orbital inclinations, so Oort cloud comets can be ruled out as a suitable source for most JF comets. The scope of this presentation is to review the capture process of new comets into HT and short-period orbits, including the possibility that some of them may become sungrazers during their dynamical evolution.

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Infinite Words - Automata, Semigroups, Logic and Games

10.1016/s0079-8169(04)x8001-6 ◽

2004 ◽

Keyword(s):

Infinite Words

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Binary diagrams of mesomorphous compounds exhibiting singular points

Annales de Physique ◽

10.1051/anphys/197803030381 ◽

1978 ◽

Vol 3 ◽

pp. 381-386 ◽

Cited By ~ 5

Author(s):

F. Hardouin ◽

G. Sigaud ◽

M.-F. Achard ◽

H. Gasparoux

Keyword(s):

Singular Points

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Dynamical properties of highly entangled polyalkylmethacrylate solutions : A comparative study

Journal de Physique IV (Proceedings) ◽

10.1051/jp4:2000765 ◽

2000 ◽

Vol 10 (PR7) ◽

pp. Pr7-321-Pr7-324

Author(s):

V. Villari ◽

A. Faraone, ◽

S. Magazù, ◽

G. Maisano ◽

R. Ponterio

Keyword(s):

Comparative Study ◽

Dynamical Properties

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Computational studies on the static and dynamical properties of one dimensional integer spin quantum magnets

10.32657/10356/143094 ◽

2020 ◽

Author(s):

◽

Ernest Teng Siang Ong

Keyword(s):

Computational Studies ◽

Quantum Magnets ◽

One Dimensional ◽

Integer Spin ◽

Spin Quantum ◽

Dynamical Properties

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Effect of proximity of the Fermi level to singular points in the band structure on the kinetic and lattice properties of metals and alloys

Uspekhi Fizicheskih Nauk ◽

10.3367/ufnr.0154.198803g.0525 ◽

1988 ◽

Vol 154 (3) ◽

pp. 525 ◽

Cited By ~ 5

Author(s):

V.P. Antropov ◽

Valentin G. Vaks ◽

M.I. Katsnel'son ◽

V.G. Koreshkov ◽

A.I. Likhtenshtein ◽

...

Keyword(s):

Band Structure ◽

Fermi Level ◽

Metals And Alloys ◽

Singular Points

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Pseudo-Anosov Theory

10.23943/princeton/9780691147949.003.0015 ◽

2017 ◽

Cited By ~ 1

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Braid Groups ◽

Intersection Numbers ◽

Branched Covers ◽

Algebraic Integers ◽

Dehn Twists ◽

Mapping Classes ◽

Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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