Auslander-Yorke type dichotomy theorems for stronger versions of $r$-sensitivity

2019 ◽  
Vol 147 (6) ◽  
pp. 2609-2617 ◽  
Author(s):  
Kairan Liu ◽  
Xiaomin Zhou
Keyword(s):  
Dichotomy Theorems

Relativization of sensitivity in minimal systems

2018 ◽  
Vol 40 (6) ◽  
pp. 1715-1728
Author(s):  
TAO YU ◽  
XIAOMIN ZHOU
Keyword(s):  
Relative Sensitivity ◽  
Dichotomy Theorems ◽  
Proximal Extension ◽  
Proximal Relation

Let $\unicode[STIX]{x1D70B}:(X,T)\rightarrow (Y,S)$ be an extension between minimal systems; we consider its relative sensitivity. We obtain that $\unicode[STIX]{x1D70B}$ is relatively $n$-sensitive if and only if the relative $n$-regionally proximal relation contains a point whose coordinates are distinct; and the structure of $\unicode[STIX]{x1D70B}$ which is relatively $n$-sensitive but not relatively $(n+1)$-sensitive is determined. Let ${\mathcal{F}}_{t}$ be the families consisting of thick sets. We introduce notions of relative block ${\mathcal{F}}_{t}$-sensitivity and relatively strong ${\mathcal{F}}_{t}$-sensitivity. Let $\unicode[STIX]{x1D70B}:(X,T)\rightarrow (Y,S)$ be an extension between minimal systems. Then the following Auslander–Yorke type dichotomy theorems are obtained: (1) $\unicode[STIX]{x1D70B}$ is either relatively block ${\mathcal{F}}_{t}$-sensitive or $\unicode[STIX]{x1D719}:(X,T)\rightarrow (X_{\text{eq}}^{\unicode[STIX]{x1D70B}},T_{\text{eq}})$ is a proximal extension where $(X_{\text{eq}}^{\unicode[STIX]{x1D70B}},T_{\text{eq}})\rightarrow (Y,S)$ is the maximal equicontinuous factor of $\unicode[STIX]{x1D70B}$. (2) $\unicode[STIX]{x1D70B}$ is either relatively strongly ${\mathcal{F}}_{t}$-sensitive or $\unicode[STIX]{x1D719}:(X,T)\rightarrow (X_{d}^{\unicode[STIX]{x1D70B}},T_{d})$ is a proximal extension where $(X_{d}^{\unicode[STIX]{x1D70B}},T_{d})\rightarrow (Y,S)$ is the maximal distal factor of $\unicode[STIX]{x1D70B}$.

scholarly journals Dichotomy theorems for random matrices and closed ideals of operators on (⊕n=1∞ℓ1n)c0

2012 ◽  
Vol 86 (1) ◽  
pp. 235-258 ◽  
Author(s):  
N. J. Laustsen ◽  
E. Odell ◽  
Th. Schlumprecht ◽  
A. Zsák
Keyword(s):  
Random Matrices ◽  
Dichotomy Theorems

scholarly journals On function field Mordell–Lang and Manin–Mumford

2016 ◽  
Vol 16 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Franck Benoist ◽  
Elisabeth Bouscaren ◽  
Anand Pillay
Keyword(s):  
Model Theory ◽  
Function Field ◽  
Positive Characteristic ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Morley Rank ◽  
Main Interest ◽  
Dichotomy Theorems ◽  
Finite Morley Rank ◽  
Crucial Ingredient

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see text] where [Formula: see text] is a saturated separably closed field.

KNIGHT'S MODEL, ITS AUTOMORPHISM GROUP, AND CHARACTERIZING THE UNCOUNTABLE CARDINALS

2002 ◽  
Vol 02 (01) ◽  
pp. 113-144 ◽  
Author(s):  
GREG HJORTH
Keyword(s):  
Automorphism Group ◽  
Partial Information ◽  
Countable Model ◽  
Dichotomy Theorems ◽  
Analytic Sets ◽  
Uncountable Cardinals ◽  
Countable Structure

We show that every ℵα(α<ω1) can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1but whose automorphism group fails the topological Vaught conjecture on analytic sets.We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knight's model.

Some dichotomy theorems for isomorphism relations of countable models

2001 ◽  
Vol 66 (2) ◽  
pp. 902-922 ◽  
Author(s):  
Su Gao
Keyword(s):  
Linear Orderings ◽  
Open Questions ◽  
Dichotomy Theorems ◽  
Countable Models

AbstractStrengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an Lω1ω-interpretability theory. We also give a survey of this theory.

The graph-theoretic approach to descriptive set theory

2012 ◽  
Vol 18 (4) ◽  
pp. 554-575 ◽  
Author(s):  
Benjamin D. Miller
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Theoretic Approach ◽  
Chromatic Numbers ◽  
Graph Theoretic ◽  
Graph Theoretic Approach ◽  
Dichotomy Theorems ◽  
Descriptive Set

AbstractWe sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.

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