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Polish group actions and computability [article]

Aleksander Ivanov, Barbara Majcher-Iwanow
2009 arXiv   pre-print
Let G be a closed subgroup of the group of all permutations of a countably infinite set. Let X be a Polish G-space with a countable basis A of clopen sets.  ...  We consider computable complexity of f and some related questions.  ...  The aim of our paper is to show that this idea extends the approach of computability theory to Polish group actions and nice topologies (introduced in [2] ).  ... 
arXiv:0903.1070v2 fatcat:d4gbtwtcdzfh3nzazr3cdil7li

Topology and descriptive set theory

Alexander S. Kechris
1994 Topology and its Applications  
of Polish groups and Bore1 equivalence relations.  ...  In the second we discuss Polish groups, and in the third the basic theory of their Bore1 actions. The last part concentrates on Bore1 equivalence relations.  ...  Let G be a Polish group, X a Polish space and (g, x) I-+ g.x a continuous action of G on X.  ... 
doi:10.1016/0166-8641(94)00146-4 fatcat:rywvqgedbrbnthj54rqxqljiem

Page 5263 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews  
Let G denote a Polish group. A Polish G-space is a Polish space on which G acts continuously.  ...  A Polish group G is called tame if for every Polish G-space ¥ we have that E2 is Borel. A group that is not tame is called wild.  ... 

What is ... a Borel reduction?

Matthew Foreman
2018 Notices of the American Mathematical Society  
Polish group actions More generally, many classification problems are given as orbit equivalences of Polish group actions. Commonly the group action is some form of conjugacy.  ...  group action. ℓ 1 -actions Every action of an abelian Polish group can be reduced to an action of the abelian group ℓ 1 (ℕ) with pointwise addition, hence to the maximal ℓ 1 -action.  ... 
doi:10.1090/noti1747 fatcat:hetyfvpzp5bwdnvmpj2t653u3a

Page 1542 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews  
group actions on Polish spaces.  ...  Hjorth has isolated the dynamical prop- erty of group actions, called turbulence.  ... 

Polish Metric Spaces: Their Classification and Isometry Groups

John D. Clemens, Su Gao, Alexander S. Kechris
2001 Bulletin of Symbolic Logic  
In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces.  ...  The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.  ...  Our first result computes the complexity of the isometric classification of Polish metric spaces as being exactly that of the universal equivalence relation induced by a Borel action of a Polish group.  ... 
doi:10.2307/2687754 fatcat:j5kabnrlzra7rnpfhu5hepvu6y

Borel actions of Polish groups

Howard Becker, Alexander S. Kechris
1993 Bulletin of the American Mathematical Society  
We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory  ...  of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture.  ...  Then since t consists of Borel sets in X and gives a Polish topology, it gives rise to the original Borel structure of X. The continuity of the action may be checked by direct computation.  ... 
doi:10.1090/s0273-0979-1993-00383-5 fatcat:7som7i4ctng3fmfl65hk5dtlue

Borel actions of Polish groups [article]

Howard Becker, Alexander S. Kechris
1993 arXiv   pre-print
We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory  ...  of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture.  ...  We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory  ... 
arXiv:math/9304208v1 fatcat:n37klzbwprhz7eklss7742alom

Non-Archimedean TSI Polish groups and their potential Borel complexity spectrum [article]

Shaun Allison
2020 arXiv   pre-print
In this paper, the Hjorth analysis of Polish group actions is applied to the special case of a non-Archimedean TSI Polish group acting continuously on a Polish spaces.  ...  not classifiable by non-Archimedean TSI Polish groups.  ...  Introduction In this paper, we study continuous (or Borel) actions of non-Archimedean TSI Polish groups on Polish spaces.  ... 
arXiv:2010.05085v1 fatcat:mxfuwgiih5cl3otkehqv32gn4y

Polish G-spaces, the generalized model theory and complexity [article]

A. Ivanov, B. Majcher-Iwanow
2019 arXiv   pre-print
Given Polish space Y and continuous language L we study the corresponding logic Iso( Y)-space Y_L.  ...  Polish group actions. A Polish space (group) is a separable, completely metrizable topological space (group).  ...  Moreover it is suited to the general setting of computable Polish group actions presented in the recent paper [41] .  ... 
arXiv:1909.12613v2 fatcat:hyk7i2i4l5dnhmvpg4n5kenjim

Universal minimal flows of extensions of and by compact groups [article]

Dana Bartošová
2021 arXiv   pre-print
As an application, we show that the phase space of M(G) for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, Cantor set 2^ℕ,  ...  Moreover, if either the left and right uniformities on G coincide or G≅ G/K⋉ K, we also recover the action, in the latter case extending a result of Kechris and Sokić.  ...  Polish totally disconnected locally compact (t.d.l.c.) groups.  ... 
arXiv:2103.10991v1 fatcat:ryxy3vyfvrbrhmt3i5ic5ac3pm

The complexity of the Structure and Classification of Dynamical Systems [article]

Matthew Foreman
2022 arXiv   pre-print
So an object is in the region Polish group actions if it can be reduced to a Polish group action.  ...  Dynamical Systems Broadly speaking a dynamical system is any group action φ : G × X → X. However, to be interesting, the group actions are taken to preserve some structure on X.  ...  General Polish Group actions We first give an example of a Polish group action that is not reducible to an S ∞ -action but is still Borel. Example.  ... 
arXiv:2203.10655v1 fatcat:xohi37xy2bej7avkub46cupl4q

The definable content of homological invariants I: Ext lim^1 [article]

Jeffrey Bergfalk, Martino Lupini, Aristotelis Panagiotopoulos
2021 arXiv   pre-print
To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover.  ...  Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem ℛ(Aut(Λ)↷Ext(Λ,ℤ)) of classifying all group extensions  ...  Many classification problems are induced by continuous actions of Polish group actions. Definition 2.5.  ... 
arXiv:2008.08782v3 fatcat:kc6umrkqiffzfac6phou4iziim

A Polish metric space whose group of isometries induces a universal relation for Polish group actions

Julien Melleray
2017 Fundamenta Mathematicae  
We show that there exists a Polish metric space (X, d) such that the action of its isometry group on X produces an equivalence relation which is universal for relations induced by a Borel action of a Polish  ...  group on a standard Borel space.  ...  There exists a Polish metric space X such that the action of the isometry group Iso(X) on X is universal among all relations induced by a Borel action of a Polish group on a Polish space.  ... 
doi:10.4064/fm341-1-2017 fatcat:ouxofq4bv5d4ngtqkucww62emi

The Prospects of E-commerce in Poland

Katarzyna Bredzel-Skowera, Tomasz Turek
2015 Procedia Computer Science  
taken by Polish e-stores.  ...  This is due to still not a very large group of e-consumers and limited use of the power of the Internet in running a business activity by estores.  ...  While pursuing global trends and following foreign e-stores, Polish e-commerce takes limited marketing actions. 6.  ... 
doi:10.1016/j.procs.2015.09.038 fatcat:f5esiph3q5h7nhiet6smo3yhki
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