29,273 research outputs found
Dynamical forcing of circular groups
In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set X ā R/Z consisting of rotation numbers Īø which can be forced by finitely presented groups is an infinitely generated Q-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number Īø is forced by a pair (G_Īø, Ī±), where G_Īø is a finitely presented group G_Īø and Ī± ā G_Īø is some element, if the set of rotation numbers of Ļ(Ī±) as Ļ ā Hom(G_Īø, Homeo^(+)(S^1)) is precisely the set {0,Ā±Īø}.
We show that the set of subsets of R/Z which are of the
form rot(X(G, Ī±)) = {r ā R/Z | r = rot(Ļ(Ī±)), Ļ ā Hom(G, Homeo^(+)(S^1))}, where G varies over countable groups, are exactly the set of closed subsets which contain 0 and are invariant under xāāx. Moreover, we show that every such subset can be approximated from above by rot(X(G_i, Ī±_i)) for finitely presented G_i.
As another application, we construct a finitely generated group Ī which acts faithfully on the circle, but which does not admit any faithful C^1 action, thus answering in the negative a question of John Franks
Closed sets of correlations: answers from the zoo
We investigate the conditions under which a set of multipartite nonlocal
correlations can describe the distributions achievable by distant parties
conducting experiments in a consistent universe. Several questions are posed,
such as: are all such sets "nested", i.e., contained into one another? Are they
discrete or do they form a continuum? How many of them are supraquantum? Are
there non-trivial polytopes among them? We answer some of these questions or
relate them with established conjectures in complexity theory by introducing a
"zoo" of physically consistent sets which can be characterized efficiently via
either linear or semidefinite programming. As a bonus, we use the zoo to
derive, for the first time, concrete impossibility results in nonlocality
distillation.Comment: 24 pages, 5 figure
Incomparable, non isomorphic and minimal Banach spaces
A Banach space contains either a minimal subspace or a continuum of
incomparable subspaces. General structure results for analytic equivalence
relations are applied in the context of Banach spaces to show that if
does not reduce to isomorphism of the subspaces of a space, in particular, if
the subspaces of the space admit a classification up to isomorphism by real
numbers, then any subspace with an unconditional basis is isomorphic to its
square and hyperplanes and has an isomorphically homogeneous subsequence
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