47 research outputs found
Topologies Refining the Cantor Topology on X ω
International audienceThe space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science. Usually, it is convenient to regard this space as a metric space, the Cantor-space. It turned out that for several purposes topologies other than the one of the Cantor-space are useful, e.g. for studying fragments of first-order logic over infinite words or for a topological characterisation of random infinite words. It is shown that both of these topologies refine the topology of the Cantor-space. Moreover, from common features of these topologies we extract properties which characterise a large class of topologies. It turns out that, for this general class of topologies, the corresponding closure and interior operators respect the shift operations and also, to some respect, the definability of sets of infinite words by finite automata
Generic Stationary Measures and Actions
Let be a countably infinite group, and let be a generating
probability measure on . We study the space of -stationary Borel
probability measures on a topological space, and in particular on ,
where is any perfect Polish space. We also study the space of
-stationary, measurable -actions on a standard, nonatomic probability
space.
Equip the space of stationary measures with the weak* topology. When
has finite entropy, we show that a generic measure is an essentially free
extension of the Poisson boundary of . When is compact, this
implies that the simplex of -stationary measures on is a Poulsen
simplex. We show that this is also the case for the simplex of stationary
measures on .
We furthermore show that if the action of on its Poisson boundary is
essentially free then a generic measure is isomorphic to the Poisson boundary.
Next, we consider the space of stationary actions, equipped with a standard
topology known as the weak topology. Here we show that when has property
(T), the ergodic actions are meager. We also construct a group without
property (T) such that the ergodic actions are not dense, for some .
Finally, for a weaker topology on the set of actions, which we call the very
weak topology, we show that a dynamical property (e.g., ergodicity) is
topologically generic if and only if it is generic in the space of measures.
There we also show a Glasner-King type 0-1 law stating that every dynamical
property is either meager or residual.Comment: To appear in the Transactions of the AMS, 49 page
Polishness of some topologies related to word or tree automata
We prove that the B\"uchi topology and the automatic topology are Polish. We
also show that this cannot be fully extended to the case of a space of infinite
labelled binary trees; in particular the B\"uchi and the Muller topologies are
not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the
proceedings of the 26th EACSL Annual Conference on Computer Science and
Logic, CSL 2017. The main addition with regard to the conference paper
consists in the study of the B\"uchi topology and of the Muller topology in
the case of a space of trees, which now forms Section
Generic Stationary Measures and Actions
Let G be a countably infinite group, and let μ be a generating probability measure on G. We study the space of μ-stationary Borel probability measures on a topological G space, and in particular on Z^G, where Z is any perfect Polish space. We also study the space of μ-stationary, measurable G-actions on a standard, nonatomic probability space.
Equip the space of stationary measures with the weak* topology. When μ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of (G, μ). When Z is compact, this implies that the simplex of μ-stationary
measures on Z^G is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on {0, 1}^G.
We furthermore show that if the action of G on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary.
Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when G has property (T), the
ergodic actions are meager. We also construct a group G without property (T) such that the ergodic actions are not dense, for some μ.
Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual
On subgroups of minimal topological groups
A topological group is minimal if it does not admit a strictly coarser
Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a
topological group is the greatest lower bound of the left and right
uniformities. A group is Roelcke-precompact if it is precompact with respect to
the Roelcke uniformity. Many naturally arising non-Abelian topological groups
are Roelcke-precompact and hence have a natural compactification. We use such
compactifications to prove that some groups of isometries are minimal. In
particular, if U_1 is the Urysohn universal metric space of diameter 1, the
group Iso(U_1) of all self-isometries of U_1 is Roelcke-precompact,
topologically simple and minimal. We also show that every topological group is
a subgroup of a minimal topologically simple Roelcke-precompact group of the
form Iso(M), where M is an appropriate non-separable version of the Urysohn
space.Comment: To appear in Topology and its Applications. 39 page