Maximally highly proximal generators of minimal flows

Topology and its Applications, 2004

We introduce a way of characterizing the linking of one-dimensional minimal sets in threedimensional flows and carry out the characterization for some minimal sets within flows modelled by templates, with an emphasis on the linking of Denjoy continua. We also show that any aperiodic minimal subshift of minimal block growth has a suspension which is homeomorphic to a Denjoy continuum.

Advances in Mathematics, 2016

We study definably amenable groups in N IP theories, focusing on the problem raised in [10] of whether weak generic types coincide with almost periodic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We give fairly definitive results in the o-minimal context, including a counterexample.

Ergodic Theory and Dynamical Systems, 2018

We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than $2^{\mathfrak{c}}$ (where $\mathfrak{c}=2^{\aleph _{0}}$) minimal left ideals is proximal isometric (PI). Then we show the existence of various minimal PI-flows with many minimal left ideals, as follows. For the acting group $G=\text{SL}_{2}(\mathbb{R})^{\mathbb{N}}$, we construct a metric minimal PI $G$-flow with $\mathfrak{c}$ minimal left ideals. We then use this example and results established in Glasner and Weiss. [On the construction of minimal skew-products. Israel J. Math.34 (1979), 321–336] to construct a metric minimal PI cascade $(X,T)$ with $\mathfrak{c}$ minimal left ideals. We go on to construct an example of a minimal PI-flow $(Y,{\mathcal{G}})$ on a compact manifold $Y$ and a suitable path-wise connected group ${\mathcal{G}}$ of a homeomorphism of $Y$, such that the flow $(Y,{\mathcal{G}})$ is PI and has $2^{\mathfrak{c}}$ minimal left ideals. F...

Duke Mathematical Journal, 1978

2012

This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of S∞. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows us to provide a partial answer to a question posed in [KPT05] by Kechris, Pestov and Todorcevic.

Revista matemática complutense, 2002

The uniqueness theorem for the ergodic maximal operator is proved in the continuous case.

Forum of Mathematics, Sigma, 2022

We show that the universal minimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation-invariant Boolean algebras of subsets of the group satisfying a higher-order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.

arXiv (Cornell University), 2022

In this note, we are interested in discussing characteristics of finite generating sets for F + , the set of all semiflows with non negative coordinates of a Petri Net. By systematically positioning these results over semirings such as N or Q + then over a field such as Q, we were able to complete the range of results about not only finite but optimal generating sets for a given family of semiflows.