The group of measure preserving transformations of the unit interval is an absolute retract

Nhu, Nguyen To

doi:10.1090/s0002-9939-1990-1009997-6

Cited by 9 publications

(2 citation statements)

References 5 publications

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“…Currently many big Polish groups are known to be homeomorphic to 2 . Examples include Aut(X, µ) with the weak topology (Nhu [24]), the group of orientation-preserving homeomorphisms of the unit interval (Anderson), the isometry group of the Urysohn space (Melleray [22]), and many others.…”

Section: Full Groups Are Homeomorphic Tomentioning

confidence: 99%

Topological properties of full groups

Kittrell

Tsankov

2009

Ergod. Th. Dynam. Sys.

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Abstract. We study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups F m and F n if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated.

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Section: Full Groups Are Homeomorphic Tomentioning

confidence: 99%

Topological properties of full groups

Kittrell

Tsankov

2009

Ergod. Th. Dynam. Sys.

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“…While those questions have been answered long ago in Mathematical Analysis and Topology, and also in Ergodic Theory there is a series of papers where topological properties of the group of measure preserving bijections or homeomorphisms are investigated, see e.g. [25], [31], [15], [10], [43], [51], it seems that they even have not been asked before in Dynamical Systems. There are very few papers on topological properties of spaces of maps that have some specific dynamical properties.…”

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confidence: 99%

A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces

Kolyada

2020

Discrete &Amp; Continuous Dynamical Systems - S

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and several other mathematical departments in 2017. † Editors' note: Professor Sergiȋ Kolyada passed away on May 16, 2018. He will be missed by the mathematical community, as a mathematician and as a person. Due to his untimely death, Professor Kolyada could not implement the changes to the first version of the manuscript as suggested in the (positive) reviews. With the consent of Professor Kolyada's family, Professor L'ubomír Snoha

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Dynamical Systems on Graph Limits and Their Symmetries

Bick,

Sclosa

2024

J Dyn Diff Equat

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The collective dynamics of interacting dynamical units on a network crucially depends on the properties of the network structure. Rather than considering large but finite graphs to capture the network, one often resorts to graph limits and the dynamics thereon. We elucidate the symmetry properties of dynamical systems on graph limits—including graphons and graphops—and analyze how the symmetry shapes the dynamics, for example through invariant subspaces. In addition to traditional symmetries, dynamics on graph limits can support generalized noninvertible symmetries. Moreover, as asymmetric networks can have symmetric limits, we note that one can expect to see ghosts of symmetries in the dynamics of large but finite asymmetric networks.

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The group of measure preserving transformations of the unit interval is an absolute retract

Abstract: Abstract.The group of measure preserving transformations of the unit interval equipped with the weak topology is an absolute retract, hence is homeomorphic to a separable Hubert space.

Cited by 9 publications

References 5 publications

Topological properties of full groups

Topological properties of full groups

A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces

Dynamical Systems on Graph Limits and Their Symmetries

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