Projectivity in topological dynamics

2025/11/18 by Bal, Jashan
#37B05 (Primary) 03E15 #46M10 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2511.14944

Abstract

We study projectivity in the category of G-flows and affine G-flows for Polish groups G. We also introduce the notion of proximally irreducible extensions between affine G-flows. Using this we provide a characterization of extreme amenability, strong amenability, and amenability for closed subgroups H ≤ G in terms of certain ``dynamical irreducibility'' properties of the Samuel compactification of G/H. We then apply this to answer an open question of Zucker by proving a structure theorem for when the universal minimal proximal flow of G is metrizable or contains a comeager orbit.

Citations

• Topological groups with tractable minimal dynamics
• Amenability, proximality and higher order syndeticity
• A direct solution to the Generic Point Problem
• Metrizable universal minimal flows of Polish groups have a comeagre orbit
• Boundaries of reduced C*-algebras of discrete groups
• Relative amenability
• On relative extreme amenability
• Injectivity and Projectivity in Analysis and Topology

Related

• Topological dynamics of Polish group extensions
• Amenability, proximality and higher order syndeticity
• Topological groups with tractable minimal dynamics
• On minimal flows and definable amenability in some distal NIP theories
• Some universal constructions in abstract topological dynamics