Generation and genericity of the group of absolutely continuous homeomorphisms of the interval

2022/03/06 by Ihli, Dakota Thor
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.2203.02854

Abstract

We examine the Polish group H₊^AC of order-preserving self-homeomorphisms f of the interval [ 0,1 ] for which both f and f^-1 are absolutely continuous; in particular, we establish two results. First, we prove that H₊^AC is topologically 2-generated; in fact, it is generically 2-generated, i.e., there is a dense G_δ set of pairs ( f,g ) ∈ H₊^AC × H₊^AC for which ⟨ f,g ⟩ is dense. Secondly, we prove that H₊^AC admits a dense G_δ conjugacy class, and we explicitly characterize the elements thereof.

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