Dakota Thor Ihli | Searcharxiv

arXiv2022

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

We examine the Polish group $H_{+}^{AC}$ of order-preserving self-homeomorphisms $f$ of the interval $\left[ 0,1 \right]$ 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 $\left( f,g \right) \in H_{+}^{AC} \times H_{+}^{AC}$ for which $\left\langle f,g \right\rangle$ is dense. Secondly, we prove that $H_{+}^{AC}$ admits a dense $G_δ$ conjugacy class, and we explicitly characterize the elements thereof.

math.GR↗

arXiv2020

The structure of generic automorphisms of the random poset

We examine properties of generic automorphisms of the random poset, with the goal of explicitly characterizing them. We associate to each automorphism an auxiliary first-order structure, consisting of the random poset equipped with an infinite sequence of binary relations which encode the action of the automorphism. We then explicitly characterize generic automorphisms in terms of properties of this structure. Two notable such properties are ultrahomogeneity, and universality for a certain class of finite structures in this language. As this auxiliary structure seems to be new, we also address some model-theoretic questions. In particular, this structure fails to be saturated, and its theory neither is $ω$-categorical nor admits quantifier-elimination, in contrast to many known ultrahomogeneous structures in finite languages. We also examine orbitals -- order-convex hulls of orbits -- and their use in describing automorphisms. In particular, we introduce and use new orders on the space of orbitals.

math.LO↗