Equi-Baire One Families of M\"obius Transformations and One-Parameter Subgroups of $\mathrm{PSL}(2,\mathbb{C}$)

TL;DR

This paper investigates the Equi-Baire one property in families of M"obius transformations, providing characterizations for loxodromic maps and one-parameter subgroups in terms of dynamical and compactness conditions.

Contribution

It offers a new dynamical characterization of the Equi-Baire one property for M"obius families, linking it to the relative compactness in SL(2,C).

Findings

Iterates of loxodromic maps form orbitally Equi-Baire one families.

One-parameter subgroups are Equi-Baire one iff relatively compact in SL(2,C).

Provides a dynamical criterion for the Equi-Baire one condition.

Abstract

We study the Equi-Baire one property families of M\"obius transformations on the Riemann sphere. For a loxodromic map $f$, we show its iterates $\{f^n\}$ form an orbitally Equi-Baire one family on the attracting basin. For a one-parameter subgroup $\{f_t \}$, we prove it is Equi-Baire one on all compact sets of $\widehat{\mathbb{C}}$ if and only if the subgroup is relatively compact in $\mathrm{SL}(2,\mathbb{C})$. This provides a dynamical characterization of the Equi-Baire one condition for M\"obius families.