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

Sandipan Dutta, Vanlalruatkimi, Jonathan Ramdikpuia

Abstract

We study the Equi-Baire one property families of Möbius 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 Möbius families.

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

Abstract

We study the Equi-Baire one property families of Möbius transformations on the Riemann sphere. For a loxodromic map , we show its iterates form an orbitally Equi-Baire one family on the attracting basin. For a one-parameter subgroup , we prove it is Equi-Baire one on all compact sets of if and only if the subgroup is relatively compact in . This provides a dynamical characterization of the Equi-Baire one condition for Möbius families.
Paper Structure (13 sections, 12 theorems, 41 equations)

This paper contains 13 sections, 12 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminaries
  4. Möbius transformations and the group $\mathrm{SL}(2,\mathbb{C})$.
  5. Classification of elements of $\mathrm{SL}(2,\mathbb{C})$.
  6. Baire one functions and Equi–Baire one families.
  7. The Riemann sphere and stereographic projection.
  8. Conjugacy and fixed points.
  9. Useful Lemmas
  10. Results regarding Equi-Baire One Families of Möbius Transformations and One–Parameter Subgroups of $\mathrm{PSL}(2,\mathbb{C})$
  11. Proof of Theorem \ref{['thm main1']}
  12. Proof of Theorem \ref{['thm main2']}
  13. Conclusion

Key Result

Theorem 1.1

Let $f \in \mathrm{SL}(2,\mathbb{C})$ act on $\widehat{\mathbb{C}}$ by the corresponding Möbius map. Suppose $f$ is loxodromic, i.e. conjugate to $z \mapsto \lambda z$ with $|\lambda| \ne 1$. Then for any point $x$ in the attracting basin of the attracting fixed point $p$, the family of iterates is orbitally Equi–Baire one at $x$. Moreover, an explicit $\delta$–function witnessing this property i

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: The chordal metric
  • Definition 2.2: Attracting Basin
  • Definition 2.3: Baire-one
  • Definition 2.4: Equi-Baire one, ALI
  • Definition 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Lemma 3.2
  • ...and 7 more